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: Friday, 12/Jul/2019 | |
| 8:25am - 8:30am | Announcements |
| vonRoll, Fabrikstr. 6, 001 | |
| 8:30am - 9:30am | IP07: Kristin Lauter: Supersingular Isogeny Graphs in Cryptography |
| vonRoll, Fabrikstr. 6, 001 | |
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8:30am - 9:30am
Supersingular Isogeny Graphs in Cryptography Microsoft Research, United States of America As we move towards a world where quantum computers can be built at scale, we are forced to consider the question of what hard problems in mathematics our next generation of cryptographic systems will be based on. Supersingular Isogeny Graphs were proposed for use in cryptography in 2006 by Charles, Goren, and Lauter. Supersingular Isogeny Graphs are examples of Ramanujan graphs, which are optimal expander graphs. These graphs have the property that relatively short walks on the graph approximate the uniform distribution, and for this reason, walks on expander graphs are often used as a good source of randomness in computer science. But the reason these graphs are important for cryptography is that finding paths in these graphs, i.e. routing, is hard: there are no known subexponential algorithms to solve this problem, either classically or on a quantum computer. For this reason, cryptosystems based on the hardness of problems on Supersingular Isogeny Graphs are currently under consideration for standardization in the NIST Post-Quantum Cryptography (PQC) Competition, and have advanced to the second round of the competition. This talk will introduce these graphs, the cryptographic applications, and the various algorithmic approaches which have been tried to attack these systems. |
| 8:30am - 9:30am | IP07-streamed from 001: Kristin Lauter: Supersingular Isogeny Graphs in Cryptography |
| vonRoll, Fabrikstr. 6, 004 | |
| 9:30am - 10:00am | Coffee break |
| Unitobler, F wing, floors 0 and -1 | |
| 10:00am - 12:00pm | MS137, part 2: Symbolic Combinatorics |
| Unitobler, F005 | |
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10:00am - 12:00pm
Symbolic Combinatorics In recent years algorithms and software have been developed that allow researchers to discover and verify combinatorial identities as well as understand analytic and algebraic properties of generating functions. The interaction of combinatorics and symbolic computation has had a beneficial impact on both fields. This minisymposium will feature 12 speakers describing recent research combining these areas. (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) Mahlerian analogues of Riccati equations and proofs of hypertranscendence A Mahler function~$f(x)$ is a function whose evaluations at iterated $b$th powers of the variable, $f(x)$, $f(x^b)$, $f(x^{b^2})$, $f(x^{b^3})$, etc, are linearly dependent. The interest in them was recently renewed, when a difference-Galois-theoretic approach was developed to determine whether a Mahler function is hypertranscendental, that is, whether $f(x)$~and all its derivatives are algebraically independent. For Mahler functions defined by a linear equation of order~2, a criterion was given by Dreyfus, Hardouin, and Roques: $f(x)$~is hypertranscendental if and only if two auxiliary non-linear equations have no rational solutions. These equations are Mahlerian analogues of Riccati equations, as they encode the right-hand factors of corresponding linear Mahler equations. I will present preliminary results of a joint work with Dreyfus, Dumas, and Mezzarobba in which we develop an algorithm to solve Riccati equations for their rational solutions. Walk in the quarter plain and differential Galois theory The determination of the nature of the generating series of walks in the quarter plain is a very vivid topic. Recently, differential Galois theory gave tools to understand what are the algebraic and differential equations satisfied by the latter. In practice, we are reduced to determine whether a telescoper relation exists or not. The latter problem may be treated with computer algebra in many situation. Systems of equations for sets of permutations and limit shapes Enumerating and analyzing sets of permutations avoiding some patterns (permutation classes) is a standard question in enumerative combinatorics. One method is to use the so-called substitution operation to decompose recursively the objects: in particular, when the permutation class contains finitely many indecomposable permutations (called simple permutations), one can obtain in an automatic way a system of combinatorial equations describing the class. This system allows to sample large uniform permutation in the class and to try and describe their limit shape. We will explain how this limit shape is related to combinatorial and analytic properties of the system. Joint work with Frédérique Bassino (Paris-Nord), Mathilde Bouvel (Zurich), Lucas Gerin (École Polytechnique), Mickaël Maazoun (ENS Lyon) and Adeline Pierrot (Orsay). The location of variables in lambda-terms with bounded De Bruijn levels We consider lambda-terms with a bounded number of De Bruijn levels, say bounded by $k$, and are interested in the shape of such a term, if it is chosen uniformly at random from all such terms of a given size. The number of such terms of given is known asymptotically, and moreover that the asymptotic expression behave in a very strange way: The subexponential term in the asymptotics is different if $k$ belongs to a certain doubly exponentially growing sequence. It is conjectured that the reason lies in the distribution of the variables within a term. Under some technical condition we first show that the number of variables is asymptotically normally distributed with mean and variance asymptotically proportional to the size. Then, we investigate the number of variables in the different De Bruijn levels and thereby exhibit a so-called ``unary profile'' of a random lambda term. It turns out that almost all the variables are located in the last De Bruijn levels and the number of these levels is proportional to $loglog k$. This is joint work with Isabella Larcher. |
| 10:00am - 12:00pm | MS146, part 2: Random geometry and topology |
| Unitobler, F006 | |
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10:00am - 12:00pm
Random geometry and topology This minisymposium is meant to report on the recent activity in the field of random geometry and topology. The idea behind the field is summarized as follows: take a geometric or topological quantity associated to a set of instances, endow the space of instances with a probability distribution and compute the expected value, the variance or deviation inequalities of the quantity. The most prominent example of this is probably Kostlan, Shub and Smales celebrated result on the expected number of real zeros of a real polynomial. Random geometry and topology offers a fresh view on classical mathematical problems. At the same time, since randomness is inherent to models of the physical, biological, and social world, the field comes with a direct link to applications. More infos at: https://personal-homepages.mis.mpg.de/breiding/siam_ag_2019_RAG.html (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) Grassmann Integral Geometry In this talk, I will give an overview on the probabilistic intersection theory on the real Grassmanniann. In particular I will focus on Schubert cells. We will show how Young Tableaux can be used to describe not only the cell itself but also its tangent and normal at one point. This description allows us to use the integral geometry formula to compute the average volume of intersection of Schubert cells. Moreover we will see how one can associate to any Schubert cell - and more generally to any submanifold of a homogeneous space - a particular convex body (zonoid) on the cotangent bundle together with a law of multiplication that form a probabilistic graded ring. I will present all the numerous questions that arise with this new point of view on integral geometry in homogeneous spaces. Topology of Gaussian Random Fields We present the space of smooth Gaussian Random Fields on a smooth manifold and discuss the notion of narrow convergence (or convergence in law) for sequences of such fields, which provides a good language to investigate a class of common problems in stochastic geometry. We will describe how narrow convergence is equivalent to smooth convergence of the covariance functions and see an application of this result to Kostlan polynomials. (This is a joint work with Antonio Lerario) Spectrum of the Laplace Operator for Random Geometric Graphs We investigate some properties of the spectrum of the normalized Laplace operator for random geometric graphs in the thermodynamic regime. This is a joint work with Antonio Lerario. Sampling from the uniform distribution on a variety This talk presents the problem of sampling from the uniform distribution on a real affine variety with finite volume, given just its defining polynomials. We can choose a point on such a variety by choosing first a hyperplane of the right codimension and then one of its intersection points with the variety. In this talk, I explain how to do this such that the chosen point is uniformly distributed. I show examples of the corresponding algorithm for sampling in action and highlight a connection to topological data analysis. This is joint work with Paul Breiding. |
| 10:00am - 12:00pm | MS181, part 2: Integral and algebraic geometric methods in the study of Gaussian random fields |
| Unitobler, F007 | |
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10:00am - 12:00pm
Integral and algebraic geometric methods in the study of Gaussian random fields Integral and algebraic geometry are at the heart of a number of contributions pertaining to the study of Gaussian random fields and related topics, not only from probabilistic and statistical viewpoints but also from the realm of interpolation and function approximation. This minisymposium will gather a team of junior researchers and established experts presenting original research results reflecting diverse challenges of geometrical and applied geometrical nature primarily involving Gaussian fields. These encompass the study of geometrical and topological properties of sets implicitly defined by random fields such as zeros of random polynomials, excursion sets, as well as integral curves stemming for instance from filament estimation. Also, Gaussian field approximations dedicated to the estimation of excursion probabilities and more general geometric questions will be tackled, as well as algebraic methods in sparse grids for polynomial and Gaussian process interpolation. (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 some Karhunen-Loève expansions related to two-point homogeneous spaces Karhunen-Loève expansions provide a powerful tool in the study of Gaussian processes. However their statistical applications are not straightforward, because whereas the existence of such developments is assured by some general theorems, explicit computations remain cumbersome. These developments involve families of orthogonal functions and the theory of classical orthogonal polynomials provides such families. Some of them are connected to families of Riemannian manifolds enjoying remarkable symmetries. By using this interplay between geometry and special functions we derive families of explicit Karhunen-Loève expansions and some of the statistical tests that can be based upon them. Geometry-driven finite-rank approximations of Gaussian random fields We investigate new approaches to uncertainty quantification on target regions that are implicitly defined by a Gaussian random field, such as level and excursion sets of the field itself or derivatives thereof. The key idea is to appeal to finite rank approximations of the field with respect to linear functionals tailored so as to best capture geometric features of interest, contrasting with the L^2 optimality property of the celebrated Karhunen-Loève expansion. The inclusion of linear forms provides a natural link to Bayesian linear inverse problems, which we will exemplify through geophysical applications. Algebraic methods in sparse grids for interpolation Sparse grids are specially construction for designs or sets of quadrature points used for polynomial interpolation and quadrature in solving differential equations with stochastic inputs. The grids are unions of tensor grids which in the so called nested case can be derived by permutation of levels from a reference grid which has a special hierarchical structure. It is shown how this structure gives rise to a monomial ideal and that the inclusion exclusion (IE) used to unravel the grid can be derived from the Hilbert series of the ideal and the coefficients use in the IE are the Betti number based on the minimal free resolution of the ideal. Remarkably, this IE structure carries over to the polynomial interpolators, not only for the reference grid but also the sparse grid itself. The considerable reduction in complexity achieved by using the algebraic method leads to the sparsity of the matrices used in the interpolation allowing the methods to be used in very high dimensions. The construction carries over to Gaussian Process interpolation when the covariance is of product type. Finally, the spacing of the grid need not be uniform but can be chosen to achieve optimal approximation. |
| 10:00am - 12:00pm | MS126, part 2: Euclidean distance geometry and its applications |
| Unitobler, F011 | |
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10:00am - 12:00pm
Euclidean distance geometry and its applications Given a natural number d and a weighted graph G=(V,E), the fundamental problem in Euclidean distance geometry is to determine whether there exists a realization of the graph G in Rd such that distances between pairs of points are equal to the corresponding edge weights. This problem naturally arises in many applications that require recovering locations of objects from the distances between these objects. Usually, measurements of the distances are noisy and there can be missing data. Examples of applications are sensor network localization, molecular conformation, genome reconstruction, robotics and data visualization. Algebraic varieties and semialgebraic sets naturally come up in Euclidean distance geometry, since distances between objects are given by polynomials. Hence questions about uniqueness and finiteness of realizations are often algebraic in nature, whereas realizations are found using semidefinite or nonconvex optimization methods. The goal of this minisymposium is to present theory and applications of Euclidean distance geometry, and connect researchers working in Euclidean distance geometry with applied algebraic geometers. (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) Rigidity theory and algebraic matroids Consider a framework consisting of fixed length bars attached at flexible joints. The central question in rigidity theory is to determine if the resulting framework is rigid or flexible. The minimally locally rigid graphs in dimension d are the bases of a matroid which can be realized as a linear matroid in joint coordinates associated to the classical "rigidity matrix" or as the algebraic matroid in (squared) distance coordinates associated to the Cayley-Menger variety. In this talk we focus on what the algebraic matroid can tell us about stresses and finite motions. This is joint work with Zvi Rosen, Louis Theran, and Cynthia Vinzant. Periodic framework enhancements A (periodic) bar-and-joint framework is a geometric (periodic) graph whose vertices are mapped to points in R^d (for a fixed dimension d) and its edges to straight-line segments between them. The framework’s configuration space consists in all the placements of the same graph which retain the edge lengths (and the abstract periodicity). We enhance the (periodic) bar-and-joint framework structure to include faces of higher dimensions and study several scenarios that preserve or alter in a controlled manner the dimension of the original configuration space. Barvinok's Naive Algorithm in Distance Geometry In 1997, A. Barvinok gave a probabilistic algorithm to derive a feasible solution of a quadratically (equation) constrained problem from its semidefinite relaxation. We generalize this algorithm to handle matrix (instead of vector) variables and to two-sided inequalities, and derive a heuristic for the distance geometry problem. We showcase its computational performance on a set of instances related to protein conformation. Mathematics of 3D genome reconstruction in diploid organisms The 3D organization of the genome plays an important role for gene regulation. Chromosome conformation capture techniques allow one to measure the number of contacts between genomic loci that are nearby in the 3D space. In this talk, we study the problem of reconstructing the 3D organization of the genome from whole genome contact frequencies in diploid organisms, i.e. organisms that contain two indistinguishable copies of each genomic locus. In particular, we study the identifiability of the 3D organization of the genome and optimization methods for reconstructing it. This talk is based on joint work with Anastasiya Belyaeva, Lawrence Sun and Caroline Uhler. |
