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 | |
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Location: Unitobler, F007 30 seats, 59m^2 |
| Date: Tuesday, 09/Jul/2019 | |
| 10:00am - 12:00pm | MS123, part 1: Asymptotic phenomena in algebra and statistics |
| Unitobler, F007 | |
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10:00am - 12:00pm
Asymptotic phenomena in algebra and statistics Across several branches of mathematics, the following fundamental question arises: given a sequence of algebraic structures with maps between them, can the entire sequence be characterized by a finite segment? Here the maps are comprising symmetries of the objects as well as morphisms between them. An affirmative answer leads to a description of all structures by using finite data only. There is a growing body of work that establishes the desired finiteness result in varied contexts. Nevertheless, instances where stability is not well understood include:
The aim of the minisymposium is to build bridges between the varied mathematicians and the different areas investigating stability phenomena. We propose a two half-day minisymposium with 8 speakers total. The proposed speakers have all expressed interest in speaking at the symposium. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Strength and polynomial functors Define the rank of an infinite-by-infinite matrix A as the supremum of the ranks of all its finite-size submatrices. This can be finite, which implies that the matrix A is the product of an infinite-by-n and an n-by-infinite matrix. Or else it is infinite. In this case, the set of matrices that can be obtained from A by a finite number of row and column operations is Zariski-dense in the space of all matrices. Next, define the strength of a series f of fixed degree in infinitely many variables to be the minimal number of products of series of lower degree that sum up to f. Then f has a simpler description when its strength is finite. And when its strength is infinite, the set of series obtained from f by finitely many variable substitutions turns out to be dense in its ambient space. This dichotomy exists in a much more general setting: an element of the inverse limit V of a polynomial functor either is in the image of a polynomial transformation from a simpler functor or has a dense orbit in V. This talk is about the complexity measure on V that associates to an element the minimal polynomial functor from which it arises. The results are part of joint work with Jan Draisma, Rob Eggermont and Andrew Snowden. Asymptotics Proved by the Method of Cumulants The method of cumulants is closely related to the method of moments both being a classical tool to prove central limit theorems. Having a good bound on the cumulants of a sequence of random variables, one can deduce precise asymptotics for the distribution function of the properly scaled random variables, and it implies large and moderate deviations as well as so-called mod-Phi-convergence. As an example, we will study dependency graphs. FI-algebras: examples and counterexamples We link the literature on algebras with an action of the infinite symmetric group to the literature on FI-algebras by identifying mutually adjoint functors in both directions. I will discuss some interesting examples and counterexamples of Noetherian and non-Noetherian FI-algebras and modules over them. Finitely generated modules over some Noetherian FI-algebras are Noetherian, and in this setting we can describe free resolutions and Betti numbers, but for other FI-algebras this fails. This is joint work with Jan Draisma and Alexei Krasilnikov. Asymptotic behavior of chains of ideals with symmetry Chains of ideals in increasingly larger polynomial rings that are invariant under the action of symmetric groups arise in various contexts, including algebraic statistics and representation theory. In this talk, I will discuss the asymptotic behavior of some invariants of ideals in such chains, namely, the Krull dimension, the projective dimension, and the Castelnuovo-Mumford regularity. The Krull dimension is eventually a linear function whose slope can be described explicitly. We conjecture that the projective dimension and the Castelnuovo-Mumford regularity also grow eventually linearly, and provide linear bounds for these invariants. This is joint work with Uwe Nagel, Hop D. Nguyen, and Tim Römer. |
| 3:00pm - 5:00pm | MS123, part 2: Asymptotic phenomena in algebra and statistics |
| Unitobler, F007 | |
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3:00pm - 5:00pm
Asymptotic phenomena in algebra and statistics Across several branches of mathematics, the following fundamental question arises: given a sequence of algebraic structures with maps between them, can the entire sequence be characterized by a finite segment? Here the maps are comprising symmetries of the objects as well as morphisms between them. An affirmative answer leads to a description of all structures by using finite data only. There is a growing body of work that establishes the desired finiteness result in varied contexts. Nevertheless, instances where stability is not well understood include:
