Conference Agenda
Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).
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Session Overview |
| Date: Tuesday, 09/Jul/2019 | |
| 7:30am - 8:15am | Registration (if too busy, come back any time later during the conference!) |
| vonRoll, Fabrikstr. 6, Foyer | |
| 8:15am - 8:30am | Opening by the Chairs and word of welcome by Daniel Candinas, Vice-Rector for Research, University of Bern |
| vonRoll, Fabrikstr. 6, 001 | |
| 8:30am - 9:30am | IP01: Pablo A. Parrilo: Switched linear systems and infinite products of matrices |
| vonRoll, Fabrikstr. 6, 001 | |
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8:30am - 9:30am
Switched linear systems and infinite products of matrices Massachusetts Institute of Technology, United States of America Many situations of interest can be modeled as "switched linear systems", which are collections of linear difference equations, with some logical rule for switching between subsystems. Mathematically, this boils down to understanding infinite products of matrices, all of which are elements of a given finite set. Analyzing these systems is a difficult question that appears in a number of applications, including the analysis of optimization algorithms, information theory, and wavelets. Depending on whether the switching is deterministic or stochastic, different notions can be used to quantify the resulting convergence rate, like the joint spectral radius, or the Lyapunov exponent. In this talk, we provide a gentle introduction to this class of problems, their applications, and several results regarding their exact and approximate computation. |
| 8:30am - 9:30am | IP01-streamed from 001: Pablo A. Parrilo: Switched linear systems and infinite products of matrices |
| vonRoll, Fabrikstr. 6, 004 | |
| 9:30am - 10:00am | Coffee Break |
| Unitobler, F wing, floors 0 and -1 | |
| 10:00am - 12:00pm | Room free |
| Unitobler, F005 | |
| 10:00am - 12:00pm | MS143, part 1: Algebraic geometry in topological data analysis |
| Unitobler, F006 | |
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10:00am - 12:00pm
Algebraic geometry in topological data analysis In the last 20 years methods from topology, the mathematical area that studies “shapes", have proven successful in studying data that is complex, and whose underlying shape is not known a priori. This practice has become known as topological data analysis (TDA). As additional methods from topology still find their application in the study of complex structure in data, the practice is evolving and expanding, and now moreover draws increasingly upon data science, computer science, computational algebra, computational topology, computational geometry, and statistics. While ideas from category theory, sheaf theory and representation theory of quivers have driven the theoretical development in the past decade, in the last years ideas from commutative algebra and algebraic geometry have started have started to be used to tackle some theoretical problems in TDA. The aim of the minisymposium is to seize this momentum and to bring together experts in algebraic geometry and researchers in topological data analysis to explore new avenues of research and foster research collaborations. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Algebraic geometry in topological data analysis: an overview In this talk I will give an overview on how techniques and ideas from algebraic geometry have been used so far in topological data analysis, and discuss possible new avenues of research.
Applications of Groebner bases I will discuss the Groebner bases theory and its application to calculation of Hilbert series, and related invariants of quadratic algebras and operads. Examples where this technique can be used and on these bases, for example, Koszulity proved, will be given. It can be applied to persistent homology problems, to problems in neuroscience, etc. The Groebner technique itself originated in computer science, more precisely, in computer algebra, but as we will see, after proper algebraic formulation it can serve for proving some structural and homological results about algebras presented by generators and relations. This in turn can serve for the solution of applied problems, for example, by the study of neural codes as pseudo monomial ideals. Decomposition of 2-parameter persistence modules Decomposition theorems are one of the pillars of persistence theory. They yield complete discrete invariants for persistence modules, which can be used as descriptors for data in downstream applications. While the case of 1-parameter persistence modules is by now well-understood, the multi-parameter case remains mostly unexplored and appears to be much more complicated. In this talk I will focus on the 2-parameter case, and provide an overview of the current state of the art together with some perspectives. Classification of filtered chain complexes Persistent homology has proven to be a useful tool to extract information from data sets. Homology, however, is a drastic simplification and in certain situations might remove too much information. This prompts us to study filtered chain complexes. We prove a structure theorem for filtered chain complexes and list all possible indecomposables. We call these indecomposables interval spheres and classify them into three types. Two types correspond respectively to finite and infinite interval modules, while the third type is unseen by homology. The structure theorem states that any filtered chain complex can be written as the unique sum of interval spheres, up to isomorphism and permutation. The proof is based on a hierarchy of full subcategories of the category of filtered chain complexes. Such hierarchy suggests an algorithm for decomposing filtered chain complexes, which also retrieves the usual persistent barcodes. This approach offers a way to retrieve more geometrical aspects of data: while homology cannot tell the difference between a point and a disk, our decomposition provides a tool to count the contractible parts of the data, thus, we can obtain not only topological but also geometrical information.
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| 10:00am - 12:00pm | MS123, part 1: Asymptotic phenomena in algebra and statistics |
| Unitobler, F007 | |
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10:00am - 12:00pm
Asymptotic phenomena in algebra and statistics Across several branches of mathematics, the following fundamental question arises: given a sequence of algebraic structures with maps between them, can the entire sequence be characterized by a finite segment? Here the maps are comprising symmetries of the objects as well as morphisms between them. An affirmative answer leads to a description of all structures by using finite data only. There is a growing body of work that establishes the desired finiteness result in varied contexts. Nevertheless, instances where stability is not well understood include:
The aim of the minisymposium is to build bridges between the varied mathematicians and the different areas investigating stability phenomena. We propose a two half-day minisymposium with 8 speakers total. The proposed speakers have all expressed interest in speaking at the symposium. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Strength and polynomial functors Define the rank of an infinite-by-infinite matrix A as the supremum of the ranks of all its finite-size submatrices. This can be finite, which implies that the matrix A is the product of an infinite-by-n and an n-by-infinite matrix. Or else it is infinite. In this case, the set of matrices that can be obtained from A by a finite number of row and column operations is Zariski-dense in the space of all matrices. Next, define the strength of a series f of fixed degree in infinitely many variables to be the minimal number of products of series of lower degree that sum up to f. Then f has a simpler description when its strength is finite. And when its strength is infinite, the set of series obtained from f by finitely many variable substitutions turns out to be dense in its ambient space. This dichotomy exists in a much more general setting: an element of the inverse limit V of a polynomial functor either is in the image of a polynomial transformation from a simpler functor or has a dense orbit in V. This talk is about the complexity measure on V that associates to an element the minimal polynomial functor from which it arises. The results are part of joint work with Jan Draisma, Rob Eggermont and Andrew Snowden. Asymptotics Proved by the Method of Cumulants The method of cumulants is closely related to the method of moments both being a classical tool to prove central limit theorems. Having a good bound on the cumulants of a sequence of random variables, one can deduce precise asymptotics for the distribution function of the properly scaled random variables, and it implies large and moderate deviations as well as so-called mod-Phi-convergence. As an example, we will study dependency graphs. FI-algebras: examples and counterexamples We link the literature on algebras with an action of the infinite symmetric group to the literature on FI-algebras by identifying mutually adjoint functors in both directions. I will discuss some interesting examples and counterexamples of Noetherian and non-Noetherian FI-algebras and modules over them. Finitely generated modules over some Noetherian FI-algebras are Noetherian, and in this setting we can describe free resolutions and Betti numbers, but for other FI-algebras this fails. This is joint work with Jan Draisma and Alexei Krasilnikov. Asymptotic behavior of chains of ideals with symmetry Chains of ideals in increasingly larger polynomial rings that are invariant under the action of symmetric groups arise in various contexts, including algebraic statistics and representation theory. In this talk, I will discuss the asymptotic behavior of some invariants of ideals in such chains, namely, the Krull dimension, the projective dimension, and the Castelnuovo-Mumford regularity. The Krull dimension is eventually a linear function whose slope can be described explicitly. We conjecture that the projective dimension and the Castelnuovo-Mumford regularity also grow eventually linearly, and provide linear bounds for these invariants. This is joint work with Uwe Nagel, Hop D. Nguyen, and Tim Römer. |
| 10:00am - 12:00pm | MS177, part 1: Algebraic and combinatorial phylogenetics |
| Unitobler, F011 | |
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10:00am - 12:00pm
Algebraic and combinatorial phylogenetics Since late eighties, algebraic tools have been present in phylogenetic theory and have been crucial in understanding the limitations of models and methods and in proposing improvements to the existing tools. In this session we intend to present some of the most recent work in this area. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) An Introduction to Algebraic and Combinatorial Phylogenetics The purpose of this talk is to provide participants with the necessary background information to attend the subsequent talks in this session. As such, we will discuss preliminary definitions and key results in the field of mathematical phylogenetics. In particular, we will discuss results concerning tree and network combinatorics and introduce some important phylogenetic models. We will also describe the algebraic methods that have been used to understand these models. Inferring species networks from gene trees Phylogenetic trees for different genes from the same taxa often differ from one another, with incomplete lineage sorting and hybridization considered to be two of the most important biological reasons underlying this discordance. Inferring a hybridization network that shows species relationships from a set of gene trees is made difficult by the confounding of these two sources of conflicting signal. We present a new algorithm for this inference problem, under the Network Multispecies Coalescent model of these processes on a level-1 network. Building on a number of combinatorial insights, the topological species network estimator is statistically consistent with reasonable running time for moderate size data sets. Analyses of several simulated and empirical datasets indicate its practical value. Algebraic versus semi-algebraic conditions for phylogenetic varieties It is common to model evolution adopting a parametric statistical model which allows to define a joint probability distribution at the leaves of phylogenetic trees. When these models are algebraic, one is able to deduce polynomial relationships between these probabilities, and the study of these polynomials and the geometry of the algebraic varieties that arise from them can be used to reconstruct phylogenetic trees. However, not all points in these algebraic varieties have biological sense. In this talk, we would like to discuss the importance of studying the subset of these varieties with biological sense and explore the extent to which restricting to these subsets can provide insight into existent methods of phylogenetic reconstruction. One of our main focuses is to understand and describe these subsets of points that come from positive parameters. We are interested in the algebraic and semi-algebraic conditions that describe them and in knowing which of these conditions are relevant for topology inference. The projection into these subsets can be seen as an optimization problem and can be solved using nonlinear programming algorithms. As these algorithms do not guarantee a global solution, we use a different approach that allows us to find a global optimum. Numerical algebraic geometry and computational algebra play a fundamental role here. We will show some results on trees evolving under groups-based models and, in particular, we will explore the long branch attraction phenomenon. Trait evolution on two gene trees Models of trait evolution use a phylogenetic tree to determine the correlation structure for traits sampled from a set of species. Typically, the phylogenetic tree is estimated from genetic data from many loci, and a single tree is used to model the trait evolution, for example by assuming that the mean trait value follows a Brownian motion on the tree. Here, we model trait evolution by assuming that there are two genetic loci influencing the trait. In this case separate evolutionary trees (called gene trees) can occur for the two loci. We model the correlation structure as arising from a linear combination of Brownian motions on the two trees, and develop a model to estimate the proportion of trait evolution contributed by each gene. |
| 10:00am - 12:00pm | Room free |
| Unitobler, F012 | |
| 10:00am - 12:00pm | MS122: Tropical and combinatorial methods in economics |
| Unitobler, F013 | |
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10:00am - 12:00pm
Tropical and combinatorial methods in economics Over the past ten years, combinatorial auctions and mechanism designs have posed interesting challenges at the intersection of tropical geometry, matroid theory, discrete convex analysis and integer programming. This minisymposium features experts who work at this intersection discussing the latest developments and potential approaches to major conjectures concerning valuated matroids (also known as gross substitutes or $M^natural$-concave functions). (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) On the Construction of Substitutes Gross substitutability is a central concept in Economics and is connected to important notions in Discrete Convex Analysis, Number Theory and the analysis of Greedy algorithms in Computer Science. Many different characterizations are known for this class, but providing a constructive description remains a major open problem. The construction problem asks how to construct all gross substitutes from a class of simpler functions using a set of operations. Since gross substitutes are a natural generalization of matroids to real-valued functions, matroid rank functions form a desirable such class of simpler functions. Shioura proved that a rich class of gross substitutes can be expressed as sums of matroid rank functions, but it is open whether all gross substitutes can be constructed this way. Our main result is a negative answer showing that some gross substitutes cannot be expressed as positive linear combinations of matroid rank functions. En route, we provide necessary and sufficient conditions for the sum to preserve substitutability, uncover a new operation preserving substitutability and fully describe all substitutes with at most 4 items. Joint work with Renato Paes Leme. Connection Between Discrete Convex Analysis and Auction Theory Discrete convex analysis is a theory of discrete convexity in combinatorial optimization. In this talk we explain the connection between discrete convex analysis and auctions with multiple differentiated items. In particular, we show that computation of an equilibrium price vector can be reduced to the minimization of an L-convex function, and the existing iterative auction algorithms can be regarded as specialized implementation of the algorithms for L-convex function minimization. Unimodular schemes In 2003 we with V Danilov proved that an ample class of discrete convexity is a pure systems such that its one dimensional generators form a totally unimodular system. I explain how to glue such pure systems into a unimodular scheme and to build a class of discrete convex functions. Transversal valuated matroids As every first course in linear algebra tells us, there are a great number of ways to describe a linear subspace of K^n when K is a field. When K is the tropical semiring, many of these cease to agree, but tropical geometers have a consensus as to which of them give the correct notion of tropical linear subspace (one that does is Plücker vectors). My subject will be one of the "wrong" descriptions, namely row spaces of matrices, which give a subset of the tropical linear spaces. We obtain tropical analogues of results from the '70s on presentations of transversal matroids. Some of this work is joint with Felipe Rincón; the rest is joint with Jorge Alberto Olarte. |
| 10:00am - 12:00pm | MS182, part 1: Matrix and tensor optimization |
| Unitobler, F021 | |
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10:00am - 12:00pm