| 10:00am - 12:00pm | MS173, part 2: Numerical methods in algebraic geometry |
| Unitobler, F012 | |
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10:00am - 12:00pm
Numerical methods in algebraic geometry This minisymposium is meant to report on recent advances in using numerical methods in algebraic geometry: the foundation of algebraic geometry is the solving of systems of polynomial equations. When the equations to be considered are defined over a subfield of the complex numbers, numerical methods can be used to perform algebraic geometric computations forming the area of numerical algebraic geometry (NAG). Applications which have driven the development of this field include chemical and biological reaction networks, robotics and kinematics, algebraic statistics, and tropical geometry. The minisymposium will feature a diverse set of talks, ranging from the application of NAG to problems in either theory and practice, to discussions on how to implement new insights from numerical mathematics to improve existing methods. (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) Numerical Root Finding via Cox Rings In this talk, we consider the problem of solving a system of (sparse) Laurent polynomial equations defining finitely many nonsingular points on a compact toric variety. The Cox ring of this toric variety is a generalization of the homogeneous coordinate ring of projective space. We work with multiplication maps in graded pieces of this ring to generalize the eigenvalue, eigenvector theorem for root finding in affine space. We present a numerical linear algebra algorithm for computing the corresponding matrices, and from these matrices a set of homogeneous coordinates of the roots of the system. Several numerical experiments show the effectiveness of the resulting method, especially for solving (nearly) degenerate, high degree systems in small numbers of variables. Numerical computation of monodromy action over R The monodromy group (over the complex numbers) is a geometric invariant that encodes the structure of the solutions for a parameterized family of polynomial systems and can be computed using numerical algebraic geometry. Since a naive extension to the real numbers is very restrictive, this talk will explore a new approach over the real numbers which is computed piece-wise to obtain tiered characteristics of the real solution set. This talk will conclude with an application in kinematics to help highlight the computational method and impact on calibration. Adaptive step size control for homotopy continuation methods At the heart of homotopy continuation methods lies the numerical tracking of implicitly defined paths by a predictor-corrector scheme. For efficient path tracking the predictor step size must be chosen appropriately. We present a new adaptive step size control which changes the step size based on computational estimates of local geometric information as well as the order of the used predictor method. We also give an update on the Julia package HomotopyContinuation.jl. Numerical homotopies from Khovanskii bases Homotopies are useful numerical methods for solving systems of polynomial equations. I will present such a homotopy method using Khovanskii bases. Finite Khovanskii bases provide a flat degeneration to a toric variety, which consequentially gives a homotopy. The polyhedral homotopy, which is implemented in PHCPack, can be used to solve for points on a general linear slice of this toric variety. These points can then be traced via the Khovanskii homotopy to points on a general linear slice of the original variety. This is joint work with Michael Burr and Frank Sottile. |
| 10:00am - 12:00pm | MS141, part 1: Chip-firing and tropical curves |
| Unitobler, F013 | |
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10:00am - 12:00pm
Chip-firing and tropical curves The chip-firing game on metric graphs is a simple combinatorial model that serves as a tropical analogue of divisor theory on algebraic curves, and it has been an active and fruitful research direction over the last decade. The behaviors of chip-firing resemble, but not always completely match, the classical situation in algebraic geometry. So on one hand, chip-firing can often be used to prove results (old and new) in algebraic geometry; while on the other hand, the combinatorics of chip-firing is interesting and surprising in its own right. We will focus on three main topics: (I) Tropical analogues (or failure thereof) of classical results of algebraic curves, (II) applications of chip-firing in algebraic geometry and other subjects, and (III) complexity issues of computational problems related to chip-firing. (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 chip firing This session is intended as a preamble, giving a quick outline on chip-firing and introducing concepts likely to appear in several of the talks to follow. These include: graphs, metric graphs, divisor theory (divisors, linear equivalence, rank), Jacobian group, reduced divisors, divisorial gonality, and related concepts. We illustrate with graphs of low genus. Computing divisorial gonality is hard The (divisorial) gonality of a graph G is the smallest degree of a divisor of positive rank in the sense of Baker-Norine. In terms of the classical chip-firing game of Björner-Lovász-Shor it relates to chip configurations that result in a finite game, even after adding a chip at an arbitrary position. We show that computing the gonality of a graph is NP-hard. In fact, it cannot be approximated to within an arbitrary factor in polynomial time (unless P=NP). Recognizing hyperelliptic graphs Based on analogies between algebraic curves and graphs, a new multigraph parameter was defined. This parameter is called divisorial gonality and can be defined using a chip-firing game. In this talk we consider so-called hyperelliptic graphs, which are graphs with divisorial gonality 2. We will see that we can decide in polynomial time whether a graph is hyperelliptic or not. Graphs of gonality three In 2013, Chan classified all metric graphs of gonality two, proving that divisorial gonality and geometric gonality are equivalent in the hyperelliptic case. We show that such a classification extends to combinatorial graphs of divisorial gonality three, under certain edge-connectivity assumptions. We also give a construction for graphs of divisorial gonality three, and provide conditions for determining when a graph is not of divisorial gonality three. |
| 10:00am - 12:00pm | MS179, part 1: Algebraic methods for polynomial system solving |
| Unitobler, F021 | |
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10:00am - 12:00pm
Algebraic methods for polynomial system solving solving Polynomial system solving is at the heart of computational algebra and computational algebraic geometry. It arises in many applications ranging from computer security and coding theory (where computations must be done over finite fields) and engineering sciences such as chemistry, biology, signal theory or robotics among many others (here computations are done over inifinite domains such as complex or real numbers). The need of reliable algorithms for solving these problems is prominent because of the non-linear nature of the problems we have in hand. Algebraic methods provide a nice framework for designing efficient and reliable algorithms solving polynoial systems. This mini-symposium will cover many aspects of this topic, including design of symbolic computation algorithms as well as the use of numerical methods in this framework with an emphasis on reliability. (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) Exploiting fast linear algebra in the computation of multivariate relations We consider the problem of computing multivariate relations in a finite-dimensional setting: for a submodule M of K[x]^n such that Q = K[x]^n/M has finite dimension D as a K-vector space, and given elements f1,..,fm in Q, the problem is to compute relations between the fi's, that is, polynomials (p1,..,pm) in K[x]^m such that p1 f1 + ... + pm fm = 0 in Q. Assume that the multiplication matrices of the r variables with respect to some basis of Q are known. Then, for any monomial order, we give an algorithm for computing the reduced Gröbner basis of the module of such relations using O(r D^w log(D)) operations in the base field K, where w is the exponent of matrix multiplication. We also show that, under some assumptions, the multiplication matrices can be computed from a Gröbner basis of M within the same complexity bound, leading in particular to a change of monomial order algorithm whose complexity bound is sub-cubic in D. Contains joint work with Éric Schost and Hamid Rahkooy Certification via squaring-up We consider numerical certification of approximate solutions to N polynomial equations in n variables in the case where n < N via passing to a square subsystem. Typically the excess complex solutions of a general squaring-up are all isolated, and their number is given in terms of a birationally-invariant intersection index. This enables certification, via Smale and Shub's alpha-theory, for examples where the intersection index is known a priori, or in cases where it may be calculated algorithmically in terms of an associated Newton-Okounkov body. joint w/ Frank Sottile (Texas A&M) Efficient and complete certification of roots in solving polynomial systems A certified algorithm is an algorithm that provides a proof or certificate of the correctness of its output. A complete algorithm is one that can be correctly implemented on a computer using a bit-based computation mode l. In this talk, I will present recent work on certification methods for solving polynomial systems. I will focus on two paradigms for certifying solutions to polynomial systems: A posteriori approaches take the output of (any) computation and check the correctness of the result. A priori approaches prove that an entire computation is correct, including the output. In particular, I will discuss how interval arithmetic-based methods can be used to make certification algorithms efficient in practice. Reconstruction of an Algebraic Surface from a 2D Projection An algebraic surface is given by its equation, the zero set of a polynomial in four homogeneous variables. Its picture under central projection is computed as the zero set of its discriminant: a plane algebraic curve. Can we recover the equation of an algebraic surface by its discriminant? If the surface is nonsingular, then the answer is yes, by a result of d'Almeida. If we allow also "generic" singularities, then the there is sometimes a finite list of possibles. This talk explains the resconstruction method and discusses the ambiguities in the singular case. |
| 10:00am - 12:00pm | MS130, part 3: Polynomial optimization and its applications |
| Unitobler, F022 | |
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10:00am - 12:00pm
Polynomial optimization and its applications The importance of polynomial (aka semi-algebraic) optimization is highlighted by the large number of its interactions with different research domains of mathematical sciences. These include, but are not limited to, automatic control, combinatorics, and quantum information. The mini-symposium will focus on the development of methods and algorithms dedicated to the general polynomial optimization problem. Both the theoretical and more applicative viewpoints will be covered. (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) Limitations on the expressive power of convex cones without long chains of faces Recently Averkov showed that various convex cones related to nonnegative polynomials do not have K-lifts (representations as projections of linear sections of another convex cone K) where K is a Cartesian product of positive semidefinite cones of "small" size. In this talk I'll explain how to extend Averkov's approach to show that cones with certain neighborliness properties do not have K-lifts whenever K is a Cartesian product of cones, each of which does not have any long chains of faces (such as smooth cones, low-dimensional cones, and cones defined by hyperbolic polynomials of low degree). On the exactness of Lasserre relaxations and pure states over real closed fields Consider a finite system of non-strict real polynomial inequalities and suppose its solution set $SsubseteqR^n$ is convex, has nonempty interior and is compact. Suppose that the system satisfies the Archimedean condition, which is slightly stronger than the compactness of $S$. Suppose that each defining polynomial satisfies a second order strict quasiconcavity condition where it vanishes on $S$ (which is very natural because of the convexity of $S$) or its Hessian has a certain matrix sums of squares certificate for negative-semidefiniteness on $S$ (fulfilled trivially by linear polynomials). Then we show that the system possesses an exact Lasserre relaxation. High-dimensional estimation via sum-of-squares proofs Estimation is the computational task of approximately recovering a hidden parameter x associated with a distribution D_x given a draw y from the distribution D_x. Numerous interesting questions in statistics, machine learning, and signal processing are captured in this way, for example, sparse linear regression, Gaussian mixture models, topic models, and stochastic block models. In many cases, there is currently a large gap between the statistical guarantees of computationally efficient algorithms and the guarantees of computationally inefficient methods; it is an open question if this gap is inherent in these cases or if better computationally efficient estimation algorithms exist. In this talk, I will present a meta-algorithm for estimation problems based on the sum-of-squares method of Shor, Parrilo, and Lasserre. For some problems, e.g., learning mixtures of spherical Gaussians, this meta-algorithm is able to close previous long-standing gaps and achieve nearly optimal statistical guarantees. Furthermore, it is plausible that, for a wide range of estimation problems, the statistical guarantees that this meta-algorithm achieves are best possible among all efficient algorithms. This talk is based on an ICM proceedings article with Prasad Raghavendra and Tselil Schramm. Exact Optimization via Sums of Nonnegative Circuits and Sums of AM/GM ExponentialsLog-concave polynomials, entropy, and approximate counting We provide two hybrid numeric-symbolic optimization algorithms, computing exact sums of nonnegative circuits (SONC) and sums of arithmetic-geometric-exponentials (SAGE) decompositions. Moreover, we provide a hybrid numeric-symbolic decision algorithm for polynomials lying in the interior of the SAGE cone. Each framework, inspired by previous contributions of Parrilo and Peyrl, is a rounding-projection procedure. For a polynomial lying in the interior of the SAGE cone, we prove that the decision algorithm terminates within a number of arithmetic operations, which is polynomial in the degree and number of terms of the input, and singly exponential in the number of variables. We also provide experimental comparisons regarding the implementation of the two optimization algorithms. |
| 10:00am - 12:00pm | MS124, part 3: The algebra and geometry of tensors 1: general tensors |
| Unitobler, F023 | |
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10:00am - 12:00pm