The aim of the minisymposium is to build bridges between the varied mathematicians and the different areas investigating stability phenomena. We propose a two half-day minisymposium with 8 speakers total. The proposed speakers have all expressed interest in speaking at the symposium. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Quantitative Properties of Ideals arising from Hierarchical Models We will discuss hierarchical models and certain toric ideals as a way of studying these objects in algebraic statistics. Some algebraic properties of these ideals will be described and a formula for the Krull dimension of the corresponding toric rings will be presented. One goal is to study the invariance properties of families of ideals arising from hierarchical models with varying parameters. We will present classes of examples where we have information about an equivariant Hilbert series. This is joint work with Uwe Nagel. Bounding degrees of generators for sequences of ideals By the celebrated Hilbert's basis theorem an ideal in a polynomial ring has a finite number of generators - in particular, there exists a bound on the degree of the generators. Varieties however often come to us in sequences and it may be highly nontrivial to establish a uniform degree bound. The questions one asks can have different flavour: one can ask for a set-, scheme- or ideal-theoretic description, an explicit or existential bound. We will report on several conjectures and theorems inspired by applied algebra, in particular, algebraic phylogenetics. Asymptotic Phenomena in the homology groups of graph configuration spaces A graph is a 1-dimensional compact, connected CW-complex. Given a graph G we define its n-fold configuration space UConf_n(G) to be the topological space of n distinct and unlabeled points on G. The study of the asymptotic behaviors of graph configuration spaces have taken two distinct paths in the literature. The first, and more classically flavored, involves fixing the graph G and increasing the number of points. In this case, Work of An, Drummond-Cole, and Knudsen, as well as independent work of the speaker, have shown that the homology groups can be equipped with the structure of finitely generated modules over a certain polynomial ring associated to the graph G. The second approach, appearing in work of Lutgehetmann, White, Proudfoot, and the speaker, involves fixing the number of points being configured and allowing the graph to vary in some regular way. In this case one once again recovers finite generation results for the homology groups, although describing what they are finitely generated over requires one to introduce concepts from the representation theory of categories. In this talk we will outline the state of the art with regards to both of these approaches, as well as how these two types of asymptotic stability can sometimes be related to one another. Mirror spaces and stability in the homology of Vandermonde varieties The level sets of the first d Newton power sums in R^k for some d ≤ k have been called Vandermonde varieties by Arnold and Giventhal. These varieties have a natural action of the symmetric group, which induces an action on their cohomology groups. By using a formula of Solomon we can study the decomposition of the resulting S_k-module and generalise some of the results obtained by Arnold and Giventhal on the homology modules of such varieties. These results in particular also yield some insight into the representational stability in the homology modules. (Based on joint works with Saugata Basu.)
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| Date: Wednesday, 10/Jul/2019 | |
| 10:00am - 12:00pm | MS200, part 1: From algebraic geometry to geometric topology: Crossroads on applications |
| Unitobler, F007 | |
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10:00am - 12:00pm
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) Momentum of vortex tangles by weighted area information A method based on the interpretation of classical linear and angular momentum of vortex dynamics in terms of weighted areas of projected graphs of filament tangles has been introduced to provide an accurate estimate of physical information when no analytical description is available [1,2]. The method is implemented here for defects governed by the Gross-Pitaevskii equation [3]. New results based on direct application of this method to determine the linear momentum associated with interacting vortex rings, links and knots are presented and discussed in detail. The method can be easily extended and adapted to more complex systems, providing a useful tool for real-time diagnostics of dynamical properties of networks of filamentary structures. This is joint work with Simone Zuccher (U. Verona). [1] Ricca, R.L. (2008) Momenta of a vortex tangle by structural complexity analysis. Physica D 237, 2223-2227. [2] Ricca, R.L. (2013) Impulse of vortex knots from diagram projections. In Topological Fluid Dynamics: Theory and Applications (ed. H.K. Moffatt et al.), pp. 21-28. Procedia IUTAM 7, Elsevier. [3] Zuccher, S. & Ricca, R.L. (2019) Momentum of vortex tangles by weighted area information. Submitted. Alexandrov spaces and topological data analysis Alexandrov spaces (with curvature bounded below) are metric generalizations of complete Riemannian manifolds with a uniform lower sectional curvature bound. In this talk I will discuss the geometric and topological properties of these metric spaces and how they arise in the context of topological data analysis. Geometrical and topological analysis of chromosome conformation capture data Despite the impressive development of methods to analyze Chromosome Conformation Capture (CCC) data, the topology of any genome still remains unknown. The output of a CCC experiment is a matrix of pairwise