Matrix and tensor optimization Matrix and tensor optimization has important applications in the context of modern data analysis and high dimensional problems. Specifically, low rank approximations and spectral properties are of interest. Due to their multilinear parametrization, sets of low rank matrices and tensors form sets with interesting, and sometimes challenging, geometric and algebraic structures. Studying such sets of tensors and matrices in the context of algebraic geometry is therefore not only helpful but also necessary for the development of efficient optimization algorithms and a rigorous analysis thereof. In this respect, the area of matrix and tensor optimization relates to the field applied algebraic geometry by the addressed problems and some of the employed concepts. In this minisymposium, we wish to bring the latest developments in both of these aspects to attention. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Tensorized Krylov subspace methods Tensorized Krylov subspace methods are a versatile tool in numerical linear algebra for addressing large-scale applications that involve tensor product structure. This includes the discretization of high-dimensional PDEs, the solution of linear matrix equations, as well as low-rank updates and Frechet derivatives for matrix functions. This talk gives an overview of such methods, with an emphasis on theoretical properties and their connection to multivariate polynomials. Critical points of quadratic low-rank optimization problems The absence of spurious local minima in certain non-convex minimization problems, e.g. in the context of recovery problems in compressed sensing, has recently triggered much interest due to its important implications on the global convergence of optimization algorithms. One example is low-rank matrix sensing under rank restricted isometry properties. It can be formulated as a minimization problem for a quadratic cost function constrained to a low-rank matrix manifold, with a positive semidefinite Hessian acting like a perturbation of identity on cones of low-rank matrices. We present an approach to show strict saddle point properties and absence of spurious local minima for such problems under improved conditions on the restricted isometry constants. This is joint work with André Uschmajew. Matrix product states from an algebraic geometer’s point of view Matrix product states and uniform matrix product states play a crucial role in quantum physics and quantum chemistry. They are used, for instance, to compute the eigenstates of the Schrödinger equation. Matrix product states provide a way to represent special tensors in an efficient way and uniform matrix product states are partially symmetric analogs of matrix product states. Computation of the norm of a nonnegative tensor The norm of a tensor can be computed by finding the maximal eigenvalue of a polynomial mapping. This problem is NP-hard in general. We present a nonlinear generalization of the Perron-Frobenius theorem which guarantees that the norm of tensors with nonnegative entries can be computed with a higher-order variant of the power method. This iterative algorithm has global optimal guarantees and a linear convergence rate. We discuss applications in nonconvex optimization and in the computation of centralities measure of multiplex networks. |
| 10:00am - 12:00pm | Room free |
| Unitobler, F022 | |
| 10:00am - 12:00pm | MS142: Algebraic geometry of low-rank matrix completion |
| Unitobler, F023 | |
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10:00am - 12:00pm
Algebraic geometry of low-rank matrix completion In a matrix completion problem, one is presented with a subset of entries of a matrix and wishes to find values for the remaining entries so that the completed matrix has a particular property. For example, one may want the completed matrix to have low rank or to be positive semidefinite. Such problems abound in application areas ranging from recommender systems (e.g. the "Netflix problem"), to rigidity theory, to compressed sensing, to maximum likelihood estimation for graphical models. Matrix completion problems also motivate many questions that can be considered fundamental within algebraic geometry. For example, studying low-rank matrix completion motivates the question: which coordinate projections of a given determinantal variety are dominant? What changes when one restricts to the real part of this determinantal variety? This minisymposium aims to bring together researchers who study algebraic aspects of matrix completion, both from theoretical and applied perspectives. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Real geometry of matrix completion I will discuss matrix completion problems from the point of view of real algebraic geometry. This means that coordinate projections of rank varieties of matrices are the central object of study. Important invariants are called generic rank, typical rank, and maximum likelihood threshold. Low algebraic dimension matrix completion In the low rank matrix completion (LRMC) problem, the low rank assumption means that the columns (or rows) of the matrix to be completed are points on a low-dimensional linear algebraic variety. We extend this thinking to cases where the columns are points on a low-dimensional nonlinear algebraic variety, a problem we call Low Algebraic Dimension Matrix Completion (LADMC). Matrices whose columns belong to a union of subspaces are an important special case. We propose a LADMC algorithm that leverages existing LRMC methods on a tensorized representation of the data. This approach succeeds in many cases where traditional LRMC is guaranteed to fail because the data are low-rank in the tensorized representation but not in the original representation. We also provide a formal mathematical justification for the success of our method. In particular, we show bounds of the rank of these data in the tensorized representation, and prove sampling requirements on the number of observed entries per column necessary and sufficient to guarantee uniqueness of the completion. We also provide experimental results showing that the new approach significantly outperforms existing state-of-the-art methods for matrix completion in many situations. The tropical Cayley-Menger variety Varieties that arise in the algebraic study of matrix completion come embedded in a vector space whose coordinates are indexed by some graph. A recurring problem is to find combinatorial descriptions, in terms of these graphs, of the algebraic matroids underlying these varieties. Tropical geometry can be used to solve such problems. In this talk, I will show that the tropicalization of the Cayley-Menger variety of points in the plane has a simplicial complex structure that can be described in terms of rooted trees. Then, I will show how one can use this to obtain a new proof of Laman's theorem, a celebrated theorem from rigidity theory giving a combinatorial description of the algebraic matroid underlying the Cayley-Menger variety of points in the plane. Unlabelled global rigidity and low-rank matrix completion A graph $G$ with $n$ is said to be generically globally rigid in dimension $d$ if we can reconstruct an unknown, but generic, set of $n$ points in $d$-dimensional Euclidean space from the pairwise distance measurements indexed by the edges of $G$. Perhaps surprisingly, even when $G$ is an unknown generically globally rigid graph, it is still possible to reconstruct the original points from the distance measurements. Despite the fact that global rigidity is a special case of low-rank matrix completion, the analogous unlabelled matrix completion problem is mostly open. I’ll talk about both problems and some key differences. This talk is based on joint work with Shlomo Gortler and Dylan Thurston. |
| 10:00am - 12:00pm | MS148, part 1: Algebraic neural coding |
| Unitobler, F-105 | |
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10:00am - 12:00pm
Algebraic Neural Coding Neuroscience aims to decipher how the brain represents information via the firing of neurons. Place cells of the hippocampus have been demonstrated to fire in response to specific regions of Euclidean space. Since this discovery, a wealth of mathematical exploration has described connections between the algebraic and combinatorial features of the firing patterns and the shape of the space of stimuli triggering the response. These methods generalize to other types of neurons with similar response behavior. At the SIAM AG meeting, we hope to bring together a group of mathematicians doing innovative work in this exciting field. This will allow experts in commutative algebra, combinatorics, geometry and topology to connect and collaborate on problems related to neural codes, neural rings, and neural networks. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Flexible Motifs in Threshold-Linear Networks Threshold-linear networks (TLNs) are popular models of recurrent networks used to model neural activity in the brain. The state space in these networks is naturally partitioned into regions defined by an associated hyperplane arrangement. The combinatorial properties of this arrangement, as captured by an oriented matroid, provide strong constraints on the network's dynamics. In recent work, we have studied how the graph of a TLN constrains the possible fixed points of the network by providing constraints on the combinatorics of the hyperplane arrangement. Here we study the case of flexible motifs, where the graph allows multiple possibilities for the set of fixed points FP(W), depending on the choice of connectivity matrix W. In particular, we find that mutations of oriented matroids correspond naturally to bifurcations in the dynamics. Flexible motifs are interesting from a neuroscience perspective because they allow us to study the effects of sensory and state-dependent modulation on the dynamics of neural ensembles. Robust Motifs in Threshold-Linear Networks Networks of neurons in the brain often exhibit complex patterns of activity that are shaped by the intrinsic structure of the network. How does the precise connectivity structure of the network influence these patterns of activity? We address this question in the context of threshold-linear networks, a commonly used model of recurrent neural networks. We identify constraints on the dynamics that arise from network architecture and are independent of the specific values of connection strengths. By appealing to an associated hyperplane arrangement, we find families of robust motifs, which are graphs where the collection of fixed points of the corresponding networks is fully determined by the graph structure, irrespective of the particular connection strengths. These motifs provide a direct link between network structure and function, and provide new insights into how connectivity may shape dynamics in real neural circuits. An Algebraic Perceptron and the Neural Ideals Feedforward neural networks have been widely used in machine-learning and theoretical neuroscience. The paradigm of "deep learning", that makes use of many consecutive layers of feedforward networks, has achieved impressive engineering success in the past two decades. However, a theoretical understanding of many-layer feedforward networks is still mostly lacking. While each layer of a feedforward network can be understood via the geometry of an hyperplane arrangement, satisfactory understanding the mathematical properties of many-layered networks remains elusive. Properties of Hyperplane Neural Codes The firing patterns of neurons in sensory systems give rise to combinatorial codes, i.e. subsets of the boolean lattice. These firing patterns represent the abstract intersection patterns of subsets of a Euclidean space, and an open problem is identifying the combinatorial properties of neural codes which distinguish the geometric properties of the corresponding subsets. We introduce the polar complex, a simplicial complex associated to any combinatorial code, and relate its associated Stanley-Reisner ring to the ring of $mathbb{F}_2$-valued functions on the code to identify some distinguishing characteristics of codes arising from feed-forward neural networks. In particular, we show the associated ring is Cohen-Macaulay, and make connections to other questions in the study of boolean functions. |
| 10:00am - 12:00pm | MS151, part 1: Cluster algebras and positivity |
| Unitobler, F-106 | |
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10:00am - 12:00pm
Cluster algebras and positivity Cluster algebras are commutative rings whose generators and relations can be defined in a remarkably succinct recursive fashion. Algebras of this kind, introduced by Fomin and Zelevinsky in 2000, are equipped with a powerful combinatorial structure frequently appearing in many mathematical contexts such as Lie theory, triangulations of surfaces, Teichmueller theory and beyond. Coordinate rings of Grassmannians and related invariant rings are well-studied examples of algebras of this type. One important aspect arising from the intrinsic combinatorial structure of cluster algebras is that it uncovers systematic, intriguing and complex positivity properties in these families of rings. For instance, it is expected that for each cluster algebra there is a distinguished basis, such that all elements can be expressed as a "positive" linear combination of basis vectors. Seemingly elementary claims of this type, so far proved only in certain cases, have triggered important developments in research areas at the intersection of geometry, algebra and combinatorics. In this session, we glimpse at recent developments in this field and discuss open questions. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Toric degenerations of cluster varieties and cluster duality This talk will review some aspects of mirror symmetry for generalised flag varieties G/P and its interaction with the representation theory of G. For a cominuscule homogeneous space G/P there is an expression for the mirror LG model W in terms of coordinates which, by the geometric Satake correspondence in representation theory, are naturally identified with cohomology classes of G/P (joint works with Marsh, Pech, and Williams) leading to some combinatorially very attractive formulas. A relationship between the critical points of W and its tropicalisation, and representation theory (work of Judd) may also be discussed. On mirror symmetry for homogeneous spaces Cluster varieties are a particularly nice class of log Calabi-Yau varieties-- the non-compact analogue of usual Calabi-Yaus. They come in pairs (A,X), with A and X built from dual tori. The punchline of this talk will be that compactified cluster varieties are a natural progression from toric varieties. Essentially all features of toric geometry generalize to this setting in some form, and the objects studied remain simple enough to get a hold of and do calculations. Compactifications of A and their toric degenerations were studied extensively by Gross, Hacking, Keel, and Kontsevich. These compactifications generalize the polytope construction of toric varieties-- a construction which is recovered in the central fiber of the degeneration. Compactifications of X were introduced by Fock and Goncharov and generalize the fan construction of toric varieties. Recently, Lara Bossinger, Juan Bosco Frías Medina, Alfredo Nájera Chávez, and I introduced the notion of an X-variety with coefficients, expanded upon the notion of compactified X-varieties, and for each torus in the atlas gave a toric degeneration where each fiber is a compactified X-variety with coefficients. We showed that these fibers are stratified, and each stratum is again a compactified X-varieties with coefficients. In the central fiber, we recover the toric variety associated to the fan in question, and we show that strata of the fibers degenerate to toric strata. This talk is based on arXiv:1809.08369. Generalised friezes and the weak Ptolemy map Frieze patterns, introduced by Conway, are infinite arrays of numbers where neighbouring numbers satisfy a local arithmetic rule. Frieze patterns with positive integer values are of a special interest since they are in one-to-one correspondence with triangulations of polygons by Conway--Coxeter. Remarkably, this established a connection to cluster algebras–predating them by 30 years– and to cluster categories. Several generalisations of frieze patterns are known. Joint with Jørgensen, we associated frieze patterns to dissections of polygons where the entries are over a (commutative) ring. Furthermore, we introduced an explicit combinatorial formula for the entries of these friezes by generalising the 'T-path formula' of Schiffler which was introduced to give explicit formulas for cluster variables for cluster algebras of type A. Perfect matching modules for dimer algebras The theory of dimer models, or bipartite graphs on surfaces, first arose in theoretical physics, and later found diverse applications in geometry and representation theory. Recently, there has been much interest in dimer models on the disk, particularly those arising from Postnikov diagrams, and their relationship to Grassmannian cluster algebras and categories. Perfect matchings of dimer models play a central role in the theory. In joint work with İlke Çanakçı and Alastair King, we provide an algebraic viewpoint on these objects, by defining and studying a module for the dimer algebra for each perfect matching. As an application, we explain the relationship between combinatorial and homological formulae for computing Grassmannian cluster variables. |
| 10:00am - 12:00pm | MS140, part 1: Multivariate spline approximation and algebraic geometry |
| Unitobler, F-107 | |
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10:00am - 12:00pm