The algebra and geometry of tensors 1: general tensors Tensors are ubiquitous in mathematics and science. Tensor decompositions and approximations are an important tool in artificial intelligence, chemometrics, complexity theory, signal processing, statistics, and quantum information theory. These topics raise challenging computational problems, but also the theory behind them is far from fully understood. Algebraic geometry has already played an important role in the study of tensors. It has shed light on: the ill-posedness of tensor approximation problems, the generic number of decompositions of a rank-r tensor, the number and structure of tensor eigen- and singular tuples, the number and structure of the critical points of tensor approximation problems, and on the sensitivity of tensor decompositions among many others. This minisymposium focuses on recent developments on the geometry of tensors and their decompositions, their applications, and mathematical tools for studying them, and is a sister minisymposium to "The algebra and geometry of tensors 2: structured tensors" organized by E. Angelini, E. Carlini, and A. Oneto. (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) Apolarity for border rank For a fixed polynomial or tensor F, the standard apolarity lemma provides a correspondence between the set of points realising the Waring rank or tensor rank of F and the apolar ideal of F. Thus it is very useful for calculating the rank of the polynomial or tensor. I will present a non-saturated version of apolarity, which instead is useful for calculating the border rank of T. The general theory is going to be illustrated by several classes of examples. Based on a joint work with Weronika Buczyńska. Symmetric tensor decompositions on varieties Interesting tensors always have rich geometric structure, which can be helpful when we decompose these tensors. For instance, the signal separation problem in signal process corresponds to the so-called Vandermonde decomposition of a tensor. In this talk, we will introduce the problem of decomposing a tensor on a given algebraic variety. We will first discuss some basic properties and then we will present an algorithm to decompose a symmetric tensor such that each summand lies on a given algebraic variety. In particular, we will see how our proposed algorithm can be applied to study the Vandermonde decomposition of a tensor. If time permits, we will exhibit some numerical examples. This talk is based on a joint work with Jiawang Nie and Lihong Zhi. Tensors under the congruence action Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry and numerical optimization to this group action. Given a tensor in the orbit of another tensor, we compute a matrix which transforms one to the other. Our primary application is an inverse problem from stochastic analysis: the recovery of paths from their third order signature tensors. This talk is based on joint work with Max Pfeffer and Bernd Sturmfels. Rank additivity for small three-way tensors I will present some result from the joint work with J. Buczyński and E. Postinghel, where we investigate the problem of the additivity of the tensor rank. That is for two independent tensors we study if the rank of their direct sum is equal to the sum of their individual ranks. A positive answer to this problem was previously known as Strassen's conjecture until recent counterexamples were proposed by Shitov (2017). The latter are not very explicit, and they are only known to exist asymptotically for very large tensor spaces. We proved that for some small three-way tensors the additivity holds. For instance, if the rank of one of the tensors is at most 6. In addition we also treat some cases of the additivity of border rank of such tensors. |
| 10:00am - 12:00pm | MS174, part 2: Algebraic aspects of biochemical reaction networks |
| Unitobler, F-105 | |
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10:00am - 12:00pm
Algebraic aspects of biochemical reaction networks ODE models for biochemical reaction networks usually give rise to dynamical systems defined by polynomial or rational functions. These systems are often high-dimensional, very sparse, and involve many parameters. This minisymposium deals with recent progress on applying and adapting techniques from (real) algebraic geometry and computational algebra for analyzing such systems. The minisymposium consists of three parts focusing on positive steady states, multistationarity and the corresponding parameter regions, and dynamical aspects. (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) Expected number of positive real solutions to systems of polynomial equations arising from reaction networks Given a chemical reaction network that exhibits multistationarity, a natural question is how to determine the region in the parameter space where the network is multistationary or to divide the parameter region according to the number of positive steady states. We introduce a new approach based on the Kac-Rice formula and Monte-Carlo integration to approximate the multistationarity region in the parameter space. Furthermore, we apply our method to solve two related questions. First, we provide a measure to compare two points in the multistationarity region and decide what choice is more robust to small variations of the parameters. Second, we address the problem of finding a point in a given box in the parameter space for which the network is multistationary. We apply our approach to two relevant reaction networks, namely a simple model of a hybrid histidine-kinase and the 2-site sequential distributive phosphorylation network. Finally we compare our approach with the former existing methods of studying the multistationarity region. This is a joint work with Elisenda Feliu. Absolute concentration robustness: an algebraic perspective How do cells maintain homeostasis in fluctuating environments? Investigations into this question led Shinar and Feinberg to introduce in 2010 the concept of absolute concentration robustness (ACR). A biochemical system exhibits ACR in some species if the steady-state value of that species does not depend on initial conditions. Thus, a system with ACR can maintain a constant level of one species even as the environment changes. Despite a great deal of interest in ACR in recent years, the following basic question remains open: How can we determine quickly whether a given biochemical system has ACR? Although various approaches to this problem have been proposed, we show in this talk that they are incomplete. Accordingly, we present a new method for deciding ACR, which uses computational algebra. We illustrate our results on several biochemical signaling networks. On the Stability of the Steady States in the n-site Futile Cycle The multiple or n-site futile cycle is a biological process that resides in the cell. Specifically, it is a phosphorylation system in which a molecular substrate might be phosphorylated sequentially n times by means of an enzymatic mechanism. The system has been studied mathematically using reaction network theory and ordinary differential equations. In its standard form it has 3n+3 variables (concentrations of species) and 6n parameters. It is known that the system might have at least as many as 2[n/2]+1 steady states (where [x] is the integer part of x) for particular choices of parameters. Furthermore, for the simple futile cycle (n=1) there is only one steady state which is globally stable. For the dual futile cycle (n=2) the stability of the steady states has been determined in the following sense: There exist parameter values for which the dual futile cycle admits two asymptotically stable and one unstable steady state. For general n, evidence that the possible number of asymptotically stable steady states increases with n has been given, which has led to the conjecture that parameter values can be chosen such that [n/2]+1 out of 2[n/2]+1 steady states are asymptotically stable and the remaining steady states are unstable. We prove this conjecture here by first reducing the system to a smaller one, for which we find a choice of parameter values that give rise to a unique steady state with multiplicity 2[n/2]+1. Using arguments from geometric singular perturbation theory, and a detailed analysis of the centre manifold of this steady state, we achieve the desired result. The work is joint with Alan Rendall (Mainz) and Elisenda Feliu (Copenhagen). The DSR graph and dynamical properties of reaction networks A significant body of recent work in mathematical biology focuses on dynamical properties of biochemical reaction networks that are a function of the network topology alone (and are therefore independent of kinetics or parameter values). One way to represent network topology is via the DSR graph, introduced by Craciun and Banaji. The structure of the DSR graph may allow conclusions on dynamical properties of the network, including multistationarity, stability of steady states, and stability under delays. In this talk we survey some old and new results on the DSR graph, and discuss connections with other graphs stemming from reaction networks. |
| 10:00am - 12:00pm | MS164, part 2: Algebra, geometry, and combinatorics of subspace packings |
| Unitobler, F-106 | |
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10:00am - 12:00pm
Algebra, geometry, and combinatorics of subspace packings Frame theory studies special vector arrangements which arise in numerous signal processing applications. Over the last decade, the need for frame-theoretic research has grown alongside the emergence of new methods in signal processing. Modern advances in frame theory involve techniques from algebraic geometry, semidefiniteprogramming, algebraic and geometric combinatorics, and representation theory. This minisymposium will explore a multitude of these algebraic, geometric, and combinatorial developments in frame theory. The theme of the second session is "Equiangular lines." (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) Equiangular tight frames from group divisible designs An equiangular tight frame (ETF) is a type of optimal packing of lines in a real or complex Hilbert space. In the complex case, the existence of an ETF of a given size remains an open problem for many choices of parameters. We discuss how many of the known constructions of ETFs are of one of two types. We further provide a new method for combining a given ETF of one of these two types with an appropriate group divisible design (GDD) in order to produce a larger ETF of the same type. By applying this method to known families of ETFs and GDDs, we obtain several new infinite families of ETFs. Using Biangular Gabor Frames to Construct Equiangular Tight Frames Biangular Gabor Frames can be described by a system of real polynomial equations of many variables. Numerical computations suggests this system is usually one dimensional. Furthermore, a vector containing all ones leads to a biangular Gabor frame. This gives the idea of following the curve to an equiangular tight frame somewhere along this curve. We present numerical results of finding equiangular tight frames using this method. Doubly transitive lines: Symmetry implies optimality Since the work of Tóth on regular figures, it has been widely observed that optimal solutions to packing problems frequently display extraordinary symmetries. For instance, spheres centered on points in the Leech lattice give an optimal packing in 24 dimensions, while lines through through antipodal vertices of an icosahedron give an optimal packing in two-dimensional projective space. In this talk, we demonstrate an extreme case of this phenomenon for line packings: symmetry can be a sufficient condition for optimality. Specifically, consider n lines spanning a space of dimension d < n. If the lines have a doubly transitive automorphism group, then they are optimally packed in projective space. In fact, unit norm representatives for the lines reach equality in the Welch bound to create an equiangular tight frame. We will explain this phenomenon, and then discuss progress toward a classification of all doubly transitive lines. This is joint work with Dustin G. Mixon. Equiangular lines in $\mathbb R^{17}$ and the characteristic polynomial of a Seidel matrix A system of lines through the origin of $mathbb R^d$ for which the angle between any pair of lines is a constant is called equiangular. A Seidel matrix, which can be interpreted as a variation of the adjacency matrix of a graph, is a tool for studying equiangular line systems. In this talk we present our recent improvement on the upper bound for the cardinality of an equiangular line system in $mathbb R^{17}$. A crucial ingredient for this improvement is a new restriction on the characteristic polynomial of a Seidel matrix. This talk is based on joint work with Pavlo Yatsyna. |
| 10:00am - 12:00pm | MS125: Efficient algorithms for geometric invariant theory |
| Unitobler, F-107 | |
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10:00am - 12:00pm
Efficient algorithms for geometric invariant theory Recently, motivated by the polynomial identity testing problem from computer science, and by questions arising in quantum information theory, efficient numerical algorithms for solving the null cone problem from geometric invariant theory have been proposed. The goal of the minisymposium is to review this progress and to report on recent advances. (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) Algorithms for the separation of orbits of matrices We consider two group actions on the set of m-tuples of n by n matrices, namely the simultaneous conjugation action of GL(n) and the left-right action of SL(n) x SL(n). An invariant can separate two orbits of m-tuples if and only if the closures of these orbits are disjoint. In both cases we present a polynomial time algorithm that decides whether the orbit closures of two m-tuples intersect. This is joint work with Visu Makam. Analytic algorithms for the null cone problem Null cone is a fundamental object in invariant theory. It is the variety defined by all the homogeneous invariant polynomials for a particular group action. This talk will be focused on the computational complexity of testing membership in the null cone. From a purely algebraic point of view, this problem seems hard, since except in a few cases, one does not have any nice description of the invariant polynomials. However, we will see that there is a good reason to believe that analytic methods can provide provably efficient algorithms for the null cone and we will discuss several examples where this has already been achieved, the most notable being bipartite matching, linear programming, non-commutative rank etc. The analytic approach goes via the Kempf-Ness criterion and connects to the exciting area of geodesically convex optimization. Non-commutative rank of linear matrices, related structures and applications The non-commutative rank of a matrix with homogeneous linear entries is the rank when we consider the variables as elements of the appropriate free division algebra. This is the same as the maximum size of a square sub-matrix such that the tuple of coefficient matrices is not in the null-cone of the polynomial invariants for the left-right action of the suitable special linear group. There is a "dual" characterization in terms of large common rectangular zero blocks (after appropriate changes of bases).We will outline a deterministic polynomial time algorithm for computing the non-commutative rank in a two-fold constructive way. Firstly, it computes a polynomial invariant for an appropriate sub-matrix together with a non-vanishing substitution of values from a division algebra of relatively small dimension (actually, even from a matrix algebra) into the variables. Then, in the non-full rank case, common zero blocks with matching parameters are revealed. We will also discuss related common "echelon forms" for the coefficient matrices and some applications. The algorithm for finding the large zero blocks works along certain flags of subspaces that further "echelonize" the coefficient matrices. These flags are analogous to the alternating forests in algorithms for finding maximum matchings in bipartite graphs. Interestingly, their behavior explain certain special cases when the non-commutative rank coincide with or approximates the usual, "commutative" one (i.e., with the rank when the variables are considered as elements of a commutative field). Some of these special cases will be discussed as well. Analytic algorithms for the moment polytope Moment polytopes are convex bodies associated to certain group actions on manifolds. When the manifold is a projective variety invariant under the action of a reductive Lie group, it is known that the moment polytope encodes asymptotic information about the irreducible representations of the group occurring in the coordinate ring of the variety. This talk concerns the computational complexity of deciding moment polytope membership. Existing methods include enumerating the (potentially exponentially many) facets and evaluating highest weight polynomials (of potentially exponential degree), but we will discuss analytic algorithms that do not seem to face the same hurdles. These analytic algorithms are also notable for their simplicity and their ability to compute preimages under the moment map, a problem of practical interest. We will discuss how alternating minimization, one of the simplest approaches in optimization, leads to analytic algorithms for Horn's problem and the quantum marginal problem. Among the conceptual tools in the analysis of these algorithms are "reductions" to the null cone problem and lower bounds for Kempf-Ness functions via representation theory. |