contact probabilities between genomic loci from which a map of distances, called distance map, can be obtained. In this work we use distance geometry and random knotting arguments to derive some rigorous results for the interpretation of distance maps. In particular we provide a rigorous characterization of the distance map of a knot and of some of its symmetries. We end by presenting a key result that shows that in the presence of noise the topology of a chromosome cannot be recovered from a distance map. Joint work with: K. Ishihara, K. Lamb, M. Pouokam, K. Shimokawa, and M. Vazquez Asymptotic behavior of the homology of random polyominoes In this talk we study the rate of growth of the expectation of the number of holes (the rank of the first homology group) in a polyomino with uniform and percolation distributions. We prove the existence of linear bounds for the expected number of holes of a polyomino with respect to both the uniform and percolation distributions. Furthermore, we exhibit particular constants for the upper and lower bounds in the uniform distribution case. This results can be extended, using the same techniques, to other polyforms and higher dimensions. |
| 3:00pm - 5:00pm | MS200, part 2: 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) Privileged topologies of self-assembling molecular knots The self-assembly of objects with a set of desired properties is a major goal of material science and physics. A particularly challenging problem is that of self-assembling structures with a target topology. Here we show by computer simulation that one may design the geometry of string-like rigid patchy templates to promote their efficient and reproducible self-assembly into a selected repertoire of non-planar closed folds including several knots. In particular, by controlling the template geometry, we can direct the assembly process so as to strongly favour the formation of constructs tied in trefoil or pentafoil, or even of more exotic knot types. A systematic survey reveals that these "privileged", addresable topologies are rare, as they account for only a minute fraction of the simplest knot types. This knot discovery strategy has recently allowed for predicting complex target topologies [1,2,3], some of which have been realized experimentally [4,5].
References
1) G. Polles et al. "Self-assembling knots of controlled topology by designing the geometry of patchy templates", Nature Communications, 2015
self-assembly video demonstration available at this link
2) G. Polles et al. "Optimal self-assembly of linked constructs and catenanes via spatial confinement", Macro Letters (2016)
3) M. Marenda, et al. “Discovering privileged topologies of molecular knots with self-assembling models", Nature Communications, 2018
4) J. Danon et al. "Braiding a molecular knot with eight crossings.", Science (2017)
5) Kim et al. "Coordination-driven self-assembly of a molecular knot comprising sixteen crossings", Angew. Chem. Int. Ed. (2018)
Why are there knots in proteins? There are now more than 1700 protein chains that are known to contain some type of topological knot in their polypeptide chains in the protein structure databank. Although this number is small relative to the total number of protein structures solved, it is remarkably high given the fact that for decades it was thought impossible for a protein chain to fold and thread in such a way as to create a knotted structure. There are four different types of knotted protein structures that contain 31, 41, 52 and 61 knots and over the past 15 years there has been an increasing number of experimental and computational studies on these systems. The folding pathways of knotted proteins have been studied in some detail, however, the focus of this talk is to address the fundamental question “Why are there knots in proteins?” It is known that once formed, knotted protein structures tend to be conserved by nature. This, in addition to the fact that, at least for some deeply knotted proteins, their folding rates are slow compared with many unknotted proteins, has led to the hypothesis that there are some properties of knotted proteins that are different from unknotted ones, and that this had resulted in some evolutionary advantage over faster folding unknotted structures. In this talk, I will review the evidence for and against this theory. In particular, how a knot within a protein chain may affect the thermodynamic, kinetic, mechanical and cellular (resistance to degradation) stability of the protein will be discussed. The study of 2-stratifolds as models for applications (Part 1) In physics, the morphological structure of granular samples in mechanical equilibrium has been modeled by graphs and analyzed by using the first Betti number (S. Ardanza-Trevijano et al.). in TDA, persistent homology is used to study graphs arising from sampling point clouds. Graphs can be viewed as 1-dimensional stratified spaces and possibly more information could be obtained by modeling with 2-dimensional stratified spaces, since these provide more topological invariants. For example, in Physics, the study of singularities of soap films (E. Goldstein et al.) and in Chemistry and Biology, the study of cyclo-octane energy landscapes (S. Martin et al.) led to 2-dimensional complexes that consist of unions of 2-manifolds intersecting along a curve. These 2-complexes are special cases of 2-dimensional stratified spaces. In TDA, techniques have been developed (Bendich et al.) for organizing, visualizing