Multivariate spline approximation and algebraic geometry The focus of the proposed minisymposium is on problems in approximation theory that may be studied using techniques from commutative algebra and algebraic geometry. Research interests of the participants relevant to the minisymposium fall broadly under multivariate spline theory, interpolation, and geometric modeling. For instance, a main problem of interest is to study the dimension of the vector space of splines of a bounded degree on a simplicial complex; recently there have been several advances on this front using notions from algebraic geometry. Nevertheless this problem remains elusive in low degree; the dimension of the space of piecewise cubics on a planar triangulation (especially relevant for applications) is still unknown in general. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Algebraic Approaches to Spline Theory In this talk we will give a brief overview of some of the algebraic methods which are used in spline theory. We will give particular attention to the pioneering work of Billera, in which homological methods were introduced for the calculation of dimension formulas. These methods have proved very fruitful for splines on all types of subdivisions; we will attempt to give a flavor for the various results that have been obtained this way, the questions that remain open, and the connections to algebraic geometry that result from these methods. Computing dimension formulas is only a beginning in spline theory - time permitting we will address how the gluing data for splines may be solved in some cases to give basis functions. Polynomial splines of non-uniform degree: Combinatorial bounds on the dimension Polynomial splines on triangulations and quadrangulations have myriad applications and are ubiquitous, especially in the fields of computer aided design, computer graphics and computational analysis. Meaningful use of splines for these purposes requires the construction and analysis of a suitable set of basis functions for the spline spaces. In turn, the computation or estimation of their dimensions is useful which, following the definition of smooth splines, depends on an interplay between algebra and geometry. We consider the general case of splines with polynomial pieces of differing degrees. The flexibility of such splines would allow design of complex shapes with fewer control points, i.e., cleaner and simpler designs; while the same would also lead to more efficient engineering analysis. Using homological techniques, introduced by Billera (1988), we analyze the dimension of splines on triangulations and T-meshes. Specifically, we generalize the frameworks presented in Mourrain and Villamizar (2013) and Mourrain (2014) to the setting of both mixed polynomial degrees and mixed smoothness. Combinatorial bounds on the dimension are presented. Several examples are provided to illustrate application of the theory developed. Approximation power of C1-smooth isogeometric functions on trivariate two-patch domains Bases and dimensions of trivariate spline functions possessing first order geometric continuity on two-patch domains were studied in (Birner, Jüttler, Mantzaflaris, Graph. Mod. 2018). It was shown that the properties of the spline space depend strongly on the type of the gluing data that is used to specify the relation between the partial derivatives along the interface between the patches. Locally supported bases were shown to exist for trilinear geometric gluing data (that corresponds to piecewise trilinear domain parameterizations) and sufficiently high degree. In this talk we discuss the approximation properties of these spline functions. In particular, we perform numerical experiments with L2 projection in order to explore the approximation power. Despite the existence of locally supported bases, we observe a reduction of the approximation order for low degrees, and we provide a theoretical explanation for this phenomenon. This is joint work with Bert Jüttler and Angelos Mantzaflaris. Splines, representations, and the Stanley-Stembridge conjecture Splines can be used to construct the (equivariant) cohomology of certain algebraic varieties. We describe one such construction, and how the action of certain permutations on the splines relates to a longstanding open question in combinatorics called the Stanley-Stembridge conjecture. We also discuss certain steps towards resolving the conjecture in special cases. |
| 10:00am - 12:00pm | MS149, part 1: Stability of moment problems and super-resolution imaging |
| Unitobler, F-111 | |
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10:00am - 12:00pm
Stability of moment problems and super-resolution imaging Algebraic techniques have proven useful in different imaging tasks such as spike reconstruction (single molecule microscopy), phase retrieval (X-ray crystallography), and contour reconstruction (natural images). The available data typically consists of (trigonometric) moments of low to moderate order and one asks for the reconstruction of fine details modeled by zero- or positive-dimensional algebraic varieties. Often, such reconstruction problems have a generically unique solution when the number of data is larger than the degrees of freedom in the model. Beyond that, the minisymposium concentrates on simple a-priori conditions to guarantee that the reconstruction problem is well or only mildly ill conditioned. For the reconstruction of points on the complex torus, popular results ask the order of the moments to be larger than the inverse minimal distance of the points. Moreover, simple and efficient eigenvalue based methods achieve this stability numerically in specific settings. Recently, the situation of clustered points, points with multiplicities, and positive-dimensional algebraic varieties have been studied by similar methods and shall be discussed within the minisymposium. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Introductory talk: stability of moment problems and super-resolution imaging I shall provide a brief overview of the minisymposium topics and the contributions. Non-ideal Super-resolution and Variations on a Theme Super-resolution is a well studied topic and deals with recovery of spikes from low-pass projections in the Fourier domain. This is a common problem of interest that finds applications across several areas of science and engineering. When it comes to practice, however, this model may not be applicable as it is. Based on several motivating examples from experimental setups, in this talk, we cover the topic of non-ideal super-resolution. To this end, we discuss some new variations on the theme. This includes the question of (a) “essential bandwidth” selection or super-resolution with optimal number of trigonometric moments under noise, (b) super-resolution with time-varying pulses, (c) super-resolution with the unlimited sensing framework, and (4) a general theory of super-resolution that goes beyond the Fourier domain. Clustered Super-Resolution Consider the problem of continuous super resolution, which is taken here as the reconstruction of a spike train signal (linear combination of shifted delta-functions), from noisy Fourier measurements limited to the band [−Ω,Ω]. We discuss some geometrical aspects of this problem, and the related problem of stability of Vandermonde matrices, in the case where the nodes form several clusters. For a single cluster of size h, we analyse the structure of the inverse image of a cube of size epsilon in the measurement space, which we call the epsilon-error set. It is shown that the inverse image has very different scaling along certain directions that depends mainly on the size of the super resolution factor SRF = 1/Ωh, and the noise level epsilon. This description is then extended to several clusters. We describe the effects of decimation (reducing of the sampling rate) on the geometry of the solution set. Specifically we examine aliasing and stability of such solutions. Joint work with: Yosef Yomdin and Dima Batenkov. Geometry of Error Amplification in Spike-train Fourier Reconstruction We consider Fourier Reconstruction of spike-train signals (i.e. of linear combinations of delta-functions). In an important case when some of the nodes nearly collide, while the measurements are noisy, a dramatic error amplification may occur in the process of reconstruction. At least in part, this error amplification reflects the geometric nature of the problem itself, and does not depend on the choice of the solution method. Our approach is based on the following observation: If the nodes near-collide, then the possible error-affected reconstructions are not distributed uniformly, but rather tightly follow certain algebraic-geometric patterns, known a priori (“Prony varieties"). We believe that understanding the geometry and singularities of the Prony varieties can improve our understanding of the geometry of the error amplification. In turn, this can be used in order to improve the overall reconstruction accuracy. We plan to present some our results in case where all the nodes form a single cluster. Here the Prony varieties are defined by the initial equations of the classical Prony system. This fact strongly simplifies their algebraic-geometric study. Next we plan to extend our results to the case of several “well-separated” node clusters. Here the one-cluster case serves as a model for the multi-cluster situation. Basically, here the relevant geometric-algebraic objects are the Cartesian products of the “local (at each cluster) Prony varieties”. |
| 10:00am - 12:00pm | MS197, part 1: Numerical differential geometry |
| Unitobler, F-112 | |
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10:00am - 12:00pm
Numerical Differential Geometry The profound theory of differential geometry have interacted with the computational and statistical communities in the past decades, yielding fruitful outcomes in a wide range of fields including manifold learning, Riemannian optimization, and geometry processing. This minisymposium encourages researchers from applied differential geometry, optimization, manifold learning, and geometry processing to share their perspectives and technical tools on problems lying in the intersection of geometry and computations. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Introduction to Numerical Differential Geometry It is quite a common phenomenon that data sets have special geometric strucutres, hence it is natural and convenient to model the problem on manifolds. In this talk, we provide an overview of the emerging impact of numerical differential geometry in areas of modern mathematical data science, including but not limited to manifold learning, Bayesian optimization, and geometry processing. As an illustrative example, we will present the framework and application of Riemannian optimization, with an emphasis on differential geometry as a guiding principle in the design and analysis of optimization algorithms. A Riemannian Proximal Gradient Descent Method with Optimal Convergence Rate We consider solving nonconvex and nonsmooth optimization problems with Riemannian manifold constraints. Such problems have received considerable attention due to many important applications such as sparse PCA, sparse blind deconvolution, robust matrix completion. In the Euclidean setting, proximal gradient method is an excellent method for solving nonconvex nonsmooth problems. However, in the Riemannian setting, the related work is still limited. In this talk, we briefly review the existing Riemannian proximal gradient methods and give an accelerated Riemannian proximal gradient with convergence analysis. Numerical experiments are used to demonstrate the performance of the proposedmethod. This is joint work with Ke Wei at Fudan University. Semi-Riemannian Manifold Optimization We introduce a manifold optimization framework that utilizes semi-Riemannian structures on the underlying smooth manifolds. Unlike in Riemannian geometry, where each tangent space is equipped with a positive definite inner product, a semi-Riemannian manifold allows the metric tensor to be indefinite on each tangent space, i.e., possessing both positive and negative definite subspaces; differential geometric objects such as geodesics and parallel-transport can be defined on non-degenerate semi-Riemannian manifolds as well, and can be carefully leveraged to adapt Riemannian optimization algorithms to the semi-Riemannian setting. In particular, we discuss the metric independence of manifold optimization algorithms, and illustrate that the weaker but more general semi-Riemannian geometry often suffices for the purpose of optimizing smooth functions on smooth manifolds in practice. In addition, for many interesting matrix manifolds, closed-form expressions for geodesics and parallel-transports are much easier to obtain under the semi-Riemannian metric. |
| 10:00am - 12:00pm | Room free |
| Unitobler, F-113 | |
| 10:00am - 12:00pm | MS172, part 1: Algebraic statistics |
| Unitobler, F-121 | |
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10:00am - 12:00pm
Algebraic Statistics Algebraic statistics studies statistical models through the lens of algebra, geometry, and combinatorics. From model selection to inference, this interdisciplinary field has seen applications in a wide range of statistical procedures. This session will focus broadly on new developments in algebraic statistics, both on the theoretical side and the applied side. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Testing model fit for networks: algebraic statistics of mixture models and beyond We consider statistical models for relational data that can be represented as a network. The nodes in the network are individuals, organizations, proteins, neurons, or brain regions, while edges---directed or undirected--- specific types of relationships between the nodes such as personal or organizational affinities or other social/financial relationships, or some physical or functional links such as co-activation of brain regions. One of the key open problems in this area is testing whether a proposed statistical model fits the data at hand. Algebraic statistics is known to provide theoretically reliable tools for testing model fit for a class of models that are log-linear exponential families; let's call these log-linear ERGMs. In this talk, we will discuss how the machinery can be extended to mixtures of log-linear ERGMs and other general linear exponential-family models that need not be log-linear, and what the hurdles are that need to be overcome in order for this set of tools to be generalizable, scalable and practical. Oriented Gaussoids An oriented gaussoid is a combinatorial structure that captures the possible signs of correlations among Gaussian random variables. We introduce this concept and present approaches to the classification and construction of oriented gaussoids, drawing parallels to oriented matroids, which capture the possible signs of dependencies in linear algebra. Ideals of Gaussian Graphical Models Gaussian graphical models are semialgebraic subsets of the cone of positive definite matrices. We will report on recent results trying to characterize the vanishing ideals of these models, in particular situations where they are generated by determinantal constraints. Combinatorial matrix theory in structural equation models Many operations on matrices can be viewed from a combinatorial point of view by considering graphs associated to the matrix. For example, the determinant and inverse of a matrix can be computed from the linear subgraphs and 1-connections of the Coates digraph associated to the matrix. This combinatorial approach also naturally takes advantage of the sparsity structure of the matrix, which makes it ideal for applications in linear structural equation models. Another advantage of these combinatorial methods is the fact that they are often agnostic on whether the mixed graph contains cycles. As an example, we obtain a symbolic representation of the entries of the covariance matrix as a finite sum. In general, this sum will become similar to the well known trek rule, but where each half of the trek is a 1-connection instead of a path. This method of computing the covariance matrix can be easily implemented in computer algebra systems, and scales extremely well when the mixed graph has few cycles. |
| 10:00am - 12:00pm | MS134, part 1: Coding theory and cryptography |
| Unitobler, F-122 | |
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10:00am - 12:00pm
Coding theory and cryptography The focus of this proposal is on coding theory and cryptography, with emphasis on the algebraic aspects of these two research fields.Error-correcting codes are mathematical objects that allow reliable communications over noisy/lossy/adversarial channels. Constructing good codes and designing efficient decoding algorithms for them often reduces to solving algebra problems, such as counting rational points on curves, solving equations, and classifying finite rings and modules. Cryptosystems can be roughly defined as functions that are easy to evaluate, but whose inverse is difficult to compute in practice. These functions are in general constructed using algebraic objects and tools, such as polynomials, algebraic varieties, and groups. The security of the resulting cryptosystem heavily relies on the mathematical properties of these. The sessions we propose feature experts of algebraic methods in coding theory and cryptography. All levels of experience are represented, from junior to very experienced researchers. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Ferrers Diagram Codes: Constructions and Proportion Ferrers diagram codes play a crucial role in the construction of large subspace codes. They are defined as rank-metric codes where all matrices have support in a given Ferrers diagram. A Singleton-like bound for such codes is known, but to this day it is an open problem whether the bound can be attained for all possible parameter sets. In this talk some constructions leading to optimal Ferrers diagram codes (codes attaining the bound) will be presented. Thereafter, the proportion of optimal Ferrers diagram codes within the set of all codes with the same Ferrers shape and the same dimension will be discussed. We will see that for certain shapes, optimal Ferrers diagram codes are dense (for growing field size) in the space of all codes with the same shape and dimension, while for other shapes the limiting proportion is known to be quite small, but not zero. Subspace designs and majority logic decoding In [Rudolph 1967], a simple decoding method for linear codes based on majority decision using combinatorial designs is presented. This method is called "one-step majority logic decoding" and it's attraction lies in the easy realization in hardware. It requires that the dual code has to contain the blocks of a t-design as codewords. Ever since then, people studied the linear codes generated by the blocks of t-designs. For a good code it is desirable that the rank of the block-point incidence matrix of the design is small over some finite field. The famous Hamada conjecture states that so called "classical or geometric designs" which consist of the set of all k-flats in PG(v,q) or AG(v,q), minimize the p-rank for a prime power q. The codes generated by these designs are called Euclidean Geometry (EG) codes and Projective Geometry (PG) codes. In case p = q = 2, the codes are the well known Reed-Muller codes. We report a few observations on the codes generated by "subspace designs" - also known as q-analogs of designs. The blocks of these designs generate essentially the same linear codes as the geometric designs but with sometimes much improved complexity of the one-step majority logic decoder. This may be of interest when implementing error correction with nano-scale technologies. Further, we will look at Chen's two-step majority logic decoder and the connection to rank metric codes. Bounds on the complexity of computing Groebner bases for HFE systems I will discuss some recent joint work with Christophe Petit and Daniela Mueller, in which we give upper bounds for the complexity of computing a Groebner basis of the polynomial system associated to the HFE (Hidden Field Equations) cryptosystem. Post-quantum key agreement from commutative group actions The present-day method for setting up a secure communication channel over the internet makes use of the Diffie-Hellman key exchange protocol, which is based on exponentiation in groups. However its security breaks down if an adversary would be given access to a large universal quantum computer. It is unclear whether such a device will see the light of day in the near future, but the threat alone is enough reason to make the transition to so-called "post-quantum key exchange", which is an actively ongoing process. One attractive line of thought is to replace exponentiation in groups by other commutative group actions. Currently, the only working such proposal goes back to Couveignes and uses the CM torsor, which is an action of the class group of an imaginary quadratic ring on a certain set of elliptic curves. I will explain this idea and report on a tweak called CSIDH, which was recently developed in collaboration with Lange, Martindale, Panny and Renes and leads to a considerable speed-up, from minutes to milliseconds. |