| 10:00am - 12:00pm | MS169, part 1: Applications of Algebraic geometry to quantum information |
| Unitobler, F-111 | |
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10:00am - 12:00pm
Applications of Algebraic geometry to quantum information Quantum information science attempts to use quantum phenomena as non-classical resources to perform new communication protocols and develop new computational paradigms. The theoretical advantages of quantum communication and quantum algorithms were proved in the 80-90’s and nowadays experimentalists are working on making that technology available. One of the quantum phenomena responsible for the speed up of quantum algorithms and the security of quantum communication is entanglement. A system of m-particules (a multipartite quantum state) is said to be entangled when the state of a particle of the system cannot be described independently of the others. Entanglement is a consequence of the superposition principle in quantum physics which mathematically translates to the fact that the Hilbert space of a composite system is the tensor product of the Hilbert space of each part. Algebraic geometry entered the study of entanglement of multipartite systems when it was both noticed in the early 2000s that the rank of tensors could be interpreted as a measure of entanglement and also that invariant theory could be used to distinguish different classes of entanglement. Since then a large amount of research has been produced in the mathematical-physics literature to classify and/or measure entanglement using techniques from classical invariant theory, representation theory, and geometric invariant theory. Because of the exponential growth of the dimension of the multipartite Hilbert spaces, when the number of factors increases, only a few examples of explicit classifications are known. Therefore to study entanglement in larger Hilbert spaces, techniques from tensor decomposition and asymptotic geometry of tensors have been recently introduced. These techniques establish new connections between entanglement and algebraic complexity 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) Tensor rank, border rank, multiplicativity and entanglement Matrix rank has several different generalizations to the setting of tensors which are natural measures of the entanglement of the quantum state described by the tensor. Some recent results show that, unlike the classical matrix rank, these generalizations are not multiplicative under the operation of tensor product. We describe this phenomenon and some of its consequences in a general geometric framework, which allows for further generalizations. Hyperdeterminants form $E_8$ Projective duality can be used to study singularities. A matrix is singular precisely when its determinant vanishes, or equivalently, when it belongs to the projective dual to rank-one matrices, the Segre variety. A higher order tensor is singular when its hyperdeterminant vanishes, i.e. when it belongs to the dual of a higher order Segre product. Efficient expressions for hyperdeterminants are mostly unknown and they are difficult to compute. We describe a connection to the exceptional Lie algebra $E_8$. This gives an interpretation of certain hyperdeterminants (of formats $2times 2times 2times 2$ and $3times 3times 3$) and certain discriminants (of the Grassmannians $Gr(3,9)$ and $Gr(4,8)$) as sparse $E_8$-discriminants. We give expressions of these high degree invariants in terms of lower degree fundamental invariants, which allow evaluation, and may be useful for Quantum Information Theory as measures of entanglement. This is joint work with Frédéric Holweck. Tensor network representations from the geometry of entangled states Tensor network states provide successful descriptions of strongly correlated quantum systems with applications ranging from condensed matter physics to cosmology. Any family of tensor network states possesses an underlying entanglement structure given by a graph of maximally entangled states along the edges that identify the indices of the tensors to be contracted. Recently, more general tensor networks have been considered, where the maximally entangled states on edges are replaced by multipartite entangled states on plaquettes. Both the structure of the underlying graph and the dimensionality of the entangled states influence the computational cost of contracting these networks. Using the geometrical properties of entangled states, we provide a method to construct tensor network representations with smaller effective bond dimension. We illustrate our method with the resonating valence bond state on the kagome lattice. Tensor scaling, quantum marginals, and moment polytopes Given a collection of quantum states of individual particles, are they compatible with a global quantum state? I will give an introduction to the mathematics of this "quantum marginal problem" (which has applications from entanglement theory to quantum chemistry), explain its connection to geometric invariant theory, and present an efficient algorithmic solution. Our numerical algorithm applies more generally to the problems of deciding semistability and computing moment polytopes. |
| 10:00am - 12:00pm | MS128, part 1: Symbolic-numeric methods for non-linear equations: Algorithms and applications |
| Unitobler, F-112 | |
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10:00am - 12:00pm
Symbolic-numeric methods for non-linear equations: Algorithms and applications Modeling real-world systems or processes in areas such as control theory, geometric modeling, biochemistry, coding theory, cryptology, and so on, almost certainly involves non-linear equations. Higher degree equations are the first step away from linear models. Available tools for recovering their solutions range from numerical methods such as Newton-Raphson, homotopy continuation algorithms, subdivision-based solvers, to symbolic tools such as Groebner bases, border bases, characteristic sets and multivariate resultants. There is continuous progress in combining symbolic methods and numerical solving, in order to devise new algorithms with varying blends of exactness, stability and robustness as well as computational complexity, that are tailored for different applications. Among the challenges which occur in the process is reliable root isolation, certification and approximation, treatment of singular solutions, the exploitation of structure coming from specific applications as well as efficient interpolation. (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) Multilinear systems, determinantal resultants and the multiparameter eigenvalue problem Multivariate resultant matrices characterize the roots of a polynomial system and reduce their computation to an eigenvalue problem. However, determinantal formulas (i.e. without extraneous factors) for the resultant do not exist for arbitrary systems. They have been constructed mostly for unmixed systems, that is, systems of polynomial equations with a common Newton polytope of special structure. In this talk we derive determinantal formulations for the multivariate resultant of structured systems with distinct supports per equation. We focus on mixed multilinear polynomial systems, that is multilinear systems with different supports per equation. These systems have applications to the Multiparameter Eigenvalue Problem (MEP). Algorithmic aspects of the rational interpolation problem The Rational Interpolation Problem is an extension of the classical Polynomial Interpolation one, and there are several approaches to it: Euclidean algorithm, Linear Algebra with structured matrices, barycentric coordinates, orthogonal polynomials, computation of syzygies,.... We will review some of these methods along with some recent bounds in the complexity of their computation. Computing Gröbner basis for sparse polynomial systems Gröbner bases are one the most powerful tools in algorithmic non-linear algebra. Their computation is an intrinsically hard problem with complexity at least single exponential in the number of variables. However, in most of the cases, the polynomial systems coming from applications have some kind of structure. For example, several problems in computer-aided design, robotics, vision, biology, kinematics, cryptography, and optimization involve sparse systems where the input polynomials have a few non-zero terms. An approach to exploit sparsity is to embed the systems in a semigroup algebra and to compute Gröbner bases over this algebra. Prior to our work, the algorithms that follow this approach benefited from the sparsity only in the case where all the polynomials have the same sparsity structure, that is the same Newton polytope. In this talk, I will present the first algorithm that overcomes this restriction. Under regularity assumptions, it performs no redundant computations. Further, I will use it to compute Gröbner basis in the standard algebra and solve sparse polynomials systems over the torus. The complexity of the algorithm depends on the Newton polytopes and it is similar to the complexity of the solving techniques involving the sparse resultant. Hence, this algorithm closes a 25-years gap between the strategies to solve systems using resultants and Gröbner bases. Additionally, for particular families of sparse systems, I will use the multigraded Castelnuovo-Mumford regularity to improve the complexity bounds. Real solving polynomial systems with interval method Given an ideal I in a polynomial ring K[x_1, x_2, ... ,x_n], let its i-th elimination ideal be the ideal I_i= I cap K[x_1, x_2, ..., x_{n-i}]. The standard method for computing elimination ideals is by computing the Groebner basis G for I with respect to an elimination order and due to the elimination property of Groebner bases, the Groebner basis of I_i is G cap K[x_1, x_2, ..., x_{n-i}]. In this work we are interested in computing the i-th elimination ideal, without computing G, the Groebner basis of I, first. We do that by using the resultant system. The resultant system, introduced in van der Waerden's Modern Algebra, is a polynomial system having the properties we expect from a resultant. The resultant system contains a large number of polynomials, and thus it is not computationally efficient. We present an analysis of how we can improve the computation of the resultant system. Elimination ideals is a useful tool when dealing with parametric polynomial systems. Such systems often appear in applications and it was one of our original motivations, especially applications in combinatorics and motion planning. |
| 10:00am - 12:00pm | MS180, part 2: Network coding and subspace designs |
| Unitobler, F-113 | |
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10:00am - 12:00pm
Network coding and subspace designs This symposium collects presentations about results on codes for linear network coding, either in the rank metric or in the subspace metric. Codes in the rank metric are usually subsets of the matrix space F_q^{m x n}, where F_q is a finite field; codes in the subspace metric are usually subsets of a finite Grassmann variety. Many interesting questions arise in this topic, e.g., about good packings in these two spaces, as well as fast encoding and decoding algorithms. (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) Sum-Rank Codes and Linearized Reed-Solomon Codes The sum-rank metric naturally extends both the Hamming and rank metrics in Coding Theory. In this talk, we will present some of their applications and general properties. We will also present linearized Reed-Solomon codes, which constitute the first general family of maximum sum-rank distance (MSRD) linear codes whose field sizes are subexponential in the code length. These codes are tightly connected to skew Reed-Solomon codes, and are natural hybrids between generalized Reed-Solomon codes and Gabidulin codes. On some automorphisms of polynomial rings and their applications in rank metric codes Recently, there is a growing interest in the study of rank metric codes. These codes have applications in network coding and cryptography. In this talk, I investigate some automorpshisms on polynomial rings over finite fields. We will show how the linear operators from these automorphisms can be used to construct some maximum rank distance (MRD) codes. First we will work on rank metric codes over arbitrary extension and then we will reduce these to finite fields extension. Some particular constructions give MRD codes which are not equivalent to twisted Gabidulin codes. Another application is to use these linear operators to construct some optimal rank metric codes from some Ferrers diagram. In fact we will give some examples of Ferrers diagrams for which there was no known construction of optimal rank metric codes. Invariants of rank-metric codes via Galois group action Codes in the rank metric have been introduced in 1978, but only in the last ten years they significantly gained interest due to their many applications in communications and security. These codes are linear subspaces of the space of matrices over a finite field, but they can be also seen as vectors over an extension field. There are not many explicit constructions of families of rank-metric codes with good parameters, and finding new ones hase become an important ongoing research question. However, when one considers new rank-metric codes constructions, it is important to check whether the new codes are equivalent to any other known construction. For this purpose, one wants to develop some criteria to check code equivalence. In this talk we introduce a new series of invariants of rank-metric codes obtained via the action of the Galois group of the underlying field extension. In particular, we consider the subspaces generated by the code and the application of several automorphisms to itself and show that their dimensions are invariant under code equivalence. This tool provides an easy checkable criterion for determining code inequivalence. We derive lower bounds on the number of equivalence classes of Gabidulin and twisted Gabidulin codes using this new invariant. In some special cases, the exact number of such equivalence classes is provided. |
| 10:00am - 12:00pm | MS198: Positive and negative association |
| Unitobler, F-121 | |
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10:00am - 12:00pm