and analyzing point cloud data that has been sampled from or near a 2-dimensional stratified space. There is no topological classification of these spaces. A systematic study of 2-dimensional stratifiedspaces without boundary curves or 0-dimensional singularities, the 2-stratifolds, was begun by W. Heil, F.J. González-Acuña and the speaker. In this talk, we wil explore 2-stratifolds with trivlal fundamental group. The study of 2-stratifolds as models for applications (Part 2) In physics, the morphological structure of granular samples in mechanical equilibrium has been modeled by graphs and analyzed by using the first Betti number (S. Ardanza-Trevijano et al.). in TDA, persistent homology is used to study graphs arising from sampling point clouds. Graphs can be viewed as 1-dimensional stratified spaces and possibly more information could be obtained by modeling with 2-dimensional stratified spaces, since these provide more topological invariants. For example, in Physics, the study of singularities of soap films (E. Goldstein et al.) and in Chemistry and Biology, the study of cyclo-octane energy landscapes (S. Martin et al.) led to 2-dimensional complexes that consist of unions of 2-manifolds intersecting along a curve. These 2-complexes are special cases of 2-dimensional stratified spaces. In TDA, techniques have been developed (Bendich et al.) for organizing, visualizing and analyzing point cloud data that has been sampled from or near a 2-dimensional stratified space. There is no topological classification of these spaces. A systematic study of 2-dimensional stratified spaces without boundary curves or 0-dimensional singularities, the 2-stratifolds, was begun by J.C. Gómez-Larrañaga, F.J. González-Acuña and the speaker. In this talk, we will describe an efficient algorithm on the labeled graph of a 2-stratifold that determines its homotopy type and an efficient algorithm that determines if its fundamental group is infinite cyclic. Also, we will discuss embeddings of 2-stratifolds as 3-manifold spines and talk about the solvability of the word problem for 2-stratifold groups. |
| Date: Thursday, 11/Jul/2019 | |
| 10:00am - 12:00pm | MS181, part 1: 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) Asymptotic normality for the Volume of the nodal set for Kostlan-Shub-Smale polynomial systems We study the asymptotic variance and the CLT, as the degree goes to infinity, of the normalized volume of the zero set of a rectangular Kostlan-Shub-Smale random polynomial system. Euler characteristic and bicovariogram of random excursions The Euler characteristic of a planar compact smooth set is a privileged tool of geometric analysis as it is a local quantity carrying information on various macroscopic features of the set topology. We indicate here how to compute it directly from the covariograms of the set, i.e. the volumes of the intersection of the set with translated copies of itself. This approach works under hypotheses of $mathcal{C}^{1,1}$ regularity, and can be applied to excursions of $mathcal{C}^{1,1}$ bivariate functions. In the realm of random sets or fields, this identity gives the mean Euler characteristic in terms of the third order marginals. Bayesian approach to filament estimation with a latent Gaussian random field model Persistent Betti numbers are a major tool in persistent homology, a subfield of topological data analysis. Many tools in persistent homology rely on the properties of persistent Betti numbers considered as a two-dimensional stochastic process. We survey existing results and present expansions of these results to larger classes of underlying sampling schemes and to the multivariate case. On the universality of roots of random polynomials A classic question in random polynomial theory is to determine whether the distribution of the roots depends on the distribution of the random coefficients. We will explore this question both for algebraic and trigonometric models and point out some important differences between the two regimes. |
| 3:00pm - 5:00pm | MS200, part 3: 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) Time-reversal homotopical properties of concurrent systems Directed topology was introduced as a model of concurrent programs, where the flow of time is described by distinguishing certain paths in the topological space representing such a program. Algebraic invariants which respect this directedness have been introduced to classify directed spaces. In this talk we study the properties of such invariants with respect to the reversal of the flow of time in directed spaces. Known invariants, natural homotopy and homology, have been shown to be unchanged under this time-reversal (as first noticed by e.g. K. Hess and L. Fajstrup). We show that these can be equipped with additional algebraic structure witnessing this reversal. Specifically, when applied to a directed space and to its reversal, we show that these refined invariants yield dual objects.These refined invariants, natural systems with a composition pairing, enjoy lots of interesting properties, as first noticed by Timothy Porter. We further refine natural homotopy by introducing a notion of relative directed homotopy and showing the existence of a long exact sequence of natural homotopy systems.Joint work with Philippe Malbos and Cameron Calk. Efficient computation of multiparameter persistent homology We will present an efficient implementation of an algorithm to compute multiparametric persistent homology. The algorithm