| 10:00am - 12:00pm | MS132, part 1: Polynomial equations in coding theory and cryptography |
| Unitobler, F-123 | |
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10:00am - 12:00pm
Polynomial equations in coding theory and cryptography Polynomial equations are central in algebraic geometry, being algebraic varieties geometric manifestations of solutions of systems of polynomial equations. Actually, modern algebraic geometry is based on the use of techniques for studying and solving geometrical problems about these sets of zeros. At the same time, polynomial equations have found interesting applications in coding theory and cryptography. The interplay between algebraic geometry and coding theory is old and goes back to the first examples of algebraic codes defined with polynomials and codes coming from algebraic curves. More recently, polynomial equations have found important applications in cryptography as well. For example, in multivariate cryptography, one of the prominent candidates for post-quantum cryptosystems, the trapdoor one-way function takes the form of a multivariate quadratic polynomial map over a finite field. Furthermore, the efficiency of the index calculus attack to break an elliptic curve cryptosystem relies on the effectiveness of solving a system of multivariate polynomial equations. This session will feature recent progress in these and other applications of polynomial equations to coding theory and cryptography. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Free resolutions of test sets and their applications to coding theory To each linear code defined over a finite field one can define its associated matroid and its generalized Hamming weights which are the same as those of the code. Johnsen and Verdure. showed that the generalized Hamming weights of a matroid are determined by the graded Betti numbers of the Stanley-Reisner ring of the simplicial complex whose faces are the independent set of M. In this talk we go a step further: our practical results indicate that the generalized Hamming weights of a linearcode can be obtained from the monomial ideal associated with a test-set for the code. Algebraic geometry codes from del Pezzo surfaces In this talk, we consider the problem of constructing codes with good parameters from algebraic surfaces. We start from two constatations. The first one, due to Voloch and Zarzar in a 2007 article, is that surfaces with a small Picard rank, in particular those with Picard rank one, seem to be interesting candidates to provide good codes. The second one, is that several nice examples in the literature of surfaces yielding good codes can be understood in a unified context : that of del Pezzo surfaces. We will study the classification of del Pezzo surfaces over finite fields and consider their anticanonical codes. Such surfaces can be classified by the action of the Frobenius on the (geometric) Picard lattice, which gives many properties such as the (arithmetic) Picard number or the number of rational points. This rich structure of del Pezzo surfaces permits to obtain fine estimates of the parameters of these codes and even to compute their automorphism groups in some cases. This investigation led to the discovery of new codes whose parameters beat the best known codes listed in the database codetables.de. This is a collaboration with Blache, Hallouin, Madore, Nardi, Rambaud and Randriam. An Approach to Density Problems in Coding Theory We give a new perspective on extremal codes. We obtain upper and lower bounds on the density functions of a number of families of codes within a larger family. In many cases, these bounds have expressions involving polynomials in indeterminate q, where q is the size of the underlying scalar field. We use these expressions to obtain precise asymptotic estimates of these quantities and hence the density functions for some families of codes possessed of a particular extremal property. We introduce the idea of a partition-balanced family of codes, and show how the combinatorial invariants of such families can be used to obtain estimates on the number of codes satisfying a particular property. In particular, we show that the MRD matrix codes are not dense in the family of all matrix codes of a given fixed dimension, unlike the MRD vector codes. Multivariate Signatures In this talk, we will present the designs of multivariate signatures. The focus will be on the schemes submitted to the 2017 NIST post-quantum standard submissions. We will present the key security analysis tools and the main challenges for these schemes. |
| 1:30pm - 2:30pm | IP02: Tamara G. Kolda: Efficient Computation of Low-Rank Approximations to Higher-Order Moments |
| vonRoll, Fabrikstr. 6, 001 | |
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1:30pm - 2:30pm
Efficient Computation of Low-Rank Approximations to Higher-Order Moments Sandia National Laboratories, United States of America We consider the problem of decomposing a data tensor that is naturally expressed as the sum of p symmetric outer products of vectors of length n. For instance, a dth-order empirical moment tensor has such an expression, and there have been examples of this structure arising in machine learning problems. Our goal is to find the best approximate decomposition that is the sum of r symmetric outer products with r « p. We reduce the work and storage from exponential to linear in n, breaking the curse of dimensionality. When p is massive or the data is streaming, we show that stochastic sampling methods can be used to further reduce the complexity. We also show some intriguing finding on the rank of random tensors. This is joint work with PhD candidate Samantha Sherman at the University of Notre Dame. |
| 1:30pm - 2:30pm | IP02-streamed from 001: Tamara G. Kolda: Efficient Computation of Low-Rank Approximations to Higher-Order Moments |
| vonRoll, Fabrikstr. 6, 004 | |
| 2:30pm - 3:00pm | Coffee break |
| Unitobler, F wing, floors 0 and -1 | |
| 3:00pm - 5:00pm | Room free |
| Unitobler, F005 | |
| 3:00pm - 5:00pm | MS165, part 1: Multiparameter persistence: algebra, algorithms, and applications |
| Unitobler, F006 | |
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3:00pm - 5:00pm
Multiparameter persistence: algebra, algorithms, and applications Multiparameter persistent homology is an area of applied algebraic topology that studies topological spaces, often arising from complex data, simultaneously indexed by multiple parameters. In the usual setting, persistent homology studies a single-parameter filtration associated with a topological space. The homology of such a filtration is a persistence module, which can be conveniently described by its barcode decomposition. In many applications, however, a single-parameter filtration is not adequate to encode the structures of interest in complex data; two or more filtrations may be required. Multiparameter persistence studies the homology of spaces equipped with multiple filtrations. The homological invariants of these spaces are far more complicated than in the single-parameter setting, requiring new algebraic, computational, and statistical techniques. This work has deep connections to representation theory and commutative algebra, with compelling applications to data analysis. Recent years have seen considerable advances in multiparameter persistent homology, including algorithms for working with large multiparameter persistence modules, software for computing and visualizing invariants, statistical techniques, and applications. This minisymposium will highlight recent work in multiparameter persistence. Talks will include including theoretical results, algorithmic advances, and applications to data analysis. As many important questions remain to be answered in order to advance the theory and to increase the applicability of multiparameter persistence, this minisymposium seeks to cultivate discussion and collaboration that will lead to new results in the practical use of multiparameter persistent homology. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Multiparameter persistence: brief background and current challenges Persistent homology is a popular tool in topological data analysis, providing a method for discerning the shape of complex data. Applied in areas including computer graphics, biology, neuroscience, and signal processing, persistent homology produces easily-visualized algebraic invariants, called barcodes, which convey information about the topological structure of data. A multiparameter variant of persistent homology is particularly desirable for working with data simultaneously indexed by multiple parameters, but its algebraically complexity poses challenges in practice. This talk with introduce multiparameter persistent homology, with emphasis on mathematical foundations of this subject. We will see how multifiltered topological spaces arise from real-world data scenarios. We will introduce multiparameter persistence modules and see how their algebraic complexity poses challenges for their use in practice. We will briefly consider recent work, applications, and open questions in multiparameter persistent homology. Computing minimal presentations and bigraded Betti numbers of 2-parameter persistent homology Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm assumes that $M$ is given implicitly: It takes as input a short chain complex of free bipersistence modules [Faxrightarrow{ma} Fb xrightarrow{mb} Fc] such that $Mcong ker{mb}/im{ma}$. The algorithm runs in time $O(sum_i |F^i|^3)$ and requires $O(sum_i |F^i|^2)$ storage, where $|F^i|$ denotes the size of a basis of $F^i$. Given the presentation, the bigraded Betti numbers of the module are readily computed. We also present a different but related algorithm, based on Koszul homology, which computes the bigraded Betti numbers without computing a presentation, with these same complexity bounds. These algorithms have been implemented in RIVET, a software tool for the visualization and analysis of two-parameter persistent homology. In preliminary experiments on topological data analysis problems, our approach outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin. A kernel for multi-parameter persistent homology and its computation Kernels for one-parameter persistent homology have been established to connect persistent homology with machine learning techniques. In this talk, we discuss a kernel construction for multi-parameter persistence and why this kernel can provably be useful in applications. Morse inequalities for multiparameter persistence Discrete Morse theory is a combinatorial version of Morse theory which has proved to be an incredibly useful tool with applications in a large variety contexts. Intuitively speaking, discrete Morse theory allows to reduce a combinatorial cell complex (for example, a simplicial or cubical complex) to a subset of its cells, called critical, that carry all the homological information. The relation between the number of critical cells and the Betti numbers is described by the so-called Morse inequalities. So far, the connection between persistent homology and discrete Morse theory has been studied mainly with the purpose of simplifying complexes and speeding up the algorithms that compute the persistence modules. In this sense, only the reduction aspect of discrete Morse theory has been leveraged in connection to persistence. In this talk we show the possibility of establishing Morse inequalities for persistence. To this aim, we consider a filtration of a cell complex that varies according to multiple parameters, the associated multiparameter persistence module, and the critical cells of a discrete gradient vector field compatible with the multifiltration. Our goal is to derive Morse inequalities relating the number of critical cells of the given vector field to the multigraded Betti numbers of the persistence module. Thisrequires the use of specific tools from homological algebra, which we briefly illustrate. |
| 3:00pm - 5:00pm | MS123, part 2: Asymptotic phenomena in algebra and statistics |
| Unitobler, F007 | |
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3:00pm - 5:00pm
Asymptotic phenomena in algebra and statistics Across several branches of mathematics, the following fundamental question arises: given a sequence of algebraic structures with maps between them, can the entire sequence be characterized by a finite segment? Here the maps are comprising symmetries of the objects as well as morphisms between them. An affirmative answer leads to a description of all structures by using finite data only. There is a growing body of work that establishes the desired finiteness result in varied contexts. Nevertheless, instances where stability is not well understood include:
The aim of the minisymposium is to build bridges between the varied mathematicians and the different areas investigating stability phenomena. We propose a two half-day minisymposium with 8 speakers total. The proposed speakers have all expressed interest in speaking at the symposium. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Quantitative Properties of Ideals arising from Hierarchical Models We will discuss hierarchical models and certain toric ideals as a way of studying these objects in algebraic statistics. Some algebraic properties of these ideals will be described and a formula for the Krull dimension of the corresponding toric rings will be presented. One goal is to study the invariance properties of families of ideals arising from hierarchical models with varying parameters. We will present classes of examples where we have information about an equivariant Hilbert series. This is joint work with Uwe Nagel. Bounding degrees of generators for sequences of ideals By the celebrated Hilbert's basis theorem an ideal in a polynomial ring has a finite number of generators - in particular, there exists a bound on the degree of the generators. Varieties however often come to us in sequences and it may be highly nontrivial to establish a uniform degree bound. The questions one asks can have different flavour: one can ask for a set-, scheme- or ideal-theoretic description, an explicit or existential bound. We will report on several conjectures and theorems inspired by applied algebra, in particular, algebraic phylogenetics. Asymptotic Phenomena in the homology groups of graph configuration spaces A graph is a 1-dimensional compact, connected CW-complex. Given a graph G we define its n-fold configuration space UConf_n(G) to be the topological space of n distinct and unlabeled points on G. The study of the asymptotic behaviors of graph configuration spaces have taken two distinct paths in the literature. The first, and more classically flavored, involves fixing the graph G and increasing the number of points. In this case, Work of An, Drummond-Cole, and Knudsen, as well as independent work of the speaker, have shown that the homology groups can be equipped with the structure of finitely generated modules over a certain polynomial ring associated to the graph G. The second approach, appearing in work of Lutgehetmann, White, Proudfoot, and the speaker, involves fixing the number of points being configured and allowing the graph to vary in some regular way. In this case one once again recovers finite generation results for the homology groups, although describing what they are finitely generated over requires one to introduce concepts from the representation theory of categories. In this talk we will outline the state of the art with regards to both of these approaches, as well as how these two types of asymptotic stability can sometimes be related to one another. Mirror spaces and stability in the homology of Vandermonde varieties The level sets of the first d Newton power sums in R^k for some d ≤ k have been called Vandermonde varieties by Arnold and Giventhal. These varieties have a natural action of the symmetric group, which induces an action on their cohomology groups. By using a formula of Solomon we can study the decomposition of the resulting S_k-module and generalise some of the results obtained by Arnold and Giventhal on the homology modules of such varieties. These results in particular also yield some insight into the representational stability in the homology modules. (Based on joint works with Saugata Basu.)