Positive and negative association Positive and negative association form strong and useful conditions on probability distributions that appear in several applications. Algebraic and combinatorial methods have led to methods for understanding and sampling from important classes of these distributions. This session aims to explore some of the recent breakthroughs and applications of positive and negative association. (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) Negative dependence and sampling Negative dependence occurs in various forms in probability and machine learning. A prominent example with applications in both probabilistic modeling and randomized algorithms are Determinantal Point Processes (DPPs). DPPs belong to the class of Strongly Rayleigh measures that are characterized by real stable polynomials and exhibit strong notions of negative dependence. For practical applications, it is important that procedures such as sampling can be performed efficiently; recent work suggests that negative dependence can enable exactly that. In this talk, I will summarize selected applications of Determinantal Point Processes and other Strongly Rayleigh measures, and then show results on fast mixing of Markov chains for those measures. Several of these results rely on the connections to real stable polynomials. Log-concave polynomials: Polynomials that a drunkard can (almost) evaluate A central question in algorithm design is what kind of distributions can we sample from efficiently? On the continuous side, uniform distributions over convex sets and more generally log-concave distributions constitute the main tractable class. We will build a parallel theory on the discrete side, that yields tractability for important discrete distributions such as uniform distributions over matroids, generalizations of determinantal point processes, and some regime of the random cluster model. The hammer enabling these algorithmic advances is the introduction and the study of a class of polynomials, that we call completely log-concave. Sampling from discrete distributions becomes equivalent to approximately evaluating associated multivariate polynomials, and we will see how we can use very simple random walks to perform both tasks. This is based on joint work with Kuikui Liu, Shayan Oveis Gharan, and Cynthia Vinzant. Total positivity in structured binary distributions We study estimation in totally positive binary distributions, in particular quadratic exponential families of these. Using results from convex optimization we show how the restriction of total positivity induces conditional independence restrictions on the estimated distributions. Also, we give necessary and sufficient conditions for the maximum likelihood estimate to exist within the corresponding exponential family and develop a globally convergent algorithm for its computation. This represents joint work with Caroline Uhler and Piotr Zwiernik. Geometric problems in non-parametric statistics We examine maximum likelihood estimation for probability densities that are both log-concave and totally positive. The solution to this optimization problem is intriguing. One ingredient is the study of convex polytopes that are closed under coordinate-wise maximum and minimum. This is joint work with Elina Robeva, Ngoc Tran, and Caroline Uhler. |
| 10:00am - 12:00pm | MS185, part 2: Algebraic Geometry Codes |
| Unitobler, F-122 | |
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10:00am - 12:00pm
Algebraic Geometry Codes The problem of finding good codes is central to the theory of error correcting codes. For many years coding theorists have addressed this problem by adding algebraic and combinatorial structure to C. In the early 80s Goppa used algebraic curves to construct linear error correcting codes, the socalled algebraic geometric codes (AG codes). The construction of an AG code with alphabet a finite field Fq requires that the underlying curve is Fq-rational and involves two Fq-rational divisors D and G on the curve. In this minisymposium we will present results on Algebraic Geometry codes and their performances. (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 Geometric Codes on Hirzebruch surfaces This talk presents results about Goppa codes over minimal Hirzebruch surfaces. Hirzebruch surfaces being toric surfaces, they are endowed with a polynomial coordinate ring, named the Cox ring. They also have a pleasant description as quotient spaces. These features enable us to define Goppa codes via the evaluation of polynomials similarly to projective Reed-Muller codes. Beside an easy implementation, considering polynomials makes us benefit from algebraic tools, such as Gröbner basis, to handle the parameters of these codes. Explicit formula for the dimension and the minimum distance of a Goppa code associated to a divisor D are computed as functions of the Picard class of D. The parameters are given for any size of the alphabet, even when the evaluation map is not injective. The minimum distance thus provides an upper bound of the number of rational points of a non-filling curve on a Hirzebruch surface. Moreover, the geometry of Hirzebruch surfaces, notably their ruling, leads to nice local decoding properties for these codes. Codes and gap sequences of Hermitian curves Hermitian functional and differential codes are AG-codes defined on a Hermitian curve. To ensure good performance, the divisors defining such AG-codes have to be carefully chosen, exploiting the rich combinatorial and algebraic properties of the Hermitian curves. In this paper, the case of differential codes CΩ(D,mT) on the Hermitian curve Hq^3 defined over Fq^6 is worked out, where supp(T):=Hq^3(Fq^2), the set of all Fq^2-rational points of Hq^3, while D is taken, as usual, to be the sum of the points in the complementary set D = Hq^3(Fq^6}) Hq(Fq^2). For certain values of m, such codes CΩ(D,mT) have better minimum distance compared with true values of 1-point Hermitian codes. The automorphism group of CL(D,mT), m≤q^3-2, is isomorphic to PGU(3,q). On the weight distribution of dual AG codes from the GK curve Let X be an algebraic curve defined over the finite field of order q. The parameters of the AG codes associated with X strictly depend on the underlying curve X. In general, curves with many rational places with respect to their genus give rise to AG codes with good parameters. For this reason maximal curves, that are curves attaining the Hasse-Weil upper bound, have been widely investigated in the literature. In this work, we focus our attention on the GK curve, which is a maximal curve constructed by Giulietti and Korchmáros which cannot be covered by the Hermitian curve whenever q is odd. In particular we investigate the minimum distance and the weight distribution of dual AG codes arising from the Giulietti-Korchmáros maximal curves. In most cases, the weight distribution of a given code is hard to be computed. Even the problem of computing codewords of minimum weight can be a difficult task, apart from specific cases. We do so using the link between the weight of the codewords of such codes and the geometry of the curve. We compute the maximal number of intersections that the GK curve can have with plane curves of low degree and we use this fact to determine the actual minimum distance and the number of minimum weight codewords of dual one-point AG codes arising from the GK curve. Subcovers and codes on a class of trace-defining curves In this talk, we will discuss explicit subcovers of a class of trace-defining curves over a finite field. It turns out that all such subcovers have a distiguished rational point P, for which the Weierstrass semigroup H(P) can often be determined. This will lead us to the construction of the corresponding one-point AG codes with very good parameters. In particular, we will present improvements on the parameters of at least 108 codes from the MinT table. |
| 10:00am - 12:00pm | MS145, part 3: Isogenies in Cryptography |
| Unitobler, F-123 | |
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10:00am - 12:00pm
Isogenies in Cryptography The isogeny graph of elliptic curves over finite fields has long been a subject of study in algebraic geometry and number theory. During the past 10 years several authors have shown multiple applications in cryptology. One interesting feature is that systems built on isogenies seem to resist attacks by quantum computers, making them the most recent family of cryptosystems studied in post-quantum cryptography. This mini-symposium brings together presentations on cryptosystems built on top of isogenies, their use in applications, and different approaches to the cryptanalysis, including quantum cryptanalysis. (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) Superspecial genus 2 curves in cryptography Isogenies can be defined between algebraic groups different from elliptic curves. In a joint work with Castryck and Smith, we construct a genus 2 version of the Charles-Goren-Lauter hash function based on isogenies. We will discuss the technical difficulties that arise from adapting the elliptic curve case. Quantum algorithms for finding isogenies between supersingular elliptic curves. We will present joint work with Jao and Sankhar on a quantum algorithm for finding an isogeny between two given supersingular elliptic curves. In general, it runs in time O(p^1/4), but it has subexponential run time if both curves are defined over Fp. We will also discuss improvements to this method obtained in collaboration with Iezzi and Jacobson. Our method consists in performing a quantum search within possible paths originating from the given curves to attain curves defined over Fp. Then we find an isogeny between curves defined over Fp by naturally exploiting the action of the class group of the endormorphism ring of these curves similarly to the work of Childs Jao and Soukharev. Further improvements to this method focus on the cost of the evaluation of the action of the class group. Horizontal isogeny graphs A horizontal isogeny graph is a graph whose vertices represent abelian varieties which all share the same endomorphism ring, and edges represent isogenies between them. They are an important tool to study the discrete logarithm problem on these abelian varieties, and allow to construct promising post-quantum public key cryptosystems. We discuss the analytic methods that allow to study the "mixing" properties of these graphs (a short random walk rapidly converges to a uniformly distributed vertex), with applications for cryptography. Isogeny Graphs of Ordinary Abelian Surfaces and Endomorphism Rings Building on some recent joint work with Brooks and Wesolowski, we recall a recent construction of certain l-power isogeny graphs of principally polarizable ordinary abelian varieties and study the structure of these graphs using the theory of Bruhat-Tits buildings for symplectic groups. Our results have implications in various problems from computational number theory and mathematical cryptology, most notably, the question of computing endomorphism rings as well as constructing hyperelliptic curves over finite fields whose Jacobians have a fixed characteristic polynomial of Frobenius and maximal endomorphism rings (the CM method in genus 2). This work is joint with Gaetan Bisson and Alexey Zykin (in memoriam). |
| 1:30pm - 2:30pm | IP08: Jeremy Gunawardena: Some mathematical aspects of gene regulation |
| vonRoll, Fabrikstr. 6, 001 | |
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1:30pm - 2:30pm
Some mathematical aspects of gene regulation Harvard Medical School, United States of America The “linear framework” describes biochemical systems under timescale separation in terms of a finite directed graph with labelled edges. When applied to gene regulation, the framework gives a gene's input-output response as a rational function of the graph labels. The sharpness of the response, or the sensitivity of output to changes in inputs, is important for understanding how gene-regulatory mechanisms control the the development of the organism during ontogeny as well as how such mechanisms evolve during phylogeny. We outline some mathematical problems relating to the sharpness of genetic input-output responses, with a focus on the role of energy expenditure away from thermodynamic equilibrium. |
| 1:30pm - 2:30pm | IP08-streamed from 001: Jeremy Gunawardena: Some mathematical aspects of gene regulation |
| vonRoll, Fabrikstr. 6, 004 | |
| 2:30pm - 3:00pm | Coffee break |
| Unitobler, F wing, floors 0 and -1 | |
| 3:00pm - 5:00pm | MS131, part 1: Computations in algebraic geometry |
| Unitobler, F005 | |
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3:00pm - 5:00pm
Computations in algebraic geometry This minisymposium highlights the use of computation inside algebraic geometry. Computations enter algebraic geometry in several different ways including numerical strategies, symbolic calculations, experimentation, and simply as a fundamental conceptual tool. Our speakers will showcase many of these aspects together with some 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) Regularity of S_n-invariant monomial ideals Consider a polynomial ring R in n variables with the action of the symmetric group Sn by coordinate permutations. I will describe an explicit recipe for computing the graded components of the modules Ext(I,R), when I is an arbitrary Sn-invariant monomial ideal, as well as the maps induced by inclusions of such ideals. As a consequence, this gives explicit formulas for the the regularity of Sn-invariant monomial ideals. A homological approach to numerical Godeaux surfaces Numerical Godeaux surfaces provide the first case in the geography of minimal surfaces of general type. By work of Miyaoka and Reid it is known that the torsion group of such a surface is cyclic of order at most 5, a full classification has been given for the cases where this order is 3,4, or 5. In my talk, I will discuss recent progress by Isabel Stenger towards the classification of numerical Godeaux surfaces with a trivial torsion group. Following a suggestion by Frank-Olaf Schreyer, the starting point of Stenger's work is a syzygy-type approach to the study of the canonical ring of such a surface. Particular attention is paid to the hyperelliptic curves arising in the fibration induced by the bicanonical system. Asymptotic syzygies for products of projective space We will discuss results describing the asymptotic syzygies of products of projective space, in the vein of the explicit methods of Ein, Erman, and Lazarsfeld’s non-vanishing results on projective space.
Where can toric syzygies live? Syzygies of toric varieties admit a natural grading by the character lattice of the corresponding torus. I will give some results on the the regions in the character lattice in which toric syzygies can be supported. This is joint work with Castryck and Lemmens. |
| 3:00pm - 5:00pm | MS189, part 2: Geometry and topology in applications. |
| Unitobler, F006 | |
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3:00pm - 5:00pm
Geometry and topology in applications. This symppsium will bring together leading practitioners, mid-carreer scientists as well as PhD students and postdoctoral fellows who are interested in the theory and practice of the applications of geometry and topology in real life problems. The spectrum of possible applications is very wide, and covers the sciences, biology, medicine, materials science, and many others. The talks will address the theoretical foundations of the methodology as well as the state of the art of geometric and topological modelling. (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) Persistent Betti numbers of random Cech complexes We study the persistent homology of random Cech complexes. Generalizing a method of Penrose for studying random geometric graphs, we first describe an appropriate theoretical framework in which we can state and address our main questions. Then we define the k-th persistent Betti number of a random Cech complex and determine its asymptotic order in the subcritical regime. This extends a result of Kahle on the asymptotic order of the ordinary k-th Betti number of such complexes to the persistent setting. This is joint work with Ulrich Bauer (TU Munich). Topological Analyses of Time Series Measurements from real-world systems produce time series, or sequential scalar-valued data, that contain information about complicated higher dimensional dynamics of the underlying system. Extracting this information from time series is often done by frequency analyses and statistics which demand linearity and stationarity. We present topological methods for investigating dynamics from nonlinear, non-stationary time series in application to TMS-EEG data. On the Robustness of the Homological Scaffold Abstract: Homological Scaffold has been firstly introduced by Petri, Expert, Vaccarino et al. in 2014 in studying the effects on the functional connectome of the human brain under the effect of psilocybin. At that time it was defined empirically by using javaplex. In this talk, we will present two new principled definitions of the scaffold and the results of a comparison of the three scaffolds on simulated and real data. Joint work with A. De Gregorio, M.Guerra and G.Petri Stable and discriminative topological invariants In this talk I will describe a framework that allows to compute a new class of stable invariants for multi-parameter persistence. The key element of our approach is defining metrics induced by so called ‘noise systems’. Such metrics generalize the classical notion of interleaving distance. At the same time, in the one parameter case, they allow to overcome the usual dichotomy interpreting short bars in a barcode as noise and long bars as relevant information. I will then focus on one of the proposed invariants, the stable rank, address its statistical properties and show how we can improve classification by adapting the noise system to the task. |