uses algebraic techniques and was originally proposed by Chacholski, Scolamiero, and Vaccarino. During the talk, we will explain the different reformulations of the definition of multiparametric persistence that give rise to the algorithm we corrected and implemented. This is joint work with Oliver Gäfvert (KTH) and Nina Otter (UCLA). Classification of Streamline Topologies for Hamiltonian vector fields and its applications to Topological Flow Data Analysis We have developed a classification theory for structurally stable Hamiltonian vector fields on multiply connected planar domains in the presence of a uniform flow, which is a model of two-dimensional incompressible fluid flows. The theory enables us to assign a unique sequence of letters and a tree structure, called maximal words, and Reeb graphs, to every topological streamline structure of the Hamiltonian vector fields. They are intuitively interpretable to those who are not familiar with mathematics. An automatic conversion algorithm is now available as a computer software, and it is thus applicable to massive flow pattern data obtained by numerical simulations and/or physical measurements in fluid science, engineering and medical studies. By extracting global topological information from flow data, one is expected to figure out latent knowledge that are not recognized by experts in those fields so far. In addition, we have also developed a mathematical theory describing all possible global transitions of streamline topologies, without exceptions, through marginal structurally unstable Hamiltonian vector fields in terms of the changes of the sequence of letters. By simply comparing them, we predict the change of global flow patterns that could possibly happen in future. Robot motion planning and equivariant cohomology The task of creating software AI for autonomous robots has an interesting topological aspect. A motion planning algorithm for a mechanical system can be represented by a section of a specific fibration and the complexity of such a section is measured by numerical invariants TC(X) and TC_r(X). Here X denotes the configuration space of the system and r>1 is an integer. I will describe recent results about computing the numbers TC(X) and TC_r(X) in the case when the space X is aspherical, i.e. has vanishing higher homotopy groups. The talk will include some results obtained jointly with S. Mescher, M. Grant, J.Oprea and G. Lupton. |
| Date: Friday, 12/Jul/2019 | |
| 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. |
| 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. |
| Date: Saturday, 13/Jul/2019 | |
| 10:00am - 12:00pm | MS171, part 1: Grassmann and flag manifolds in data analysis |
| Unitobler, F007 | |
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10:00am - 12:00pm
Grassmann and flag manifolds in data analysis A number of applications in large scale geometric data analysis can be expressed in terms of an optimization problem on a Grassmann or flag manifold.The solution of the optimization problem helps one to understand structure underlying a data set for the purposes such as classification, feature selection, and anomaly detection. For example, given a collection of points on a Grassmann manifold, one could imagine finding a Schubert variety of best fit corresponds to minimizing some function on the flag variety parameterizing the given class of Schubert varieties. A number of different algorithms that exist for points in a linear space have analogues for points in a Grassmann or flag manifold such as clustering, endmember detection, self organized mappings, etc. The purpose of this minisymposium is to bring together researchers who share a common interest in algorithms and techniques involving Grassmann and Flag varieties applied to problems in 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) PCA Integral Invariants for Manifold Learning Local integral invariants at scale based on principal component analysis have recently been shown to provide estimators of curvature information at every point of a manifold. These can thus be applied to perform manifold learning from point clouds sampled from embedded Riemannian manifolds, and to inform optimization and geometry processing methods in arbitrary dimension, e.g. feature detection at scale. In particular, regular curves in Euclidean space are completely characterized up to rigid motion by the EVD of the PCA covariance matrix at every point, which reproduces the Frenet-Serret apparatus asymptotically with scale. We will also introduce the general result that establishes a dictionary in the limit between these statistical integral invariants and the classical differential-geometric curvature, in the form of a generalized Darboux-Ricci frame, providing an algorithm to estimate the Riemann curvature tensor for embedded manifolds of arbitrary dimension. Subspace Averaging in Multi-Sensor Array Processing In this talk we address the problem of averaging on the Grassmann manifold, with special emphasis placed on the question of estimating the dimension of the average. The solution to this problem provides a simple order fitting rule based on thresholding the eigenvalues of the average projection matrix, and thus it is free of penalty terms or other tuning parameters commonly used by other information-theoretic criteria for model order estimation such as the minimum description length (MDL) criterion. The proposed rule appears to be particularly well suited to problems involving