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| 3:00pm - 5:00pm | Room free |
| Unitobler, F011 | |
| 3:00pm - 5:00pm | MS160, part 1: Numerical methods for structured polynomial system solving |
| Unitobler, F012 | |
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3:00pm - 5:00pm
Numerical methods for structured polynomial system solving Improvements in the understanding of numerical methods for dense polynomial system solving led to the complete solution of Smale's 17th problem. At this point, it remains an open challenge to achieve the same success in the solution of structured polynomial systems: explain the typical behavior of current algorithms and devise polynomial-time algorithms for computing roots of polynomial systems. In this minisymposium, researchers will present the current progress on applying numerical methods to structured polynomial systems. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Introductory Talk This talk will provide a brief overview of the state of the art in numerical methods for polynomial system solving, and introduce the goals of the mini-symposium. On the condition number of some algebraic problems. In this talk we will introduce a geometric framework to analyze the condition number of some classic algebraic problems. We will compute the average value of the squared condition number, when the input is taken at random, for a large family of these problems. In particular, we will show that the polynomial eigenvalue problem is quite well conditioned. Numerical irreducible decomposition with one homotopy A numerical irreducible decomposition (NID) of an algebraic set V includes a set of witness points (approximations of generic points) on each irreducible component of V, along with various auxiliary data. The computation of an NID typically involves a sequence of homotopies. Pairing together the machinery of excess intersection and isosingular sets, we show how to compute an NID with only one homotopy. This is joint work with David Eklund, Jonathan Hauenstein, and Chris Peterson. Computing the Homology of arbitrary Semialgebraic Sets We describe recent advances regarding the computation of the homology groups of arbitrary semialgebraic sets. These advances follow the line of results obtained for the particular cases of smooth projective varieties and basic semialgebraic sets. Coauthors are Peter Buergisser and Josue Tonelli-Cueto. |
| 3:00pm - 5:00pm | MS138: Computational aspects of tropical geometry |
| Unitobler, F013 | |
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3:00pm - 5:00pm
Computational aspects of tropical geometry The aim of this session is to demonstrate the effictive use of tropical geometry to tackle problems from optimization and various applications. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Condition numbers of stochastic mean payoff games and what they say about nonarchimedean semidefinite programming Semidefinite programming can be considered over any real closed field, including fields of Puiseux series equipped with their nonarchimedean valuation. Nonarchimedean semidefinite programs encode parametric families of classical semidefinite programs, for sufficiently large values of the parameter. Recently, a correspondence has been established between nonarchimedean semidefinite programs and stochastic mean payoff games with perfect information. This correspondence relies on tropical geometry. It allows one to solve generic nonarchimedean semidefinite feasibility problems, of large scale, by means of stochastic game algorithms. Computing tropical hypersurface intersections A tropical hypersurface is a balanced polyhedral complex of codimension 1 while a tropical prevarity is a finite intersection of such hypersurfaces. The dynamic enumeration strategy for prevarity computation proposed by Mizutani, Takeda and Kojima (2006) appears in the context of polyhedral homotopy continuation in polynomial system solving. It produces the prevariety for zero-dimensional intersections in general position. In contrast, the non-generic situation is more involved as the intersection may be non-pure and numerical exactness becomes critical. Extending joint work with Sommars and Verschelde we propose a dynamic decomposition strategy for reducing the number of vertices in the enumeration tree. We report on experimental results with our implementation. Algebraic systems and exterior semi-algebras In this talk, we describe negation maps and ``systems,'' and their application to Grassmann (exterior) algebra in a rather general framework that includes tropical algebra, hyperfields, and fuzzy rings. Tropical volume by tropical Ehrhart polynomials We are motivated by the problem of intrinsically defining a volume concept for tropical polytopes. In Euclidean space, the number of integral points contained in a large dilate of a given polytope approximates its volume. Even more, by Ehrhart's theorem, the function that counts integral points in dilates of an integral polytope is a polynomial in the dilation factor, whose leading coefficient equals the volume of the polytope. After introducing a suitable and natural concept of a tropical lattice, we aim to establish an analogous result for tropical lattice polytopes based on their covector decomposition. |
| 3:00pm - 5:00pm | MS191, part 1: Algebraic and geometric methods in optimization. |
| Unitobler, F021 | |
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3:00pm - 5:00pm
Algebraic and geometric methods in optimization. Recently advanced techniques from algebra and geometry have been used to prove remarkable results in Optimization. Some examples of the techniques used are polynomial algebra for non-convex polynomial optimization problems, combinatorial tools like Helly's theorem from combinatorial geometry to analyze and solve stochastic programs through sampling, and using ideal bases to find optimality certificates. Test-set augmentation algorithms for integer programming involving Graver sets for block-structured integer programs, come from concepts in commutative algebra. In this sessions experts will present a wide range of results that illustrate the power of the above mentioned methods and their connections to applied algebra and geometry. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Integer optimization from the perspective of subdeterminants For an integer optimization problem (IP), one important data parameter is the maximum absolute value among all square submatrices of the constraint matrix. We present recent developments about this topic. The Minimum Euclidean-Norm Point in a Convex Polytope: Wolfe's Combinatorial Algorithm is Exponential The complexity of Philip Wolfe's method for the minimum Euclidean-norm point problem over a convex polytope has remained unknown since he proposed the method in 1974. The method is important because it is used as a subroutine for one of the most practical algorithms for submodular function minimization. We present the first example that Wolfe's method takes exponential time. Additionally, we improve previous results to show that linear programming reduces in strongly-polynomial time to the minimum norm point problem over a simplex. This is joint work with J.A. De Loera and L. Rademacher. Matrices of bounded factor width and sums of $k$-nomial squares In 2004, Boman et al introduced the concept of factor width of a semidefinite matrix $A$. This is the smallest $k$ for which one can write the matrix as $A=VV^T$ with each column of $V$ containing at most $k$ non-zeros. The cones of matrices of bounded factor width give a hierarchy of inner approximations to the PSD cone. In the polynomial optimization context, a generalized Hankel matrix of a polynomial having factor width k corresponds to the polynomial being a sum of squares where each polynomial being squared has support at most $k$. This connection has recently been explored by Ahmadi and Majumdar to introduce SDSOS, a sum of squares hierarchy based on sums of binomial squares (sobs), but the study of sobs goes back to Robinson, Choi, Lam and Reznick and ultimately Hurwitz. In this presentation we will prove some results on the geometry of the cones of matrices with bounded factor widths and their duals, and use them to derive new results on the existence of certificates of nonnegativity by sums of k-nomial squares. Joint work with Mina Saee Bostanabad and Alexander Kovačec A friendly smooth analysis of the Simplex method The simplex method for linear programming is known for its good performance in practice, although the theoretical worst-case performance is exponential in the input size. The smoothed analysis framework of Spielman and Teng (2001) aims to explain the good practical performance. In our work, we improve on all previous smoothed complexity results for the simplex algorithm on all parameter regimes with a substantially simpler and more general proof. This is joint work with Daniel Dadush. https://arxiv.org/abs/1711.05667
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| 3:00pm - 5:00pm | MS195, part 1: Algebraic methods for convex sets |
| Unitobler, F022 | |
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3:00pm - 5:00pm
Algebraic methods for convex sets Convex relaxations are extensively used to solve intractable optimization instances in a wide range of applications. For example, convex relaxations are prominently utilized to find solutions of combinatorial problems that are computationally hard. In addition, convexity-based regularization functions are employed in (potentially ill-posed) inverse problems, e.g., regression, to impose certain desirable structure on the solution. In this mini-symposium, we discuss the use of convex relaxations and the study of convex sets from an algebraic perspective. In particular, the goal of this minisymposium is to bring together experts from algebraic geometry (real and classical), commutative algebra, optimization, statistics, functional analysis and control theory, as well as discrete geometry to discuss recent connections and discoveries at the interfaces of these fields. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) The slack variety of a polytope The slack variety of a polytope is an algebraic model for the realization space of a combinatorial class of a polytope. We establish a correspondence between realizations of a given polytope and points in the positive part of a variety of matrices of constrained rank. This allows us to apply the tools of computational algebra to a number of problems in polytope theory, such as rational realizability, projectively uniqueness, non-prescribability of faces and realizability of combinatorial polytopes. We then discuss the relationship between slack varieties, Grassmannians and Gale transforms. Spectrahedral representations of polar orbitopes Let $G$ be a connected compact Lie group. A linear representation $V$ of $G$ with $G$-invariant inner product is called polar if there is a linear subspace $Ssubset V$ that intersects every $G$-orbit orthogonally. A $G$-orbitope in $V$ is the convex hull of a $G$-orbit in $V$. We show that every orbitope in a polar representation of $G$ is a spectrahedron, and we construct an explicit spectrahedral representation. By analyzing the moment polytope we can often reduce the size of this representation. In particular, we arrive at new examples where the representation constructed has minimal size. (Joint work with Tim Kobert.) Sums of squares and quadratic persistence How does one effectively recognize sums of squares? We will focus on new bounds on the number of terms in a sum-of-squares representation for a quadratic form on a real projective subvariety. This talk is based on joint work with G. Blekherman, R. Sinn, and M. Velasco. Semialgebraic Vision In this talk I will discuss examples of problems in computer vision that are inherently semialgebraic but have not been studied from that angle thus far. This leads to interesting gaps between algebraic results and their semialgebraic versions. |
| 3:00pm - 5:00pm | MS187, part 1: Signature tensors of paths |
| Unitobler, F023 | |
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3:00pm - 5:00pm
Signature tensors of paths Given a path X in R^n, it is possible to naturally associate an infinite list of tensors, called the iterated-integral signature of X. These tensors were introduced in the 1950s by Kuo-Tsai Chen, who proved that every (smooth enough) path is uniquely determined by its signature. Over the years this topic became central in control theory, stochastic analysis and, lately, in time series analysis. In applications the following inverse problem appears: given a finite collection of tensors, can we find a path that yields them as its signature? One usually introduces additional requirements, like minimal length, or a parameterized class of functions (say, piecewise linear). It then becomes crucial to know when there are only finitely many paths having a given signature that satisfy the constraints. This problem, called identifiability, can be tackled with an algebraic-geometric approach. On the other hand, by fixing a class of paths (polynomial, piecewise linear, lattice paths, ..), one can look at the variety carved out by the signatures of those paths inside the tensor algebra. Besides identifiability, the geometry of these signature varieties can give a lot of information on paths of that class. One important class is that of rough paths. Apart from applications to stochastic analysis, its signature variety has a strong geometric significance and it exhibits surprising similarities with the classical Veronese variety. In time series analysis, it is often necessary to extract features that are invariant under some group action of the ambient space. The signature of iterated signals is a general way of feature extraction; one can think of it as a kind of nonlinear Fourier transform. Understanding its invariant elements relates to classical invariant theory but poses new algebraic questions owing to the particularities of iterated integrals. Recent developments in these aspects will be explored in this minisymposium. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Varieties of signature tensors The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is here examined through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic paths and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures. Joint work with Peter Fritz and Bernd Sturmfels. Learning paths from signature tensors We aim to recover paths from their third order signature tensors. For this, we apply methods from tensor decomposition, algebraic geometry and numerical optimization to the group action of matrix congruence. Given a signature tensor in the orbit of another tensor, we compute a matrix which transforms one to the other. We establish identifiability results, both exact and numerical, for piecewise linear paths, polynomial paths, and generic dictionaries. Numerical optimization is applied for recovery from inexact data. We also compute the shortest path with a given signature tensor. Signatures of paths: an algebraic perspective Coming from stochastic analysis, the signature of a path is the collection of all the iterated integrals of the path. It can be seen in terms of tensors or as formal power series in words, which make them more relevant in other areas such as algebraic geometry or combinatorics. In this talk, I would like to look at the signatures of paths from an algebra perspective. For that, we will look at the work done by C. Améndola, P. Friz, and B. Sturmfels about the variety defined by the signature of piecewise linear paths, as well as the work done by F. Galuppi about the variety of rough paths. As a continuation, I would like to present our work on the signature varieties of two very different classes of paths: rough paths and axis-parallel paths. This is joint work with F. Galuppi and M. Michalek. Signatures of paths transformed by polynomial maps We characterize the signature of piecewise continuously differentiable paths transformed by a polynomial map in terms of the signature of the original path. For this aim, we define recursively an algebra homomorphism between two shuffle algebras on words. This homomorphism does not depend on the path and behaves well with respect to composition and homogeneous maps. Joint work with Laura Colmenarejo. |
| 3:00pm - 5:00pm | MS152: Stochastic chemical reaction networks |
| Unitobler, F-105 | |
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3:00pm - 5:00pm
Stochastic chemical reaction networks The focus of this minisymposium is on new algebraic and analytic methods for stochastic chemical reaction networks. In contrast to deterministic models, stochastic systems cannot be described by systems of ordinary differential equations and, hence, direct application of algebraic methods is often not possible. We are interested in when the deterministic and the stochastic behaviour of chemical reaction networks diverge and how to analyse this behaviour with a combination of algebra, stochastic analysis and chemical reaction network theory. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Piecewise linear Lyapunov functions for stochastic reaction networks Stochastic reaction networks are mathematical models heavily utilized to describe the time evolution of biological systems, when few active molecules are present. In this case, the system dynamics is stochastic, and the changes of the molecules counts are described by means of a continuous time Markov chain. Despite the large use of these models, simple questions concerning the existence of a stationary distribution are hard to answer to, except for few exceptions, and constitute an active area of research. Often, in order to prove the convergence of a model to a stationary distribution, a suitable Foster-Lyapunov function is sought. I will present a novel and fast convex programming technique to check whether conditions implying the existence of a piecewise linear Lyapunov function hold. Such technique utilizes the geometry of the network to divide the state space in different regions where the calculated Foster-Lyapunov function is linear. Robust stochastic control of reaction networks Synthetic biology is a rapidly growing interdisciplinary field of science and engineering that aims to design biochemical systems which behave in a desired manner. With the recent breakthroughs in nucleic-acid-based biochemistry, arbitrary reaction networks can be experimentally implemented using solely DNA molecules, with applications to areas such as medicine, industry and nanotechnology. In this talk, I will focus on developing mathematical methods for designing reaction networks with predefined stochastic behaviors. In particular, the following fundamental problem in biochemical control theory will be considered: given any well-behaved mass-action kinetics input reaction network, whose structure and dynamics are at-most partially known, the goal is to introduce suitable additional biochemical species and reactions in a systematic manner, such that the resulting enlarged output network has a predefined stationary probability mass function (PMF). For experimental implementability in nucleic-acid-based synthetic biology, it is also required that the output stationary PMF is robust with respect to the initial conditions (ergodicity), and with respect to variations in the rate coefficients of the input network (robust perfect adaptation). I will present a solution to this problem, by embedding a faster controller network, called the stochastic morpher, into any given (slower) input network. Due to the introduced time-scale separation, it will be rigorously shown, using singular perturbation theory, that the controller overrides the firing of the input network, and morphs the input PMF into an output one with a desired form. The morphing will be performed at a lower-resolution level, by mapping the input PMF to the output one taking the form of a linear combination of Poisson PMFs, suitable for designing networks with predefined multi-modality (multi-stability). Higher-resolution morphing will also be presented, with arbitrary output PMFs. The results will be exemplified on relatively simple input networks, whose dynamics will be morphed to display noise-induced multi-modality and predefined PMFs. One-dimensional stochastic reaction networks: Classification and dynamics A crucial dynamical property to guarantee the existence of a stationary distribution is positive recurrence. However, it is not easy to provide checkable criteria for stochastic reaction networks, particularly with complex topological or graphical structures. Motivated by this need, this talk contributes to stochastic dynamics of chemical reaction networks (CRNs) with one-dimensional stoichiometric subspace. I will first present a classification of the state space of the underlying continuous time Markov chain (CTMC) associated with the CRN by identifying all types of states: absorbing (neutral and trapping) as well as escaping states, and open repelling as well as closed attracting non-singleton communicating classes, on each stochastic stoichiometric compatibility class with all large initial states. I will also mention how to use this result to discuss the diversity of long-term dynamics of stochastic CRNs, and point out that limit distributions of CRNs with absolute concentration robustness (ACR) are not necessarily Poisson, which answers a question by Anderson in 2014. Moreover, I will present checkable necessary and sufficient network conditions for various dynamical properties: Recurrence (positive and null), transience, (non)explosivity, (non)implosivity, as well as existence of moments of passage times of the associated CTMC of one-dimensional stochastic CRNs. As a byproduct, any one-dimensional weakly reversible CRN is positive recurrent, confirming the Positive Recurrence Conjecture proposed by Anderson and Kim in 2018 (in 1-d case). In addition, I will mention how to use these conditions to address a question by Anderson et al. in 2018 on non-explosivity. Finally, I will emphasize results on one-species CRNs, regarding asymptotics of tails of stationary distributions as well as approximation of an arbitrary discrete distribution by either ergodic stationary distributions or quasi-stationary distributions of one-species mass-action CRNs, and present parameter regions for consistency and inconsistency of stochastic and deterministic one-species CRNs regarding various dynamical properties aforementioned. The geometry and dynamics of spatial networks subject external noise In this talk I will introduce a way of modelling external noise in chemical reaction networks. Further, I am going to show how steady state geometry influences the modelling of external noise systems with particular regard to the multistationarity structure of the system. |