| 3:00pm - 5:00pm | MS200, part 4: From algebraic geometry to geometric topology: Crossroads on applications |
| Unitobler, F007 | |
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3:00pm - 5:00pm
From algebraic geometry to geometric topology: crossroads on applications The purpose of the Minisymposium "From Algebraic Geometry to Geometric Topology: Crossroads on Applications" is to bring together researchers who use algebraic, combinatorial and geometric topology in industrial and applied mathematics. These methods have already seen applications in: biology, physics, chemistry, fluid dynamics, distributed computing, robotics, neural networks and data analysis. (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) Reconnection in Biology and Physics Reconnection is a fundamental event in many areas of science, including the interaction of vortices in classical and quantum fluids, magnetic flux tubes in magnetohydrodynamics and plasma physics, and site-specific recombination in DNA. The helicity of a collection of flux tubes can be calculated in terms of writhe, twist and linking among tubes. We discuss that the writhe helicity is conserved under anti-parallel reconnection [1]. We will discuss the mathematical similarities between reconnection events in biology and physics, and the relationship between iterated reconnection and curve topology. We will discuss helicity and reconnection in a tangle of confined vortex circles in a superfluid. [1] Laing C.E., Ricca R.L. & Sumners D.W. (2015), Conservation of writhe helicity under anti-parallel reconnection, Nature Scientific Reports 5:9224/ DOI: 10.1038/srep09224. On the real geometric hypothesis of Banach The following is known as the geometric hypothesis of Banach: let V be an m-dimensional Banach space with unit ball B and suppose all n-dimensional subspaces of V are isometric (all the n-sections of B are affinely equivalent). In 1932, Banach conjectured that under this hypothesis V is isometric to a Hilbert space (the boundary of B is an ellipsoid). Gromov proved in 1967 that the conjecture is true for n=even and Dvoretzky derived the same conclusion under the hypothesis n=infinity. We prove this conjecture for n=5 and 9 and give partial results for an integer n of the form 4k+1. The ingredients of the proof are classical homotopic theory, irreducible representations of the orthogonal group and geometric TOMOGRAPHY. Suppose B is an (n+1)-dimensional convex body with the property that all its n-sections through the origin are affinity equivalent to a fixed n-dimensional body K. Using the characteristic map of the tangent vector bundle to the n-sphere, it is possible to prove that if n=even, then K must be a ball and using homotopical properties of the irreducible subgroups of SO(5) and SO(9), we prove that if N=5,9, then K must be a body of revolution. Finally, we prove, using geometry tomography and topology that, if this is the case, then there must be a section of B which is an ellipsoid and consequently B must be also an ellipsoid. The Cucker-Smale flocking model on manifolds: Geometric & topological effects, and flocking realizability We introduce a generalisation of the well-established Cucker-Smale model to complete Riemannian manifolds to study the influence of geometry and topology on the formation of flocks. The dynamics of the Cucker-Smale model facilitate the flocking of a group of particles in disordered motion into a coordinated one where all particle move parallelly with the center of mass. Despite their name, flocking models do not only illustrate the herding of animals but more generally the emergence of collective behaviour. The possible applications cover a broad spectrum of subjects such as linguistics, biology, opinion formation, sensor networks and robotics. While the Cucker-Smale model already received much attention over the last decade, those efforts focused on particles moving in a Euclidean space. Chi, Choi and Ha raised the flocking realizability problem: Given a manifold and a group of particles, construct a dynamical system that leads to a collective movement as a flock at least asymptotically. We establish theorems about the convergence of the particles to a flocked state under the dynamics of our generalized model. Not only does this address the flocking realizability problem but it also lays the groundwork for further investigations of topological and geometric effects on the dynamics. As an example, we already established that the presence of curvature restricts the final flocked state into specific patterns and we are looking forward to further investigations in this direction. This is joint work done with S.-Y. Ha & D. Kim (Seoul National U., Seoul). Topological modeling of local reconnection Local reconnection events are common in nature. One example is the action of recombination enzymes, and in particular of site-specific recombinases that recognize two short segments of DNA (the recombination sites), introduce two double-stranded breaks and recombine the ends. The local action of site-specific recombinases is a reconnection event which is modeled mathematically as a band surgery. The banding can be coherent or non-coherent, depending on the relative orientation of the recombination sites. Motivated by the unlinking of circular chromosomes after DNA replication, we have done extensive studies of coherent banding. In this talk I focus on more recent work that deals with non-coherent bandings. We use tools from low dimensional topology to investigate local reconnection between two sites in inverted repeats along a knot. We complement the analytical work with computer simulations. The numerical work provides a quantitative measure to distinguish among pathways of topology simplification by local reconnection, and also informs the search for bandings between specific pairs of knot or link types. This is joint work with Allison Moore, Tye Lidman, Michelle Flanner and Koya Shimokaw. |
| 3:00pm - 5:00pm | MS186, part 1: Algebraic vision |
| Unitobler, F011 | |
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3:00pm - 5:00pm
Algebraic vision There has been a burst of recent activity focused on the applications of modern abstract and numerical algebraic geometry to problems in computer vision, ranging from highly-optimized Gröbner-basis techniques, to homotopy continuation methods, to Ulrich sheaves and Chow forms, to functorial moduli theory. We will discuss this recent progress, with a focus on multiview geometry, both in theory and in practice. (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" Algebraic Vision Computer Vision is rarely done over the complex numbers. On the other hand, it does often involve semi-algebraic sets, non-genericity and finite precision arithmetic. Dealing with these complications leads to a variety of interesting and hard mathematical questions. I will talk about a few of my favorite ones. A geometric construction of the essential variety We give a new construction of the essential variety using a geometric definition of calibration. This construction recovers classical results about the essential variety while also yielding a new 2-1 cover that is strongly related to previous work of Kileel-Fløystad-Ottaviani. Classification of Point-Line Minimal Problems in Complete Multi-View Visibility We present a complete classification of all minimal problems for generic arrangements of points and lines completely observed by calibrated perspective cameras. We show that there are only 30 minimal problems in total, no problems exist for more than 6 cameras, for more than 5 points, and for more than 6 lines. For all minimal problems discovered, we present their algebraic degrees, i.e. the number of solutions, which measure their intrinsic difficulty. Our classification shows that there are many interesting new minimal problems. Our results also show how exactly the difficulty of problems grows with the number of views. Importantly, we discovered several new minimal problems with small degrees that might be practical in image matching and 3D reconstruction. |
| 3:00pm - 5:00pm | MS160, part 4: 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) Numerical Schubert Calculus via the Littlewood-Richardson Homotopy Algorithm We describe the Littlewood-Richardson homotopy algorithm, which uses numerical continuation to compute solutions to Schubert problems on Grassmannians and is based on the geometric Littlewood-Richardson rule. We provide algorithmic details and discuss its mathematical aspects. Our implementation of this algorithm can solve problem instances with tens of thousands of solutions. We also give a new and optimal formulation of Schubert problems in local Stiefel coordinates as systems of equations. Computing Verified Real Solutions of Polynomials Systems via Low-rank Moment Matrix Completion We propose a new algorithm for computing verified real solutions of polynomial systems with equalities and inequalities. We recast Lasserre's hierarchy of moment relaxations for computing real solutions of polynomial systems into finding symmetric positive semidefinite matrices of minimum nuclear norm subject to linear equality constraints, and then apply fixed point iterations together with Barzilai-Borwein technique for solving a sequence of moment matrix completion problems. Although the method based on function values and gradient evaluations cannot yield as high accuracy as interior point methods, much larger problems can be solved since no second-order information needs to be computed and stored. Finally, we apply interval arithmetic to verify the existence of real solutions of polynomial systems near to the computed approximate real solutions. The algorithm has been implemented in Matlab and is available at http://159.226.47.210:8080/verifyrealroots/tryOnline.jsp Computing the Canonical Polyadic Decomposition of Tensors with Damped Gauss-Newton Method Low rank approximation of tensors can be formulated as a structured nonlinear minimization problem. Exploiting this structure allows to improve the speed and accuracy of a damped Gauss-Newton method. A preliminary implementation of this method performed better than availble published software. A most outrageous action The cost of homotopy algorithms for sparse polynomial systems can be bounded above by an integral of a condition length (Found Comput Math (2019) 19: 1. https://doi.org/10.1007/s10208-018-9375-2). This integral depends on a toric condition number and on a distortion invariant nu. In this talk, I will show how a certain renormalization operator induces a group action on the solution variety. This action will be used to produce a renormalized algorithm, where the distortion nu is constant. Then it becomes possible to integrate the square of the condition number for normal systems. This method provides upper bounds for the expected cost of sparse homotopy. |
| 3:00pm - 5:00pm | MS167, part 3: Computational tropical geometry |
| Unitobler, F013 | |
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3:00pm - 5:00pm
Computational tropical geometry This session will highlight recent advances in tropical geometry, algebra, and combinatorics, focusing on computational aspects and applications. The area enjoys close interactions with max-plus algebra, polyhedral geometry, combinatorics, Groebner theory, and numerical algebraic 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) Tropicalized quartics and curves of genus 3 Brodsky, Joswig, Morrison and Sturmfels showed that not all abstract tropical curves of genus 3 can be realized as a tropicalization of a quartic in the euclidean space. We focus on the interior of the maximal cones in the moduli space and study all curves which can be realized as a faithful tropicalization in a tropical plane. Reflecting the algebro-geometric world, these are exactly those which are not realizably hyperelliptic. Our approach is constructive: For any not realizably hyperelliptic curve, we explicitly construct a realizable model of the tropical plane and a faithfully tropicalized quartic in it. These constructions rely on modifications resp. tropical refinements. Conversely, we prove that any realizably hyperelliptic curve cannot be embedded in such a fashion. For that, we rely on the theory of tropical divisors and embeddings from linear systems, and recent advances in the realizability of sections of the tropical canonical divisor. Tropical Jucys Covers and refined quasimodularity Hurwitz numbers count genus $g$, degree $d$ covers of the complex projective line with fixed branched locus and fixed ramification data. An equivalent description is given by factorisations in the symmetric group. Simple double Hurwitz numbers are a class of Hurwitz-type counts of specific interest. In recent years a related counting problem in the context of random matrix theory was introduced as so-called monotone Hurwitz numbers. These can be viewed as a desymmetrised version of the Hurwitz-problem. Moreover, the notion of strictly monotone Hurwitz numbers has risen in interest as it is equivalent to a certain Grothendieck dessins d'enfant count. We study monotone and strictly monotone Hurwitz numbers from a bosonic Fock space perspective. This yields a new interpretation in terms of tropical covers involving local multiplicities given by Gromov-Witten invariants. We further discuss applications of this new interpretation with regards to quasimodularity results and wall-crossing formulae. Tropical lines on tropical surfaces In 1849, Arthur Cayley and George Salmon proved that every smooth cubic surface in P3 contains exactly 27 lines. Since the early development of tropical geometry, two natural problems were to understand whether the same statement holds for smooth tropical cubic surfacss and to classify combinatorial positions of their tropical lines. The answer to the first turned out to be false, as smooth tropical surfaces might contain families of tropical lines. Moreover, classifying positions of tropical lines reveals some computational challenges due to the massive number of combinatorial types of smooth tropical cubic surfaces. In this talk we will tell this tropical story. We will introduce motifs of tropical lines on tropical surfaces and study their configurations. Throughout the talk we will highlight the computational aspects. Polyhedral tropical geometry of higher rank In a recent paper, Jell, Scheiderer, and Yu define a notion of real tropicalization for semialgebraic sets. In this talk I will discuss what happens when the semialgebraic set is a linear subspace. In this setting, the real tropicalization is a polyhedral fan that is best understood using the theory of oriented matroids. The main focus of this talk will be on understanding the topological and combinatorial properties of this fan. |
| 3:00pm - 5:00pm | MS158, part 2: Structured sums of squares |
| Unitobler, F021 | |
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3:00pm - 5:00pm