high-dimensional data and low sample support, such as the determination of the number of sources with a large array of sensors: the so-called source enumeration problem. The talk will discuss subspace averaging (SA) for source enumeration under the challenging conditions of: i) large uniform arrays with few snapshots (the small sample regime), and ii) non-white or spatially correlated noises with arbitrary correlation. As illustrated by some simulation examples, SA provides a very robust method of enumerating sources in these challenging scenarios. Variations on Multidimensional Scaling for non-Euclidean Distance Matrices Classical multidimensional scaling takes as input a distance matrix and extracts a configuration of points in a low dimensional Euclidean space whose Euclidean distances best approximate the input data. In this talk we put a twist on the classical algorithm by changing the geometry of the embedding space. Specifically, we show that pseudo-Euclidean coordinates are the natural choice when working with non-Euclidean distance data. Examples are furnished by Lie groups and homogeneous manifolds, which display characteristic signatures in pseudo-Euclidean space. |
| 3:00pm - 5:00pm | MS171, part 2: Grassmann and flag manifolds in data analysis |
| Unitobler, F007 | |
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3:00pm - 5:00pm
Grassmann and flag manifolds in data analysis A number of applications in large scale geometric data analysis can be expressed in terms of an optimization problem on a Grassmann or flag manifold.The solution of the optimization problem helps one to understand structure underlying a data set for the purposes such as classification, feature selection, and anomaly detection. For example, given a collection of points on a Grassmann manifold, one could imagine finding a Schubert variety of best fit corresponds to minimizing some function on the flag variety parameterizing the given class of Schubert varieties. A number of different algorithms that exist for points in a linear space have analogues for points in a Grassmann or flag manifold such as clustering, endmember detection, self organized mappings, etc The purpose of this minisymposium is to bring together researchers who share a common interest in algorithms and techniques involving Grassmann and Flag varieties applied to problems in 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) A dual subgradient approach to computing an optimal rank Grassmannian circumcenter This talk concerns the circumcenter of a collection of linear subspaces. When the subspaces are k-dimensional subspaces of n-dimensional Euclidean space, this can be cast as an infinity-norm minimization problem on a Grassmann manifold, Gr(k,n). For subspaces of different dimension, the setting becomes a disjoint union of Grassmannians rather than a single manifold, and the problem is no longer well-defined. However, natural geometric maps exist between these manifolds with a well-defined notion of distance for the images of the subspaces under the mappings. Solving the initial problem in this context leads to a candidate circumcenter on each of the constituent manifolds, but does not inherently provide intuition about which candidate is the best representation of the data. Additionally, the solutions of different rank are generally not nested so a deflationary approach will not suffice, and the problem must be solved independently on each manifold. In this talk we propose and solve an optimization problem parametrized by the rank of the circumcenter. The solution can be computed approximately using a dual subgradient algorithm. By scaling the objective and penalizing the information lost by the rank-k circumcenter, we jointly recover an optimal dimension, k*, and a central subspace on Gr(k*,n) that best represents the correlated subspace of the data. Low Rank Representations of Matrices using Nuclear Norm Heuristics The connection between the entries of an Euclidean distance matrix and the nuclear norm of the matrix in the positive semidefinite cone given by the one to one correspondence between the two cones. In the case when the Euclidean distance matrix is the distance matrix for a complete k-partite graph, the nuclear norm of the associated positive semidefinite matrix can be evaluated in terms of the second elementary symmetric polynomial evaluated at the partition. For k-partite graphs the maximum value of the nuclear norm of the associated positive semidefinite matrix is attained in the situation when we have equal number of vertices in each set of the partition. This result can be used to determinea lower bound on the chromatic number of the graph. Grassmann Tangent-Bundle Means Applications of geometric data analysis often involve producing collections of subspaces, such as illumination spaces for digital imagery. For a given collection of subspaces, a natural task is to find the mean of the collection. A robust suite of algorithms has been developed to generate mean representatives for a collection of subspaces of xed dimension, or equivalently, a collection of points on a particular Grassmann manifold. These representatives include the flag mean, the normal mean, and the Karcher mean. In this talk, we catalogue the types of means and present comparative heuristics for the suite of mean representatives. We respond to, and at times, challenge, the conclusions of a recent paper outlining various means built via tangent-bundle maps on the Grassmann manifold. |