| 3:00pm - 5:00pm | MS154, part 1: New developments in matroid theory |
| Unitobler, F-106 | |
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3:00pm - 5:00pm
New developments in matroid theory The interactions between Matroid Theory, Algebra, Geometry, and Topology have long been deep and fruitful. Pertinent examples of such interactions include breakthrough results such as the g-Theorem of Billera, Lee and Stanley (1979); the proof that complements of finite complex reflection arrangements are aspherical by Bessis (2014); and, very recently, the proof of Rota's log-concavity conjecture by Adiprasito, Huh, and Katz (2015). The proposed mini-symposia will focus on the new exciting development in Matroid Theory such as the role played by Bergman fans in tropical geometry, several results on matroids over a commutative ring and over an hyperfield, and the new improvement in valuated matroids and about toric arrangements. We plan to bring together researchers with diverse expertise, mostly from Europe but also from US and Japan. We are going to include a number of postdocs and junior mathematicians. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Positivity of the coefficients of G-Tutte polynomials Recently, the notions of G-Tutte polynomials and G-plexifications were introduced to build a general framework for studying hyperplane, toric, q-reduced arrangements and their "Tutte-like" polynomials (Tutte, arithmetic Tutte, characteristic (quasi-)polynomials) en masse. Like the above-mentioned (quasi-)polynomials, the G-Tutte polynomials possess Deletion-Contraction and convolution formulas, but unlike them, the G-Tutte polynomials may have negative coefficients. We are currently interested in under what conditions their coefficients all are positive? In this talk, we will propose some ideas and partial answers. This talk is based on two recent joint works with Ye Liu and Masahiko Yoshinaga. Enumerative aspects of G-Tutte polynomials G-Tutte polynomial is a generalization of arithmetic Tutte polynomial. I will discuss some results on enumerative aspects of G-Tutte polynomial. Abelian arrangements, matroids and group actions Arrangements of hyperplanes have long offered a geometric point of view on matroids - at times leading to structural advances even in the nonrealizable case. The theory of arrangements recently broadened its scope beyond the case of hyperplanes to include arrangements in the torus, in products of elliptic curves and, more generally, in Abelian Lie groups. This development spurred the search for suitable enrichments of matroid theory. In this context, I will introduce the foundations of a theory of group actions on (semi)matroids, focussing mainly on applications to the structure of intersection posets of arrangements. I will also outline how this framework relates to (arithmetic) Tutte polynomials, arithmetic matroids and G-Tutte polynomials. A further ramification of this setup will be illustrated in A. D’Alì’s talk on generalized Stanley-Reisner rings. The material I will present is partly drawn from joint works with Alessio D’Alì, Giacomo d’Antonio, Noriane Girard, Giovanni Paolini and Sonja Riedel. Group actions on generalized Stanley-Reisner rings The Stanley-Reisner correspondence, which assigns a commutative ring to each finite simplicial complex, is a useful and well-studied bridge between commutative algebra and combinatorics, yielding particularly nice results for the independence complex of a matroid. In 1987 Sergey Yuzvinsky proposed a construction that allows to see the Stanley-Reisner ring of any finite simplicial complex as the ring of global sections of a sheaf of rings on a poset. Motivated by applications in the theory of Abelian arrangements, E. Delucchi and I extend Yuzvinsky's construction to the case of (possibly infinite) finite-length simplicial posets. We show that this generalization behaves well with respect to quotients of simplicial complexes and posets by group actions such as those introduced in E. Delucchi's talk. |
| 3:00pm - 5:00pm | MS168, part 1: Riemann Surfaces |
| Unitobler, F-107 | |
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3:00pm - 5:00pm
Riemann Surfaces In the past decades, the central role played by Riemann surfaces in pure mathematics has been strengthened with their surprising appearance in string theory, cryptography and material science. This minisymposium is intended for the curve theorists and the avant-garde applied mathematician. Our emphasis will be on the computational aspects of Riemann surfaces that are prominent in pure mathematics but are not yet part of the canon of applied mathematics. Some of the subjects that will be touched upon by our speakers are integrable systems, Teichmüller curves, Arakelov geometry, tropical geometry, arithmetic geometry and cryptography of curves. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Real soliton lattices of KP-II equation and desingularization of spectral curves Planar bicolored (plabic) networks in the disk were originally introduced by A.Postnikov to parametrize positroid cells in totally nonnegative Grassmannians and used by Y. Kodama and L.Williams to explain the asymptotic behavior of real regular multiline soliton solutions (rrss) of Kadomtsev-Petviashvili II (KP) equation. In this talk based on recent papers in collaboration with P.G. Grinevich (arXiv:1801.00208, arXiv:1803.10968, arXiv:1805.05641) we explain a different relation of plabic networks with KP theory based on the spectral theory for degenerate finite-gap solutions on reducible curves by Krichever where the rrss play the role of potentials. In our construction the plabic graph is dual to a reducible curve which is the rational degeneration of a smooth M-curve of genus equal to the number of faces of the graph diminished by one. The boundary of the disk corresponds to the rational curve associated to the soliton data in the direct spectral problem and each internal vertex is a rational component. Edges are the double points where two such components are glued. We then introduce and characterize systems of edge vectors on plabic networks and use them to uniquely associate to generic soliton data a Krichever divisor satisfying the reality and regularity conditions of Dubrovin-Natanzon. Our approach is constructive and may be used to effectively desingularize curves. The case of soliton data in the positive part of Gr(2,4) is shown in detail. Conformal patterns on closed surfaces via discrete conformal maps and holomorphic differentials Using uniformization of discrete Riemann surfaces we construct conformal patterns on closed surfaces without cuts and overlapping. Arakelov invariants in the tropical limit In this talk we are interested in semistable degenerations of compact Riemann surfaces. We consider the asymptotic behavior, under such degenerations, of certain canonical metrics, Green's functions and related invariants as studied in Arakelov theory and string perturbation theory. We obtain precise expressions for the asymptotics under consideration in terms of potential theory on metric graphs, aka tropical curves. In particular, non-archimedean and tropical geometry appear naturally when studying degenerations of Riemann surfaces. Siegel modular forms and classical invariants For abelian varieties of dimension 2 and 3, Siegel modular forms for the full symplectic group can be reinterpreted as classical invariants for the action of GL2 or GL3. There are many applications and consequences of this dictionary. We will show in particular that one can decrease the number of generators for the ring of Siegel modular forms in dimension 3 obtained by Tsuyumine (1986). Joint work with Reynald Lercier. |
| 3:00pm - 5:00pm | Room free |
| Unitobler, F-111 | |
| 3:00pm - 5:00pm | MS197, part 2: Numerical differential geometry |
| Unitobler, F-112 | |
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3:00pm - 5:00pm
Numerical Differential Geometry The profound theory of differential geometry have interacted with the computational and statistical communities in the past decades, yielding fruitful outcomes in a wide range of fields including manifold learning, Riemannian optimization, and geometry processing. This minisymposium encourages researchers from applied differential geometry, optimization, manifold learning, and geometry processing to share their perspectives and technical tools on problems lying in the intersection of geometry and computations. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Anisotropic Diffusion Kernels to Compare Distributions We introduce a kernel-based Maximum Mean Discrepancy (MMD) statistic for measuring the distance between two distributions from finitely-many multivariate samples, where the kernel is anisotropic. The kernel computes the affinity between n data points and a set of nR reference points, where nR can be drastically smaller than n. When the unknown distributions are locally low-dimensional, the proposed MMD test can be more powerful to distinguish certain alternatives, which is theoretically characterized by the spectral decomposition of the kernel. The consistency of the test is proved as long as the magnitude of the distribution departure is of a higher order than n^{-1/2}, and a finite-sample lower bound of the testing power is provided. The test is applied to flow cytometry and diffusion MRI datasets, which motivate the proposed approach to compare distributions. Coupled Geometric and Topological Basis for Data-Driven Shape Reconstruction We introduce a data-driven geometry reconstruction method with provable guarantees. A key enabler for the robustness of our shape recovery, with geometric and topological fidelity, is a new coupled basis representation that combines a voxelized implicit form of the shape geometry, and a vectorized persistent diagram of the shape topology. Our method optimizes an objective function that enforces the agreement between the reconstructed shape and the input point cloud, regularizes geometry and topology of the reconstruction with data and enforces the consistency between the geometric prior and the topological prior. We show how to solve this optimization problem effectively by combing spectral initialization under the geometric representation alone and gradient-descent refinement under the coupled representation. In particular, we show that the spectral initialization does not need to be accurate, as the refinement procedure is able to improve the topology of the reconstruction. Experimental results on synthetic and real datasets justify the usefulness of our approach. Intrinsic Gaussian processes on complex constrained domains We propose a class of intrinsic Gaussian processes (in-GPs) for interpolation, regression and classification on manifolds with a primary focus on complex constrained domains or irregular-shaped spaces arising as subsets or submanifolds of R, R2, R3 and beyond. For example, in-GPs can accommodate spatial domains arising as complex sub- sets of Euclidean space. in-GPs respect the potentially complex boundary or interior conditions as well as the intrinsic geometry of the spaces. The key novelty of the proposed approach is to utilise the relationship between heat kernels and the transition density of Brownian motion on manifolds for constructing and approximating valid and computation- ally feasible covariance kernels. This enables in-GPs to be practically applied in great generality, while existing approaches for smoothing on constrained domains are limited to simple special cases. The broad utilities of the in-GP approach are illustrated through simulation studies and data examples. Locally Linear Embedding on Manifold Locally Linear Embedding(LLE), is a well known manifold learning algorithm published in Science by S. T. Roweis and L. K. Saul in 2000. In this talk, we provide an asymptotic analysis of the LLE algorithm under the manifold setup. We establish the kernel function associated with the LLE and show that the asymptotic behavior of the LLE depends on the regularization parameter in the algorithm. We show that on a closed manifold, asymptotically we may not obtain the Laplace--Beltrami operator, and the result may depend on the non-uniform sampling, unless a correct regularization is chosen. The talk is based on the joint work with Hau-tieng Wu. |
| 3:00pm - 5:00pm | MS184, part 1: Algebraic geometry for kinematics, mechanism science, and rigidity |
| Unitobler, F-113 | |
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3:00pm - 5:00pm
Algebraic geometry for kinematics, mechanism science, and rigidity Mathematicians became interested in problems concerning mobility and rigidity of mechanisms as soon as study of the subject began. Algebraists and geometers among them, notably Clifford and Study, developed tools still used today to investigate pertinent questions in the field. Recent renewed interest in techniques of algebraic geometry applied to kinematics and rigidity led to a modern classification of mechanisms, discovery of new families, development of algorithms for path planning and overall better understanding of rigid structures and configurations. A wide variety of techniques has been used in this regard and it is reasonable to expect that further influence of algebraic geometry upon kinematics and rigidity will produce deeper understanding leading to useful advancement of technology. We will focus on topics in algebraic geometry motivated by kinematics and rigidity or algebraic geometry methodology with potential application in kinematics and rigidity. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) On four-bar linkages, elliptic functions, and flexible polyhedra Darboux discovered that the (complexified) configuration space of a four-bar linkage is an elliptic curve. We present an explicit parametrization of the configuration space (in terms of the angles between the bars) by Jacobi elliptic functions and some geometric applications of this parametrization: an interpretation of Bottema's zigzag theorem, a derivation of the Dixon angle condition in the Burmester linkage, and examples of flexible quad-surfaces. Singularity distance computation for parallel manipulators of Stewart Gough Type The number of applications of parallel robots, ranging from medical surgery to astronomy, has increased enormously during the last decades due to their advantages of high speed, stiffness, accuracy, load/ weight ratio, etc. One of the drawbacks of these parallel robots are their singular configurations, where the manipulator has at least one uncontrollable instantaneous degree of freedom. Furthermore, the actuator forces can become very large, which may result in a breakdown of the mechanism. Therefore singularities have to be avoided. As a consequence the kinematic/robotic community is highly interested in evaluating the singularity closeness, but geometric a meaningful distance measure between a given manipulator configuration and the next singular configuration is still missing. We close this gap for parallel manipulators of Stewart Gough type by introducing such measures. Moreover the favored metric has a clear physical meaning, which is very important for the acceptance of this index by mechanical/constructional engineers. Every proposed singularity distance results from the solution of an algebraic system of equations, whose computational aspects are discussed on the basis of examples. Analysis of kinematic singularities through roadmap computations The analysis of kinematic singularities can be modeled as the problem of counting the connected components of a semi-algebraic set or finding a path joining two points in this set whenever such a path exists. These algorithmic problems are known to be difficult and usually tackled through the computation of a roadmap. This is an algebraic curve which will capture the connectivity of the semi-algebraic set under study. In this talk, I will review some recent progress on the state-of-the algorithms for computing roadmaps and report on their implementations which were used to analyze the kinematic singularities of some robots. Computing cognates of mechanisms A coupler cognate of a planar linkage is a different mechanism that has the same coupler curve. Roberts showed that there are 3 four-bar mechanisms that generate the same coupler curve. Dijksman provided a list of cognates for six-bar mechanisms but without proof the list was complete. This talk will describe a geometric approach to easily understand cognates which yields a simple method to generate cognates. We combine this with numerical algebraic geometry to give a method to produce a complete list of all coupler cognates. Examples on six - bar mechanisms will be shown to demonstrate the method. This is joint work with Jon Hauenstein and Charles Wampler. |
| 3:00pm - 5:00pm | MS157, part 1: Graphical models |
| Unitobler, F-121 | |
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3:00pm - 5:00pm