Structured sums of squares A description of a nonnegative polynomial as a sum of squares gives a concise proof of its nonnegativity. Computationally, such sum-of-squares decompositions are appealing because we can search for them by solving a semidefinite feasibility problem. This connection means that optimization and decision problems arising in a range of areas, from robotics to extremal combinatorics, can be reformulated as, or approximated with, semidefinite optimization problems. This minisymposium highlights the roles of various kinds of additional structures, including symmetry and sparsity, in understanding when (structured) sum of squares decompositions do and do not exist. It will also showcase interesting connections between sums of squares and a range of areas, such as extremal combinatorics, logic, dynamical systems and control, and algorithms and complexity 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) Simple Graph Density Inequalities with no Sum of Squares Proofs Establishing inequalities among graph densities is a central pursuit in extremal combinatorics. One way to certify the nonnegativity of a graph density expression is to write it as a sum of squares. In this talk, we identify a simple condition under which a graph density expression cannot be a sum of squares. Using this result, we prove that the Blakley-Roy inequality does not have a sum of squares certificate when the path length is odd. We also show that the same Blakley-Roy inequalities cannot be certified by sums of squares using a multiplier of the form one plus a sum of squares. These results answer two questions raised by Lovász. Our main tool is used again to show that the smallest open case of Sidorenko's conjectured inequality cannot be certified by a sum of squares. This is joint work with Greg Blekherman, Mohit Singh and Rekha Thomas. Symmetry and Nonnegativity I will discuss several recent results on symmetric nonnegative polynomials and their approximations by sums of squares. I will consider several types of symmetry, but the situation is especially interesting in the limit as the number of variables tends to infinity. There are diverse applications to quantum entanglement, graph density inequalities and theoretical computer science. Symmetry and the Sum of Squares Hierarchy In this talk, I will describe how symmetry can be used both to greatly reduce the size of the semidefinite program needed to implement SOS and to make proving SOS lower bounds considerably easier. In particular, I will describe how to construct semidefinite programs for SOS on symmetric problems whose size is independent of n using Razborov's flag algebras. I will also describe a sufficient condition for proving SOS lower bounds when the problem is symmetric. As an application, I will describe SOS lower bounds for the following Turan-type problem: What is the minimum possible number of triangles in a graph G with n vertices? More generally, I will describe how we can obtain SOS lower bounds almost automatically whenever our problem is symmetric and the difficulty of our problem comes from integrality arguments (i.e. arguments that a certain expression must be an integer). Note: This talk will be based on the paper "Symmetric Sums of Squares over k-Subset Hypercubes" by Raymond, Saunderson, Singh, and Thomas and my paper "Sum of Squares Lower Bounds from Symmetry and a Good Story". |
| 3:00pm - 5:00pm | MS129, part 1: Sparsity in polynomial systems and applications |
| Unitobler, F022 | |
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3:00pm - 5:00pm
Sparsity in polynomial systems and applications In this session we bring together researchers working in different areas involving sparsity in applications and sparse polynomial systems. The principle of sparsity is to represent a structure by functions, e.g., polynomials, with as few variables or terms as possible. It is ubiquitous in various areas and problems, where algebra and geometry play a key role. Recently, it has been succesfully applied to problems such as sparse interpolation, polynomial optimization, sparse elimination, fewnomial theory, or tensor decomposition. This minisymposium provides an opportunity to learn about a selection of these recent developments and explore new potential applications of sparsity. (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) Optimal Descartes' rule of signs for polynomial systems supported on circuits We will describe a refinement of the Descartes'rule of signs for polynomial systems supported on circuits which wasproposed by Bihan and Dickenstein few years ago. The main difference is that new bound is sharp for any given circuit, and is always smaller or equal to the normalized volume of the convex hull of the circuit. This is a joint work with Alicia Dickenstein and Jens Forsgard. Polyhedral Approximations to the Cone of Nonnegative Polynomials Can we always approximate a semidefinite program with a linear program? Is it possible to approximately check nonnegativity of a polynomial by just a few pointwise evaluations? These questions fall into the general framework of approximating the cone of nonnegative polynomials with polyhedral cones. In this talk, we will show inapproximability of the cone of nonnegative polynomials in the dense case, and existence of a polyhedral approximation with polynomial number of facets in the sparse case (i.e. the case of a subspace of polynomials with fixed dimension). Time permits, we will also discuss a randomized construction of the approximation cone based on a different set of tools coming from computational geometry. Nonegativity and Discriminants We study the class of nonnegative polynomials obtained from the inequality of arithmetic and geometric means, called emph{agiforms} or emph{nonnegative circuit polynomials}. They generate a full dimensional subcone $S$ of the cone of all nonnegative polynomials, which is distinct from the cone of sums of squares. Let $mathbb{R}^A$ denote the space of all real polynomials with support $A$. We describe the boundary of the cone $S cap mathbb{R}^A$ as a space stratified in real semi-algebraic varieties. In order to describe the strata, we take a journey through discriminants, polytopes and triangulations, oriented matroids, and tropical geometry. This is based on joint work with Timo de Wolff. Exploiting Sparsity for Semi-Algebraic Set Volume Computation We provide a systematic deterministic numerical scheme to approximate the volume (i.e. the Lebesgue measure) of a basic semi-algebraic set whose description follows a sparsity pattern. As in previous works (without sparsity), the underlying strategy is to consider an infinite-dimensional linear program on measures whose optimal value is the volume of the set. This is a particular instance of a generalized moment problem which in turn can be approximated as closely as desired by solving a hierarchy of semidefinite relaxations of increasing size. The novelty with respect to previous work is that by exploiting the sparsity pattern we can provide a sparse formulation for which the associated semidefinite relaxations are of much smaller size. In addition, we can decompose the sparse relaxations into completely decoupled subproblems of smaller size, and in some cases computations can be done in parallel. To the best of our knowledge, it is the first contribution that exploits sparsity for volume computation of semi-algebraic sets which are possibly high-dimensional and/or non-convex and/or non-connected. |
| 3:00pm - 5:00pm | MS127, part 1: The algebra and geometry of tensors 2: structured tensors |
| Unitobler, F023 | |
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3:00pm - 5:00pm
The algebra and geometry of tensors 2: structured tensors Tensors are ubiquitous in mathematics and science. Tensor decompositions and approximations are an important tool in artificial intelligence, chemometrics, complexity theory, signal processing, statistics, and quantum information theory. Often, due to the nature of the problem under investigation, it might be natural to consider tensors equipped with additional structures or might be useful to consider tensor decompositions which respect particular structures. Among many interesting constructions, we might think of: symmetric, partially-symmetric and skew-symmetric tensors; tensor networks; Hadamard products of tensors or non-negative ranks. This minisymposium focuses on how exploiting these additional structures from algebraic and geometric perspectives recently gave new tools to study these special classes of tensors and decompositions. This is a sister minisymposium to "The algebra and geometry of tensors 1: general tensors" organized by Y. Qi and N. Vannieuwenhoven. (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) Projective geometry and tensor identifiability A tensor rank-1 decomposition of a tensor T, lying in a given tensor space is an additive decomposition with rank one tensors. In many instances, for both pure and applied mathematics, it is interesting to understand when such a decomposition is unique, in a suitable sense. This problem translates very efficiently into geometric statements and can be attached via old and new techniques in projective geometry. In the talk, as an application, I will present some results concerning identifiability of tensors and partially symmetric tensors obtained via birational geometry techniques. A bound for the Waring rank of the determinant via syzygies The Waring rank of the 3x3 generic determinant is known to be greater than or equal to 14, and less than or equal to 20. Proofs of the lower bound of 14 were given in terms of geometric singularities or the Hilbert function of the apolar ideal. We improve the lower bound to 15 by considering higher syzygies in the minimal graded free resolution of the apolar ideal of the determinant. This is joint work with Mats Boij. On the identifiability of ternary forms I will discuss a method which in principle can determine the uniqueness (and the minimality) of any given Waring decomposition of a ternary form of any degree. The method is based on an algebraic and geometric study of the set of points representing the decomposition, and from this point of view it can be seen as an extension of the Kruskal criterion for the identifiability of tensors. In addition, the method is sensitive of the coefficients of the elementary tensors in the given decomposition. Thus, it can distinguish between identifiable and not identifiable forms in the span of a given set of powers. Real Waring Rank Geometry of Quaternary Forms We studied the real ternary forms whose real rank equals the generic complex rank, and we characterize the semialgebraic set of sums of powers representations with that rank in the previous paper [1]. The semialgebraic set is called the Space of Sums of Powers, which is naturally included in the Variety of Sums of Powers. In this talk, we will go further for quaternary forms. For quadrics, we find a simple way to characterize the Space of Sums of Powers, and characterize the behaviors of representations. For cubics, we developed an algorithm to obtain the unique Waring rank decomposition and using this, we determined which quaternary cubics has a real rank decomposition. For other cases with degrees bigger than 4, we identify some of components of the real rank boundary. And also we will present some problems related to this topic. [1] 1. Michalek, M., Moon, H., Sturmfels, B.,Ventura, E., “Real Rank Geometry of Ternary Forms”, Annali di Matematica Pura ed Applicata, June 2017, Volume 196, Issue 3, pp. 1025-1054 |
| 3:00pm - 5:00pm | Room free |
| Unitobler, F-105 | |
| 3:00pm - 5:00pm | MS154, part 4: 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) Gain matroids and their applications Let (G=(V,E)) be an undirected graph and let (k) and (ell) be two integers. (G) is said to be emph{((k,ell))-sparse} if (|E'|leq k|V'|-ell) holds for every subgraph (G'=(V',E')) of (G). The edge sets of the ((k,ell))-sparse subgraphs form a matroid on the edge set of a graph (H) called the emph{((k,ell)) count matroid}. Gain matroids are a generalisation of count matroids. Let (Gamma) be a group. Now assign an element of (Gamma) (a emph{gain}) and a reference direction to every edge in (E). The emph{gain} of a (not necessarily directed) closed walk is defined as the group element obtained by the multiplication of the gains of its edges where the inverse gain should be used for the edges used in the reverse direction. The gain group corresponding to the edge set of a connected subgraph is the set of gains of closed walks starting at one of its vertices, (v). (After some observations (v) can be dropped from the definition and it can be extended to arbitrary subgraphs.) Matroid threshold hypergraphs In this talk we introduce the notion of a matroid threshold hypergraph: a collection of bases of a matroid obtained by capping the total weight of the bases under given a function of the ground set. Focusing on the uniform matroid yields the classical theory of threshold hypergraphs. In this talk we will motivate the definition, explain a few their interesting properties and speculate about the uses of the theory. Whitney Numbers for Cones An arrangement of hyperplanes dissects space into connected components called chambers. A nonempty intersection of halfspaces from the arrangement will be called a cone. The number of chambers of the arrangement lying within the cone is counted by a theorem of Zaslavsky, as a sum of certain nonnegative integers that we will call the cone's "Whitney numbers of the 1st kind". For cones inside the reflection arrangement of type A (the braid arrangement), cones correspond to posets, chambers in the cone correspond to linear extensions of the poset, and these Whitney numbers refine the number of linear extensions. We present some basic facts about these Whitney numbers, and interpret them for two families of posets. |
| 3:00pm - 5:00pm | MS136, part 2: Syzygies and applications to geometry |
| Unitobler, F-107 | |
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3:00pm - 5:00pm
Syzygies and applications to geometry In this minisymposium, titled "Syzygies and applications to geometry”, we will focus on the striking results and applications that the study of syzygies provides in algebraic geometry, in a wide sense. Topics should include but are not limited to the study of rational and birational maps, singularities, residual intersections and the defining equations of blow-up algebras. We plan to focus on recent progress in this area that result in explicit and effective computations to detect certain geometrical property or invariant. Applications to geometric modeling are very welcome. (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) Implicitization of Tensor Product Surfaces via Virtual Projective Resolutions (Part I) In this talk, we address the implicitization problem for tensor product surfaces. A tensor product surface is defined by a parametrization (rational map) $mathbb{P}^1times mathbb{P}^1to mathbb{P}^3$. We consider the problem of computing the defining equation for the tensor product surface based on the polynomials which give its parametrization. Towards this end, we use the residual resultants developed by Busé-Elkadi-Mourrain. Our perspective is informed by the new development of virtual resolutions, which afford the derivation of the implicit equation from a smaller, more manageable algebraic construction than the more standard projective resolutions. Part I of this talk will discuss the algebraic underpinnings of our method. Implicitization of Tensor Product Surfaces via Virtual Projective Resolutions (Part II) In this talk, we address the implicitization problem for tensor product surfaces. A tensor product surface is defined by a parametrization (rational map) $mathbb{P}^1times mathbb{P}^1to mathbb{P}^3$. We consider the problem of computing the defining equation for the tensor product surface based on the polynomials which give its parametrization. Towards this end, we use the residual resultants developed by Busé-Elkadi-Mourrain. Our perspective is informed by the new development of virtual resolutions, which afford the derivation of the implicit equation from a smaller, more manageable algebraic construction than the more standard projective resolutions. Part II of this talk will discuss computational considerations and the practical implementation of our method. The Hilbert quasipolynomial of a polynomial ring and generating functions related the Frobenius complexity for various classes of singularities The talk will present some open questions on the Hilbert quasipolynomial associated to a polynomial ring over a field in finitely many indeterminates, with nonstandard grading. These questions originate in investigations regarding the Frobenius complexity for finitely generated algebras over the integers. The investigation approaches this notion of complexity by analogy to the Hilbert-Samuel and Hilbert-Kunz functions. This is joint work with Yongwei Yao. Generalized Stanley-Reisner rings Given a simplicial complex, we study the structure of the subrings of the Stanley-Reisner ring associated to the simplicial complex. These subrings can be seen as the space of spline functions defined on the simplicial complex with higher order continuity conditions accross the faces of the partition. Their ring structure becomes particularly interesting by identifying the ring of continuous splines with the equivariant cohomology ring of a space with a torus action. In the talk, we will explore this identification as well as the the geometric realizations of genealized Stanley-Reisner rings via the description of certain syzygy modules thar encode the smoothness conditions on the splines.