Graphical Models Graphical models are used to express relationships between random variables. They have numerous applications in the natural sciences as well as in machine learning and big data. This minisymposium will feature talks on several different types of graphical models, including latent tree models, max linear models, network models, boltzman machines, and non-Gaussian graphical models, each of which exploits their intrinsic algebraic, geometric, and combinatorial structure. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Brownian motion tree models are toric Felsenstein’s classical model for Gaussian distributions on a phylogenetic tree is shown to be a toric variety in the space of concentration matrices. We give an exact semialgebraic characterization of this model, and we demonstrate how the toric structure leads to exact methods for maximum likelihood estimation. Algebra and statistical learning for inferring phylogenetic networks Phylogenetic trees are graphical summaries of the evolutionary history of a set of species. In a phylogenetic tree, the interior nodes represent extinct species, while the leaves represent extant, or living, species. While trees are a natural choice for representing evolution visually, by restricting to the class of trees, it is possible to miss more complicated events such as hybridization and horizontal gene transfer. For more complete descriptions, phylogenetic networks, directed acyclic graphs, are increasingly becoming more common in evolutionary biology. In this talk, we will discuss Markov models on phylogenetic networks and explore how understanding their algebra and geometry can aid in establishing identifiability and model selection. In particular, we will describe a method for network inference that combines computational algebraic geometry and statistical learning. This is joint work with Travis Barton, Colby Long, and Joseph Rusinko. Geometry of max-linear graphical models Motivated by extreme value theory, max-linear graphical models have been recently introduced and studied as an alternative to the classical Gaussian or discrete distributions often used in graphical modeling. We present max-linear models naturally in the framework of tropical geometry. This perspective allows us to shed light on some known results and to prove others with algebraic techniques, including conditional independence statements and maximum likelihood parameter estimation. This is joint work with Claudia Klüppelberg, Steffen Lauritzen and Ngoc Tran. Maximum Likelihood Estimation of Toric Fano Varieties motivated by phylogenetics We study the maximum likelihood estimation problem for several classes of toric Fano models. We start by exploring the maximum likelihood degree for all 2-dimensional Gorenstein toric Fano varieties. We show that the ML degree is equal to the degree of the surface in every case except for the quintic del Pezzo surface with two singular points of type A1 and provide explicit expressions that allow to compute the maximum likelihood estimate in closed form whenever the ML degree is less than 5. We then explore the reasons for the ML degree drop using A-discriminants and intersection theory. Finally, we show that toric Fano varieties associated to 3-valent phylogenetic trees have ML degree one and provide a formula for the maximum likelihood estimate. We prove it as a corollary to a more general result about the multiplicativity of ML degrees of codimension zero toric fibre products, and it also follows from a connection to a recent result about staged trees. This is joint work with Carlos Amendola and Kaie Kubjas. |
| 3:00pm - 5:00pm | MS134, part 2: Coding theory and cryptography |
| Unitobler, F-122 | |
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3:00pm - 5:00pm
Coding theory and cryptography The focus of this proposal is on coding theory and cryptography, with emphasis on the algebraic aspects of these two research fields.Error-correcting codes are mathematical objects that allow reliable communications over noisy/lossy/adversarial channels. Constructing good codes and designing efficient decoding algorithms for them often reduces to solving algebra problems, such as counting rational points on curves, solving equations, and classifying finite rings and modules. Cryptosystems can be roughly defined as functions that are easy to evaluate, but whose inverse is difficult to compute in practice. These functions are in general constructed using algebraic objects and tools, such as polynomials, algebraic varieties, and groups. The security of the resulting cryptosystem heavily relies on the mathematical properties of these. The sessions we propose feature experts of algebraic methods in coding theory and cryptography. All levels of experience are represented, from junior to very experienced researchers. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Privacy and lifted codes For any linear code and abstract simplicial complex on the same ground set, we define the lift of the linear code to be the smallest code whose projection to any simplex agrees with that of the original code. The motivation for this construction comes from private information retrieval (PIR), and in particular from the so-called star-product schemes for PIR from coded storage systems with colluing servers. We study the basic combinatorial and algebraic properties of the lifted code, and relate the PIR rate of a star product scheme to the quotient of a lifted code modulo its underlying code. Decoding of 2D convolutional codes In this talk, we present a decoding algorithm for 2D convolutional codes over the erasure channel. This algorithm breaks down the decoding of the 2D convolutional code to several decoding steps with 1D convolutional codes. Moreover, we present constructions of codes, which are especially suitable for this algorithm. On the computation of the duals of certain Algebraic Geometric codes with an application to quantum codes We consider a family of smooth projective and absolutely irreducible plane curves over $mathbb{F}_q$. We compute the number of rational points and a canonical divisor for it. Thanks to it we can deduce when the associated algebraic geometric code is self-orthogonal and construct stabilizer quantum codes. This work was inspired by the work titled " Quantum error-correcting codes from Algebraic Geometry codes of Castle type." Generalization of the ball-collision algorithm Since 1978 it is known that decoding a random linear code is an NP-complete problem, this was shown by Berlekamp, McEliece and van Tilburg. One of the methods to decode a random linear code is called Information Set Decoding (ISD). Many improvements for the ISD algorithm over the binary field have been suggested, amongst them is the ball-collision algorithm by Bernstein, Lange and Peters. The problem of decoding a random linear code has recently received prominence with the McEliece cryptosystem, since ISD attacks on this cryptosystem determine the choices of secure parameters and hence the key size. Since some of the new variants of the McEliece cryptosystem involve codes over general finite fields, we present in this talk the generalization of the ball-collision algorithm to an arbitrary finite field. |
| 3:00pm - 5:00pm | MS132, part 2: Polynomial equations in coding theory and cryptography |
| Unitobler, F-123 | |
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3:00pm - 5:00pm
Polynomial equations in coding theory and cryptography Polynomial equations are central in algebraic geometry, being algebraic varieties geometric manifestations of solutions of systems of polynomial equations. Actually, modern algebraic geometry is based on the use of techniques for studying and solving geometrical problems about these sets of zeros. At the same time, polynomial equations have found interesting applications in coding theory and cryptography. The interplay between algebraic geometry and coding theory is old and goes back to the first examples of algebraic codes defined with polynomials and codes coming from algebraic curves. More recently, polynomial equations have found important applications in cryptography as well. For example, in multivariate cryptography, one of the prominent candidates for post-quantum cryptosystems, the trapdoor one-way function takes the form of a multivariate quadratic polynomial map over a finite field. Furthermore, the efficiency of the index calculus attack to break an elliptic curve cryptosystem relies on the effectiveness of solving a system of multivariate polynomial equations. This session will feature recent progress in these and other applications of polynomial equations to coding theory and cryptography. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Efficient Key Generation for Rainbow The Rainbow Signature Scheme is one of the most studied multivariate signature scheme and was accepted as a second round candidate for the NIST standardization process for post-quantum cryptosystems. However, the key generation process of the first round version was not very efficient. In this talk, we present a modified version of the Rainbow key generation process, which generates a Rainbow key pair using only matrix products, and therefore is very efficient. Furthermore, our new algorithm also allows a very efficient key generation for Rainbow variants such as cyclicRainbow. Algebraic techniques for cryptanalysis of rank-based cryptosystems In the past few years, code-based cryptography in the rank-metric has become increasingly popular mainly because of the efficiency advantages over similar constructions in the Hamming metric and the ongoing NIST post-quantum standardization process. Several new ideas have emerged - for example, the cryptosystems based on LRPC codes follow an NTRU-like design and provide an alternative to the classical rank-based cryptosystems based on Gabidulin codes. Furthermore, the security in the rank metric is now much better understood - recently it was shown that the rank syndrome decoding problem is hard. On the other hand, the known cryptosystems do not have a reduction from the hard problem. Therefore, it is interesting not only to study the practical security of the rank syndrome decoding, but also attacks that take advantage of the particular construction of the cryptosystems. In this talk I will focus on algebraic techniques in the context of rank-based cryptography. It is known that decoding problems in rank-based cryptography can be modeled as systems of (non-linear) equations, however not much attention has been devoted to this modelling. It turns out that algebraic techniques are more powerful than previously thought. I will discuss how they can be refined and used in an unexpected way and how particular structure of the cryptosytems influences their efficiency. MinRank Problems in Post-Quantum Cryptography We explore some of the variety of MinRank problem instances arising in post-quantum cryptography. We briefly review some prototypical applications in some of the post-quantum families of schemes, recall some of the computation techniques and summarize some of the complexity results for such instances. This talk should illustrate the landscape more closely investigated in the talks of Daniel Cabarcas and Ray Perlner. Rank Analysis of Cubic Multivariate Cryptosystems Multivariate cryptography is the study of public-key cryptosystems based on multivariate polynomials over a finite field. Since solving a system of multivariate nonlinear polynomials over a finite field of order 2 is proven to be NP-hard, it is considered to be secure against quantum computers. Currently, most of the multivariate schemes are based on system of quadratic polynomials, mainly because of two reasons. First, they are smaller compared to higher degree constructions and hence more efficient. Second, if f is cubic, its (symmetric) differential Df(x) = f(x+a) - f(x) - f(a) is a quadratic map that preserves some of the properties of f. In quadratic constructions, one of the most successful family of attacks is the min-rank attack. It exploits the existence of low-rank linear combination of the matrices representing the quadratic forms of the public polynomials. One natural way to avoid this attack is to use cubic polynomials. This leads to several natural questions: Is there a notion of rank for cubic forms? Can we extend the min-rank attack to cubic constructions? Is the differential attack always a vulnerability for such constructions? What are the implications of low-rank cubic constructions? In this talk, we address all these questions by taking a general perspective of cubic multivariate schemes. This is a joint work with John Baena, Daniel Cabarcas, Daniel Escudero and Javier Verbel. |
| 5:15pm - 7:30pm | PP: Welcome reception and poster session |
| vonRoll, Fabrikstr. 8, Foyer | |
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A constructive algorithm for a positive solution to a system of polynomial inequalities Texas A&M University, United States of America In this poster we present an algorithm to construct a positive point that solves a system of polynomial inequalities. The procedure uses ideas from polyhedral geometry, specifically normal fans of the Newton polytopes of the polynomials. We give an application of this algorithm to a problem from chemical reaction network theory, an area of mathematics that analyzes the behaviors of chemical processes. A major problem in this area is the stability of equilibria of dynamical systems arising from these networks. Here, we focus on a biological signaling network called the ERK network, a model for dual-site phosphorylation and desphorylation of extracellular signal-regulated kinase. The ERK network is known to be bistable and to exhibit oscillations (Rubinstein, Mattingly, Berezhkovskii and Shvarstman, 2016), but a limiting network of the ERK network- the fully processive dual-site network- is known to have a unique, stable steady state (Conradi and Shiu, 2015). We investigate the emergence of oscillations and instability by analyzing certain subnetworks of ERK. A precursor to oscillations is the existence of a Hopf bifurcation, which are characterized by sign conditions on Hurwitz-matrix determinants (Yang, 2002). Thus, finding a Hopf bifurcation amounts to finding a positive solution to a system of polynomial inequalities. We present a solution using our algorithm. A generalization of Strassen's Positivstellensatz and its application to large deviation theory Perimeter Institute for Theoretical Physics, Canada Strassen's Positivstellensatz is a powerful but little known theorem on preordered commutative semirings satisfying a boundedness condition similar to Archimedeanicity. It characterizes the relaxed preorder induced by all monotone homomorphisms to $mathbb{R}_+$ in terms of a condition involving large powers. I will present a generalization and strengthening of Strassen's result and sketch a variety of applications to probability theory, representation theory, graph theory, and information theory. As a generalization, Strassen's boundedness condition by a polynomial growth condition; as a strengthening, I will show two further equivalent characterizations of the homomorphism-induced preorder. The application to probability gives results on the asymptotic comparison of one random walk relative to another, which generalizes a (weak form of) Cramér's large deviation theorem. This constitutes the first probabilistic characterization of when the moment-generating function of one random variables dominates that of another, in the context of bounded random variables. A linear method for positive solutions to polynomial systems Univeresity of Wisconsin-Madison, United States of America We propose a linear method to parametrize the positive solutions of a class of real polynomial systems. These complex-balanced systems arise naturally from chemical reaction networks, where the polynomial system is determined by a directed graph embedded in Euclidean space. Given a positive solution, we define a linear feasibility problem to determine whether the polynomial system can be realized as a complex-balanced system. If yes, we exploit its known toric structure to obtain a monomial parametrization of its positive steady states. A module theoretic perspective on matroids University of Wisconsin Madison, United States of America Speyer recognized that matroids encode the same data as a special class of tropical linear spaces and Shaw interpreted tropically certain basic matroid constructions; additionally, Frenk developed the perspective of tropical linear spaces as modules over an idempotent semifield. All together, this provides bridges between the combinatorics of matroids, the algebra of idempotent modules, and the geometry of tropical linear spaces. The goal of this paper is to strengthen and expand these bridges by systematically developing the idempotent module theory of matroids. Applications include a geometric interpretation of strong matroid maps and the factorization theorem; a generalized notion of strong matroid maps, via an embedding of the category of matroids into a category of module homomorphisms; a monotonicity property for the stable sum and stable intersection of tropical linear spaces; a novel perspective of fundamental transversal matroids; and a tropical analogue of reduced row echelon form. Catalan-many tropical morphisms to metric trees University of Bern, Switzerland We investigate tree gonality of metric graphs, a tropical version of curve gonality (i.e. the minimum degree of a rational map to P1). Our main contribution is to construct and describe a space Φdℳtrop of tropical morphisms. As a corollary, we get a constructive method to prove that the gonality of every genus g metric graph Γ is at most ⌈ g / 2 + 1 ⌉. When g is even, we construct all the maps realizing the gonality, and a count of these using certain multiplicity gives the g/2-th Catalan number. Classification of triples of lattice polytopes with a given mixed volume Otto-von-Guericke Universität Magdeburg, Germany In this poster we present an overview of our algorithm for the classification of triples of lattice polytopes with a given mixed volume m in dimension 3. It is known that the classification can be reduced to the enumeration of so-called irreducible triples, the number of which is finite for fixed m. Following this algorithm, we were able to enumerate all irreducible triples of normalized mixed volume up to 4 that are inclusion-maximal. On algebro-geometric side this produces a classification of generic trivariate sparse polynomial systems with up to 4 solutions in the complex torus, up to monomial changes of variables. By a recent result of Esterov, this leads to a description of all generic trivariate sparse polynomial systems that are solvable by radicals. This is joint work with Gennadiy Averkov and Ivan Soprunov. Complexity of variety learning KTH Royal Institute of Technology, Sweden Extracting structure from a point cloud is a fundamental problem in data analysis, and when this structure is coming from polynomial equations we can use the machinery of algebraic geometry. We study the set of closest real algebraic hypersurfaces to a finite set of points, and investigate methods to approximate the closest hypersurface of a given degree. We investigate the complexity of this problem by considering the Euclidean Distance Degree (EDD) of the variety of point configurations lying on a hypersurface of a given degree and how the complexity depends on the number of points in the configuration. Embeddability of Markov matrices does not depend only on its principal logarithm Universitat Politècnica de Catalunya, Spain The embedding problem for Markov substitution matrices consists on determining whether the substituition process described by a given Markov matrix can be explained by homogeneous time-continuous models or not. Several results on the topic seem to point out that the embeddability of a matrix with pairwise different eigenvalues is determined by its principal logarithm. In this poster we build the first example, up to our knowledge, disproving this fact. We will work with 4x4 matrices corresponding to DNA nucleotide-substitution models.