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| 3:00pm - 5:00pm | MS193: Algebraic geometry, data science and fundamental physics |
| Unitobler, F-111 | |
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3:00pm - 5:00pm
Algebraic geometry, data science and fundamental physics There has been an increasing interaction between computational algebraic geometry, data science and fundamental theoretical physics. This is rooted in the tradition that the 2 pillars of theoretical physics- general relativity and the standard model of particle physics, as well as their best candiate unified theory of superstrings - are physical realizations of the study of gauge connections and Riemannian metrics on manifolds. In the last couple of years, problems such as mapping the Calabi-Yau landscape, translating problems in particle theory to precise problems in algebraic and differential geometry, using the latest techniques in machine-learning, etc., have taken off in the theoretical physics community. This session in SIAM AG 2019 is a perfect venue for further explorations. (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 Calabi-Yau landscape & machine learning To be completed. Machine Learning for String Vacua I will discuss complexity classes of problems encountered in string theory. Since most problems are NP-complete or undecidable, data science techniques are used to tackle them. As an example, I will present Reinforcement Learning applications to string landscape questions and demonstrate how the algorithm learns to solve the associated problem. Knot Theory and Machine Learning I will discuss various aspects of studying knot theory and knot topological invariants with machine learning. Machine-learning a virus assembly fitness landscape Realistic evolutionary fitness landscapes are notoriously difficult to construct. A recent cutting-edge model of virus assembly is based on a detailed understanding of the geometry involved and fundamental biophysical principles, which allows one to capture the contribution to fitness coming from assembly efficiency in a suitably quantitative way. This model has a virus capsid shell consisting of twelve pentagons in a dodecahedral arrangement. Furthermore, there are 12 corresponding packaging signals - features in the genome which help recruit the twelve pentagonal capsid building blocks onto the growing capsid - in three binding affinity bands. The complete assembly phenotype space consisting of 312genomes has been explored via computationally expensive stochastic ab initio assembly models on a supercomputer, giving a fitness landscape in terms of the assembly efficiency. Using machine-learning techniques, we have shown that the intensive computation can be short-circuited in a matter of minutes to astounding accuracy. There is thus a large hidden degeneracy in the detailed microphysical models, which allows one to understand general features that emerge at a higher level. This opens up the possibility of tackling more complicated models by bootstrapping, i.e. by only partially exploring the phenotype space in order to machine learn the generic features and then to use these to predict the remainder of the fitness landscape. |
| 3:00pm - 5:00pm | MS137, part 3: Symbolic Combinatorics |
| Unitobler, F-112 | |
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3:00pm - 5:00pm
Symbolic Combinatorics In recent years algorithms and software have been developed that allow researchers to discover and verify combinatorial identities as well as understand analytic and algebraic properties of generating functions. The interaction of combinatorics and symbolic computation has had a beneficial impact on both fields. This minisymposium will feature 12 speakers describing recent research combining these areas. (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) Polynomial Reduction and Super Congruences Based on a reduction processing, we rewrite a hypergeometric term as the sum of the difference of a hypergeometric term and a reduced hypergeometric term (the reduced part, in short). We show that when the initial hypergeometric term has a certain kind of symmetry, the reduced part contains only odd or even powers. As applications, we derived two infinite families of super-congruences. Diagonals, determinants, and rigidity Diagonals of rational functions occur naturally in lattice statistical mechanics and enumerative combinatorics. We find that the diagonals of certain families of rational functions can be expressed as pullbacked hypergeometric 2F1 functions. On the other hand, the enumeration of combinatorial objects is often encoded by determinants. We study several families of binomial determinants that count the number of lozenge tilings of hexagonal domains with holes. Also graphs play a prominent role in combinatorics, and we are particularly interested in the aspect of rigidity. Using a novel combinatorial algorithm for computing the number of complex realizations of a maximally rigid graph, we explore exhaustively the Laman numbers of graphs with up to 13 vertices. Central Limit Theorems from the Location of Roots of Probability Generating Functions For a discrete random variable, what information can we deduce from the roots of its probability generating function? We consider a sequence of random variables X_n taking values between 0 and n, and let P_n(z) be its probability generating function. We show that if none of the complex zeros of the polynomials P_n(z) are contained in a neighborhood of 1 in the complex plane then a central limit theorem occurs, provided the variance of X_n isn't subpolynomial in n. This result is sharp a sense that will be made precise, and thus disproves a conjecture of Pemantle and improves upon various results in the literature. This immediately improves a multivariate central limit theorem of Ghosh, Liggett and Pemantle, and has ramifications for certain variables that arise in graph theory contexts. This is based on joint work with Julian Sahasrabudhe. Periodic Pólya urns and an application to Young tableaux Pólya urns are urns where at each unit of time a ball is drawn uniformly at random and is replaced by some other balls according to its colour. We introduce a more general model: The replacement rule depends on the colour of the drawn ball and the value of the time mod p. Our key tool are generating functions, which encode all possible urn compositions after a certain number of steps. The evolution of the urn is then translated into a system of differential equations and we prove that the moment generating functions are D-finite. From these we derive asymptotic forms of the moments. When the time goes to infinity, we show that these periodic Pólya urns follow a rich variety of behaviours: their asymptotic fluctuations are described by a family of distributions, the generalized Gamma distributions, which can also be seen as powers of Gamma distributions. Furthermore, we establish some enumerative links with other combinatorial objects, and we give an application for a new result on the asymptotics of Young tableaux: This approach allows us to prove that the law of the lower right corner in a triangular Young tableau follows asymptotically a product of generalized Gamma distributions. This is joint work with Cyril Banderier and Philippe Marchal. |
| 3:00pm - 5:00pm | MS155, part 2: Massively parallel computations in algebraic geometry |
| Unitobler, F-113 | |
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3:00pm - 5:00pm
Massively parallel computations in algebraic geometry Massively parallel methods have been a success story in high performance numerical simulation, but so far have rarely been used in computational algebraic geometry. Recent developments like the combination of the parallelization framework GPI-Space with the computer algebra system Singular have made such approaches accessible to the mathematician without the need to deal with a multitude of technical details. The minisymposium aims at bringing together researchers pioneering this approach, discussing the current state of the art and possible future developments. We plan to address applications in classical algebraic geometry, tropical geometry, geometric invariant theory and links to high energy physics. (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) Tools for perturbative calculations from algebraic geometry In the last few years perturbative methods in scattering amplitudes have incorporated a lot of tools and methods from computational algebraic geometry. In this talk I will present some ideas that have been developed for reducing and calculating Feynman Integrals. A massively parallel fan traversal with applications to geometric invariant theory The GIT fan of an algebraic group action on an algebraic variety describes all GIT quotients arising from Mumford’s construction in Geometric Invariant Theory. Its computation poses a challenging hurdle due to Buchberger's algorithm with double exponential worst case complexity. We present an optimized and scalable approach for computing the GIT fan by means of computational convex geometry. By factoring out symmetries and utilizing an ultra scale computing center, we were able to traverse the GIT fan of 6-pointed stable curves of genus 0 in approximately 13 minutes, yielding 249.604 chamber orbits. Parallel algorithms for computing tropical varieties with symmetry In this talk I will report on the massively parallel computation of tropical varieties, taking their symmetry into account. For this purpose we build upon a massively parallel fan traversal method implemented by Christian Reinbold using Singular in conjunction with GPI-Space. To pass between neighboring tropical cones of the Gröbner fan we use recently developed methods by Hofmann and Ren. This is a joint work with Janko Boehm, Mirko Rahn and Yue Ren. Space sextics and their tritangents In this talk, we will briefly review the well-known algebro-geometric oddity that complex space sextic curves admit exactly 120 tritangent planes. We discuss recent works which shows that all tritangents can be totally real, as well as the current state on the question whether the 120 tritangents determine the curve uniquely. The latter context gives rise to computational algebro-combinatorial challenges in which parallelization would be of great help. This is joint work with Turku Celik, Avinash Kulkarni, Mahsa Sayyary, and Bernd Sturmfels. |
| 3:00pm - 5:00pm | MS139, part 2: Combinatorics and algorithms in decision and reason |
| Unitobler, F-121 | |
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3:00pm - 5:00pm
Combinatorics and algorithms in decision and reason Combinatorial, or discrete, structures are a fundamental tool for modeling decision-making processes in a wide variety of fields including machine learning, biology, economics, sociology, and causality. Within these various contexts, the goal of key problems can often be phrased in terms of learning or manipulating a combinatorial object, such as a network, permutation, or directed acyclic graph, that exhibits pre-specified optimal features. In recent decades, major break-throughs in each of these fields can be attributed to the development of effective algorithms for learning and analyzing combinatorial models. Many of these advancements are tied to new developments connecting combinatorics, algebra, geometry, and statistics, particularly through the introduction of geometric and algebraic techniques to the development of combinatorial algorithms. The goal of this session is to bring together researchers from each of these fields who are using combinatorial or discrete models in data science so as to encourage further breakthroughs in this important area of mathematical research. (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 Graphs of Graphical Models Graphical models, including constraint networks, Bayesian networks, Markov random fields and influence diagrams, have become a central paradigm for knowledge representation and reasoning in artificial intelligence, and provide powerful tools for solving problems in numerous application areas. Reasoning over probabilistic graphical models typically involves answering inference queries, such as computing the most likely configuration (maximum a posteriori or MAP) or evaluating the marginals or the normalizing constant of a distribution (the partition function). Exact computation of such queries is known to be intractable in general, yet, the underlying graphs of graphical models provide a powerful tool that allow exploiting the problems structure in reasoning algorithms. In this talk I will provide an overview of how such graph parameters (e.g., tree-width, height, w-cutset) interact in their role for bounding graphical models complexity. Causal Inference with Unknown Intervention Targets We consider the problem of estimating causal DAG models from a mix of observational and interventional data, when the intervention targets are partially or completely unknown. This problem is highly relevant for example in genomics, since gene knockout technologies are known to have off-target effects. In this paper, we characterize the interventional Markov equivalence class of DAGs that can be identified from interventional data with unknown intervention targets. In addition, we propose the first provably consistent algorithm for learning the interventional Markov equivalence class from such data. The proposed algorithm greedily searches over the space of permutations to minimize a novel score function. The algorithm is nonparametric, which is particularly important for applications to genomics, where the relationships between variables are often non-linear and the distribution non-Gaussian. We demonstrate the performance of our algorithm on synthetic and biological datasets. On attempts to characterize facets of the chordal graph polytope Our idea of integer linear programming approach to structural learning decomposable graphical models, which are models described by undirected chordal graphs, is to encode them by special zero-one vectors, named characteristic imsets. It leads to the study of a special polytope, defined as the convex hull of all characteristic imsets for chordal graphs we name the chordal graph polytope (Studeny and Cussens; 2017). The talk will be devoted to the attempts to characterize theoretically all facet-defining inequalities for this polytope in order to utilize that in ILP-based procedures for learning decomposable models. |
| 3:00pm - 5:00pm | MS134, part 6: 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) New results on graph-based codes Codes over graphs have become widespread in industry applications due to their excellent performance with low-complexity decoders. Since long block lengths are desirable in practice, constructing codes using lifts of well-designed base graphs has become a standard technique. In this talk, we will present some recent results on how the permutations chosen affect the parameters of the resulting codes. Large constant dimension subspace codes consisting of k-dimensional subspaces, pairwise intersecting in at least (k-2)-dimensional subspaces Within the theory of subspace codes, a constant dimension code C is a set of k-dimensional subspaces in the vector space V(n,q) of dimension n over the finite field of order q. One of the goals in the theory of subspace codes is to characterize large subspace codes, satisfying specific conditions, such as intersection conditions or lower bounds on the minimum distance. There are two types of constant dimension codes consisting of k-dimensional subspaces, pairwise intersecting in (k-1)-dimensional subspaces. They are either: (1) a sunflower: a set of k-dimensional subspaces passing through a common (k-1)-dimensional subspace, or (2) a set of k-dimensional subspaces lying in a common (k+1)-dimensional subspace. The next step would be to investigate the largest sets of k-dimensional subspaces in the vector space V(n,q), pairwise intersecting in exactly In this talk, we present classification results on the largest examples of sets of k-dimensional subspaces, pairwise intersecting in exactly (k-2)-dimensional subspaces, or pairwise intersecting in at least (k-2)-dimensional subspaces. These classification results are obtained via geometrical techniques in the corresponding (n-1)-dimensional projective space PG(n-1,q) corresponding to the n-dimensional vector space V(n,q). Algebraic properties of codes with symmetries We will illustrate some new results and properties of codes with symmetries. Whenever a linear code over K has a non-trivial group of (permutation) automorphisms G, it can be viewed as a KG-module. Many well-studied families of codes are characterized by this property: cyclic, quasi-cyclic, abelian, quasi-abelian, group codes, etc. We will show how the algebraic structure of these codes allow to deduce properties on their parameters and to construct optimal codes. Moreover, we will show new asymptotic results for group codes in odd characteristic. Quantum codes coming from J-affine variety codes We will introduce J-affine variety codes and we will give conditions for their self-orthogonality with respect to Euclidean and Hermitian inner products. Parameters of stabilizer codes coming from subfield-subcodes of J-affine variety codes will be showed. Many of these codes turn to exceed the known Gilbert-Varshamov bounds and improve some quantum codes given in the literature. Finally, we will show how to use hyperbolic codes to provide stabilizer codes with designed distance. |
| 3:00pm - 5:00pm | MS162, part 1: Applications of finite fields theory |
| Unitobler, F-123 | |
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3:00pm - 5:00pm
Applications of finite fields theory The theory of finite fields is one of the most important meeting points of Algebraic Geometry, Computer Science, and Number Theory. One of the most important challenges in the area is to develope the theory of finite fields in connection with useful applications, in particular in secure communication, coding theory, and pseudorandom number generation. In this minysimposium we plan to bring together experts from many different areas of the mathematics of communication who share the common interest towards the theory of finite fields. Our main purpose is to provide an overview of some of the cutting-edge research in the field, and to lay the fundations for new collaborations among researchers interested in applications of the theory of finite 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) Introductory Talk This is an introductory talk to this session. Using Mersenne and Fermat numbers in Cryptosystems Modern public-key cryptography is mostly based on the hardness of Factoring and computing discrete logarithms. However, due to Shor’s algorithm, large scale Quantum computer if and when they become available would put these systems at risk, with the danger of compromising the security of all computer applications. In this talk, we show the construction of new crypto algorithms based on arithmetic modulo Mersenne or Fermat numbers. We describe both a simple encryption algorithm and a fully homomorphic encryption scheme. Cryptographic attacks against filter generator using monomial mapping Filter generators are vulnerable to several attacks which have led to well-known design criteria on the Boolean filtering function (mainly Algebraic Immunity and Nonlinearity). However, Rønjom and Cid have observed that a change of the primitive root defining the LFSR leads to several equivalent generators. They usually offer different security levels since they involve filtering functions of the form F(xk) where k is coprime to (2n -1) and n denotes the LFSR length. We prove that this monomial equivalence does not affect the resistance of the generator against algebraic attacks, but usually impacts the resistance to correlation attacks. Most importantly, a more efficient attack can often be mounted by considering non-bijective monomial mappings. In this setting, a divide-and-conquer strategy applies based on a search within a multiplicative subgroup of F2n*. Moreover, if the LFSR length n is not a prime, a fast correlation attack involving a shorter LFSR can be performed. This attack is generic and uses the decomposition in the multiplicative subgroups of F2n*, leading to new design criteria for Boolean functions used in Cryptography. Permutation and complete rational functions via Chebotarev theorem for function fields Constructing permutation functions of finite fields is a task of great interest in coding theory and cryptography. Permutation polynomials over finite fields have been completely classified up to degree 6, with "ad hoc" methods for every degree. In this talk, we present a general approach for classifying permutation rational functions of any degree that exploits a refined version of Chebotarev density theorem for function fields due to Kosters. We will show how to use the method to completely classify permutation rational functions and complete rational functions of degree 3. This is joint work with Giacomo Micheli. |
| 5:15pm - 7:00pm | SI(AG)^2 business meeting |
| vonRoll, Fabrikstr. 6, 001 | |