To obtain such example we intersect: i) the polyhedral cone of principal logarithms not fulfilling sthocastic embeddability constrains, ii) the polyhedral cone of a certain family of non-principal logarithms fulfilling sthocastic embeddability constrains and iii) the set of real logarithms of Strand Symmetric Model (SSM) substitution matrices. As a result we get a semialgebraic non-empty open set containing embeddable SSM matrices whose principal logarithm is not involved on its embeddability. Furthermore, this example can be deformed in such a way that we can obtain infinitely many examples for general Markov matrices. This will be proven in a forthcoming paper jointly with Marta Casanellas and Jesús Fernández-Sánchez. Gröbner Bases for Toric Staged Tree Models Otto-von-Guericke Universität Magdeburg, Germany A toric staged tree model is a discrete statistical model defined as the set of probability distributions in the image of a certain monomial parameterization. In this poster we give an explicit description of the toric ideal associated to any toric staged tree model. Using the theory of toric fiber products we show that toric staged trees can be constructed inductively by gluing smaller components. This allows us to explicitly construct a Gröbner basis whose elements can be obtained combinatorially from the tree graph representation of the model. Hermitian Determinantal Surfaces and Three-Dimensional Spectrahedra TU Dortmund University, Germany Spectrahedra in real three-space are convex bodies whose algebraic boundary is described by a (projective) surface. Unlike in the plane, the geometric properties of the boundary are not fully understood in general. Smooth determinantal surfaces over the complex numbers are very special and have been classified by Catanese and Beauville. Here, we are interested in the case of real hyperbolic surfaces admitting a hermitian determinantal representation, with applications to spectrahedra. The results are part of a dissertation in progress. Hyperplane Sections on Real Algebraic Curves TU Dortmund, Germany Given a real algebraic curve, we bound the smallest integer N such that any divisor of degree N is linearly equivalent to a totally real effective divisor. In special cases, this problem can be more vividly interpreted as finding a hyperplane intersecting the given curve in real points only. In the presented poster, we show the main ideas for extending the known bounds for curves with many connected components. Further, we illustrate a connection between the formulated problem and a conjecture about unramified curves in odd dimensional projective spaces. The results are part of a dissertation in progress. Initial degenerations of Grassmannians University of Wisconsin - Madison, United States of America Let Gr_0(d,n) denote the open subvariety of the Grassmannian Gr(d,n) consisting of d-1 dimensional subspaces of P^(n-1) meeting the toric boundary transversely. We prove that Gr_0(3,7) is schoen in the sense that all of its initial degenerations are smooth. We use this to show that the Chow quotient of Gr(3,7) by the maximal torus in GL(7) is the log canonical compactification of the moduli space of 7 lines in P^2 in linear general position. This provides a positive answer to a conjecture of Hacking, Keel, and Tevelev from "Geometry of Chow quotients of Grassmannians." Maximum Likelihood Estimation for Linear Gaussian Covariance Models with One Sample Point North Carolina State University, United States of America We will discuss the problem of maximum likelihood estimation using a single sample point for the multivariate Gaussian model in which the covariance matrix is assumed to be a linear combination of two generic symmetric matrices. Solving the score equations in this setting leads to interesting applications of the theory of algebraic curves. In particular, we will focus on counting the number of solutions to the score equations as it depends on the size of the covariance matrix. Multistationarity in Deficiency-one Power-law Kinetic Systems with Reactant-determined Interactions De La Salle University Manila, Philippines Multistationarity or the existence of multiple equilibria in a chemical reaction network (CRN) is responsible for the switch-like behavior in the system. Generally, discerning whether a CRN has multistationarity is a difficult task as this entails finding multiple positive solutions to a system of nonlinear differential equations that frequently contains unknown parameters. For deficiency-one mass action networks satisfying certain structural properties, multistationarity can be checked using Feinberg’s Deficiency One Algorithm (DOA). This procedure translates the nonlinear problem of determining the multistationarity of a CRN into a problem that takes the form of linear inequality systems. In this contribution, we extend Feinberg’s DOA to embrace deficiency-one CRNs endowed with rate laws more general than mass action kinetics -- i.e., power-law kinetic systems with reactant-determined interactions (denoted by “PL-RDK”). These are kinetic systems with power-law rate functions whose kinetic order vectors are identical for reactions with the same reactant complex. As illustration, we implement the algorithm to a power-law approximation of a model pre-industrial carbon cycle. The application reveals multistationarity in the pre-industrial state of global carbon cycle. On new families of stable subgroups of affine Cremona groups, their tame homomorphisms and Non-commutative Cryptography. University of Maria Curie Sklodowska, Poland Non-commutative cryptography studies cryptographic primitives and systems which are based on algebraic structures like groups, semigroups and noncommutative rings. It is intensively developing area due due to efforts of G. Maze, G. Monico, J, Rosenthal, P.H. Kropholler, V. Shpilrain, A. Myasnikov, A. Ushakov, D. Kahrobaei, B. Khan, S.Blackburn, S. Galbraith and others. In the case of group the most popular way of group presentation is usage of generation and relations. Classical objects like braid groups, Tompson groups and Grigorchuk groups have been been identified as potential candidates for cryptographic post quantum applications. One of recent directions is the intersection of Non-commutative and Multivariate Cryptographies formed by studies of subgroups (subsemigroups) affine Cremona group (semigroup and semigroup given by their generators in a standard forms of multivariate maps. Some general schemes of key exchange protocols and El Gamal type cryptosystems were recently defined jn terms of stable subgroups and subsemigroups and their homomorphisms. The talk is devoted to new explicit constructions of sequences of stable subsemigroups of affine Cremona semigroup formed by transformation of degree bounded by small constant c (c=2,3) together with their special homomorphisms. Their usage as platforms for cryptographical algorithms will be presented. Parameter identifiability for ODE models via an input-output representation New York University, United States of America Systems of parametric ordinary differential equations are often used in modeling. One of the challenges in designing models using such systems is structural identifiability, which can be described as follows. The values of some select parameters might be of special interest due to their importance. Usually one tries to determine their numerical values (identify them) by collecting input and output data. However, due to the structure of the model, it can be impossible to determine the parameters from input and output data. When this happens, the parameters are said to be ``not identifiable''. In the process of model design, it is often crucial to know whether the parameters of interest in a potential model are identifiable. There have been intensive efforts on this challenge since the 1970s made within different communities (e.g. math biology, control theory, symbolic computation). One significant contribution is an approach via an input-output representation of the system proposed in the late 80s. Since then, several algorithms for assessing identifiability based on this approach were designed. DAISY and COMBOS, two modern software packages, are based on such algorithms. Recent examples by Hong, Ovchinnikov, Pogudin, and Yap show that the underlying assumption of the approach via an input-output representation (called solvability) might be violated even in small linear system. This observation poses two challenges:
In the talk, we will describe our new results in these two directions and discuss remaining open problems. This is joint work with Alexey Ovchinnikov and Peter Thompson. Probabilistic analysis on Macaulay matrices over finite fields and complexity of constructing Gröbner bases University of Bergen, Norway Problems in cryptanalysis may be reduced to solving a system of multivariate polynomial equations over a finite field $F_q$. Such systems are sometimes overdetermined. E.g., that holds for AES (Advanced Encryption Standard). Gröbner basis methods may be employed to solve the equations, but their complexity is poorly understood. A key parameter is the degree of regularity $d_{reg}$ for the leading forms of the polynomials. Let $I$ be an ideal in $R=F_q[x_1,…,x_n]/(x_i^q)$ generated by the leading forms of the polynomials. Let $I_d$ be the vector space over $F_q$ generated by the forms in $I$ of degree $d$. The degree of regularity $d_{reg}$ of $I$ is the smallest integer $d$ for which $I_d$ is equal to $R_d$. We prove that time-complexity of constructing a Groebner basis and therefore solving the system is polynomial in the number of monomials of degree ≤d_{reg}. Besides an upper bound on $d_{reg}$ for a sufficiently overdetermined system of polynomials with coefficients in $F_q$ is proved. Their leading forms are of the same degree $D$ are taken uniformly at random. We do not impose any other restrictions. The bound holds with probability tending to $1$ and depends only on $n$, $m$, $D$. Therefore almost all equation systems are solvable in polynomial time if $m$ is large enough compared to $n$. E.g., $m approx n^2/6$ random quadratic equations over $F_2$ have $d_{reg}=3$ with probability close to $1$ and may be solved in time $O(n^{14})$. Our result complies with the one by Bardet, Faugère, and Salvy of 2004. They computed $d_{reg}$ for a class of systems over $F_2$ called semiregular and conjectured that a random system is semiregular with probability tending to 1. A consequence of our result is that a random system over $F_2$ has the same degree of regularity as a semiregular one. Joint work with Igor Semaev. The colorful interior of families of convex bodies and its tropical analogue INRIA École Polytechnique, France Given convex bodies (C1,...,Cn) of Rn that we think as distinct "colors" classes, we say that a vector y in Rn is "rainbow" if every decomposition of y as non-negative linear combination of vectors of C1,...,Cn uses at least one vector in each color class. The "colorful interior" of C1,...,Cn is the set of rainbow vectors. The Configuration Space and Kinematics of the Canfield Joint University of California, Santa Barbara and NASA Glenn Research Center The Canfield joint is a novel 3 degrees-of-freedom parallel robotic manipulator is being explored by NASA's Integrated Radio and Optical Communications (iROC) project for deep space communication. As a robotic linkage, we describe its configuration space as a real algebraic set with a natural projection map to the 3-torus. Additionally, we describe a birationally equivalent space that is easier to work with which factors the projection map. Since global sections (right inverses) of this map correspond to the forward kinematics of the Canfield joint, we investigate when the fibers of this projection map can be positive-dimensional. These special points on the 3-torus would correspond to control settings in which the Canfield joint forward kinematics is undefined. These techniques and methodology can be extended naturally to similar non-traditional robotics platforms whose configuration spaces and kinematics are not yet determined. Separately, we impose a symmetry condition motivated by the practical usage of the Canfield joint and provide algorithms for the solutions of various inverse kinematic problems. In particular, we show that when considering all possible symmetric Canfield joints, the locus of distal centers that would successfully point to an object generically form a nodal cubic curve and otherwise form a sphere union a line. Limits of Voronoi Decompositions University of California, Berkeley, United States of America Voronoi diagrams of finite point sets partition space into regions. Each region contains all points which are nearest to one point in the finite point set. Voronoi diagrams (and their generalizations and variations) have been an object of interest for hundreds of years by mathematicians spanning many fields, and they have numerous applications across the sciences. Recently, Cifuentes, Ranestad, Sturmfels, and Weinstein defined Voronoi cells of varieties, in which the finite point set is replaced by a real algebraic variety. Each point y on the variety has a cell of points in the ambient space corresponding to those points which are closer to y than any other point on the variety. In this poster, we present the limiting behavior of Voronoi diagrams of finite sets, where the finite sets are sampled from the variety and the sample size increases. In this setting, we observe that many interesting features of the variety emerge. This is joint work with Maddie Weinstein. Multistationarity in the space of total concentrations for systems that admit a monomial parametrization OvGU Magdeburg, Germany We apply tools from real algebraic geometry to the problem of multistationarity of chemical reaction networks. For systems whose steady states admit a monomial parameterization we show that in the space of total concentrations multistationarity is scale invariant. Moreover, for these networks it is possible to decide about multistationarity independent of the rate constants by formulating semialgebraic conditions that involve only total concentrations. Hence quantifier elimination may give new insights into multistationarity regions in the space of total concentrations. To demonstrate this, we show that for the distributive phosphorylation of a protein at two binding sites multistationarity is only possible if the total concentration of the substrate is larger than either the concentration of the kinase or the phosphatase. This result is enabled by the chamber decomposition from polyhedral geometry. This is joint work with Carsten Conradi and Thomas Kahle (arXiv:1810.08152). Rhomboid Designs for Linear Regression with Correlated Random Coefficients OVGU Magdeburg, Germany We study a linear regression model Yi(xi) = f(xi)Tbi on the hypercube with a linear intercept where bi ~ N(β,D) and all Yi(xi) are independent, which means that there is only one observation per realisation of bi. The parameter to be estimated is β, while D is fixed. We assume that the structure of D displays an independent linear intercept and a completely symmetric covariance matrix for the random coefficients. Through a model transformation and the introduction of rhomboid designs, we see that the Kiefer-Wolfowitz equivalence theorem implies when the optimality regions of these designs are either algebraic varieties intersected with trivial constraints or semi-algebraic sets. In fact, it shows that this distinction depends on the choice of the design points. Consequently we then discuss up to dimension 4, for which covariance matrices an optimal rhomboid design is supported either completely on the vertices of the hypercube or has support points in the interior. Furthermore, we conjecture a similar result for arbitrary dimension. This is joint work with Ulrike Graßhoff, Heinz Holling and Rainer Schwabe. Selecting Minimum Explaining Variables by Pruned Primary Ideal Decomposition with Recursive Calls Kwansei Gakuin University, Japan Jarrah et al. (2007) proposed an algorithm using the primary decomposition of monomial ideals for selecting minimum wiring diagrams for biological gene networks (or input-output relationships with polynomial functions in general) from finite observations of a set of variables. However, its computational cost with computer algebra system is relatively high, preventing the practical applications to the big data. Here we implemented the algorithm in the form of recursive calls in Matlab and approximated it by pruning the search trees to consider only the cases with the minimum number of explaining variables. This speed-up enabled us to treat larger data of about 100x100 size. Species Subsets and Embedded Networks of S-systems De La Salle University, Philippines Magombedze and Mulder (2013) studied the gene regulatory system of Mycobacterium Tuberculosis (Mtb) by partitioning this into three subsystems based on putative gene function and role in dormancy/latency development. Each subsystem, in the form of S-system, is represented by an embedded chemical reaction network (CRN), defined by a species subset and a reaction subset induced by the set of digraph vertices of the subsystem. Based on the network decomposition theory initiated by Feinberg in 1987, we have introduced the concept of incidence-independent and developed the theory of C- and C*-decompositions including their structure theorems in terms of linkage classes. With the S-system CRN N of Magombedze and Mulder's Mtb model, its reaction set partition induced decomposition of subnetworks that are not CRNs of S-system but constitute independent decomposition of N. We have also constructed a new S-system CRN N for which the embedded networks are C*-decomposition. We have shown that subnetworks of N and the embedded networks (subnetworks of N*) are digraph homomorphisms. Lastly, we attempted to explore modularity in the context of CRN. TensorFox Universidade Federal do Rio de Janeiro, Brazil Computing the canonical polyadic decomposition of a tensor is currently a challenging problem. Several approaches have been suggested. Nonlinear least square (NLS) algorithms - more precisely, damped Gauss-Newton (dGN) algorithms - are known to have good convergence properties. Unfortunately they require inverting large Hessian matrices, and for this reason there are just a few implementations of dGN methods. We propose a faster implementation with lower computational and memory costs. Our software was compared against other state of the art implementations. Topological analysis of neural spike data BCAM, Spain Topological methods have drawn increasing interest in data science, since they allow to extract information from the data and to summarize it in the form of topological invariants that complement the features detected via other techniques. In particular, methods based on the computation of the homology of (a filtration of) suitable objects associated with the data have proved themselves versatile, powerful and computationally treatable. In the field of neuroscience, algebraic topology has been successfully employed for example to detect co-activation patterns of neurons, to investigate functional and structural brain networks and to understand the neural code. In this poster, we study how algebraic topological techniques can be applied to extract information from spike train data. We consider various possibilities to endow a collection of spike trains with a metric, which can be used to construct a filtration of simplicial complexes in standard ways. Then we compute the invariants provided by algebraic topology and compare them to highlight which type of information they are able to detect. We focus our study on the method of persistent homology and on the computation of Betti curves, showing that statistical properties of the invariants provided by these methods have a strong discriminative power, which can be particularly useful in neural data analysis. Among the number of possible applications in neural spike analysis suggested by our results, we devote particular attention to visual data. Torus quotient of Richardson varieties in Orthogonal and Symplectic Grassmannians INDIAN INSTITUTE OF TECHNOLOGY, KANPUR, INDIA, India For any simple, simply connected algebraic group $G$ of type $B,C$ and $D$ and for any maximal parabolic subgroup $P$ of $G$, we provide a criterion for a Richardson variety in $G/P$ to admit semistable points for the action of a maximal torus $T$ with respect to an ample line bundle on $G/P$. Unboundedness of Markov complexity of monomial curves in A^n for n≥ 4 University of Glasgow, United Kingdom Computing the complexity of Markov bases is an extremely challenging problem; no formula is known in general and there are very few classes of toric ideals for which the Markov complexity has been computed. A monomial curve C in A^3 has Markov complexity m(C) two or three. Two if the monomial curve is complete intersection and three otherwise. Our main result shows that there is no d in N such that m(C) ≤ d for all monomial curves C in A^4. The same result is true even if we restrict to complete intersections. We extend this result to all monomial curves in A^n, n ≥ 4. |
