Measure-Theoretic Dvoretzky Theorem and Applications to Data Science
- Probability,Uploaded Videos ( 1451 Views )SEPC 2021 in honor of Elizabeth Meckes. Slides from the talks and more information are available <a href="https://services.math.duke.edu/~rtd/SEPC2021/SEPC2021.html">at this link (here).</a>
Dick Hain : Hecke actions on loops and periods of iterated itegrals of modular forms
- Number Theory ( 314 Views )Hecke operators act on many invariants associated to modular curves and their generalizations. For example, they act on modular forms and on cohomology groups of modular curves. In each of these cases, they generate a semi-simple, commutative algebra. In the first part of this talk, I will recall (in friendly, elementary, geometric terms) what Hecke operators are and how they act on the standard invariants. I will then show that they also act on loops in modular curves (aka, conjugacy classes in modular groups). In this case, the Hecke operators generate a non-commutative subalgebra of the vector space generated by the conjugacy classes, which leads to a very natural non-commutative generalization of the classical Hecke algebra. In the second part of the talk will discuss why one might want do construct such a Hecke action. As a prelude to this, I will explain why this Hecke action commutes with the natural action of the absolute Galois group after taking profinite completions. And, in the unlikely event that I have sufficient time, I will also explain how (after taking the appropriate completion) this Hecke action is also compatible with Hodge theory.
Luca Di Cerbo : Extended Graph Manifolds, Dehn Fillings, and Einstein Metrics
- Geometry and Topology ( 301 Views )In this talk, I will present some new topological obstructions for solving the Einstein equations (in Riemannian signature) on a large class of closed four-manifolds. Finally, I will show that complex-hyperbolic Einstein Dehn filling compactification cannot possibly performed in dimension four. This is in striking contrast with the real-hyperbolic case, and it answers (negatively) a fifteen years old question of Michael Anderson. If time permits, I will conclude with some tantalizing open problems both in dimension four and in higher dimensions. Part of this work is joint with M. Golla (Universit\’e de Nantes).
David Schwein : Recent progress on the formal degree conjecture
- Number Theory ( 288 Views )The local Langlands correspondence is a dictionary between representations of two kinds of groups: reductive p-adic groups (such as the general linear group) and the absolute Galois groups of p-adic fields. One entry in the dictionary is a conjectural formula of Hiraga, Ichino, and Ikeda for the size of a representation of a p-adic group, its "formal degree", in terms of the corresponding representation of a Galois group. In this talk, after reviewing the broad shape of p-adic representation theory, I'll explain why the conjecture is true for almost all supercuspidals, the fundamental building blocks of the subject.
Blair Sullivan : Finding a role for structural graph theory in real-world network analysis
- Undergraduate Seminars ( 282 Views )Network science is a rapidly growing interdisciplinary field with methods and applications drawn from across the natural, social, and information sciences. Perhaps surprisingly, very few approaches use techniques from the rich literature of structural graph theory. In this talk, we discuss some first steps towards integrating what have been predominantly theoretical results into tools for scalable network analysis. Tree-like structures arise extensively in network science - for example, hierarchical structures in biology, hyperbolic routing in the internet, and core-periphery behavior in social networks. As such, this talk focuses on ways to use tree decompositions, key combinatorial objects used in graph minor theory, in tandem with k-cores and Gromov hyperbolicity to provide structural characterization of and improve inference on complex networks. We also discuss new algorithms using tree decompositions to enable scalable solution of certain graph optimization problems in a high performance computing environment.
For more information, see http://www.ornl.gov/~b7r/
Jordan S. Ellenberg : Stability and Representations
- Gergen Lectures ( 278 Views )The notion of stability --speaking loosely, "sometimes an infinite sequence of vector spaces eventually starts being constant" -- appears in many branches of mathematics, perhaps most notably topology, where Harer's theorem about the stability of the homology of mapping class groups has driven decades of work. Some natural sequences of vector spaces are evidently NOT eventually constant: for instance, the space Q_n of quadratic polynomials in n variables has dimension (1/2)n(n-1), so gets larger and larger as n goes to infinity. On the other hand, Q_n carries an action of the symmetric group S_n by permutation of coordinates. We will discuss a new framework which allows us to speak meaningfully about what it means for a sequence of representations of S_n to be stable. It turns out that the structures we define are ubiquitous, appearing in topology (e.g. homology groups of configuration spaces and of moduli spaces of curves) algebraic combinatorics (e.g. the graded pieces of diagonal coinvariant algebras) and algebraic geometry (e.g. spaces of polynomials on discriminant and rank varieties.) We prove, for instance, that all these sequences of vector spaces have dimension which is eventually a polynomial in n.
Luis Caffarelli : Degenerate ellipticity and the porous media equation
- Gergen Lectures ( 273 Views )In the first lecture I will give a brief discussion of local and non local diffusion and degenerate ellipticity and different local and non local models for compressible flows in porous media.
In the second and third lectures I will discuss some properties of the
(infinitesimal) porous media equation, a non local in space model and
equations with memory.
Miklos Racz : From trees to seeds: on the inference of the seed from large random trees
- Presentations ( 266 Views )I will discuss the influence of the seed in models of randomly growing trees; in particular, I will focus on the preferential attachment and uniform attachment models. In both of these models, different seeds lead to different distributions of limiting trees from a total variation point of view. I will discuss the differences and similarities in proving this for the two models. This is based on joint work with Sebastien Bubeck, Ronen Eldan, and Elchanan Mossel.
Nils Bruin : Prym varieties of genus four curves
- Algebraic Geometry ( 257 Views )Many arithmetic properties of hyperbolic curves become apparent from embeddings into abelian varieties, in particular their Jacobians. For special curves, particularly those that arise as unramified double covers of another curve (of genus g), the Jacobian variety itself is decomposable. This leads to Prym varieties. These are principally polarized abelian varieties of dimension g-1. Having an explicit description of these varieties is an essential ingredient in many computational methods. We discuss an explicit construction for g equal to 4. This is joint work with Emre Can Sertoz.
Joseph Rabinoff : From Diophantine equations to p-adic analytic geometry
- Presentations ( 254 Views )A Diophantine equation is a polynomial equation in several variables, generally with integer coefficients, like x3 + y3 = z3. Provably finding all integer solutions of a Diophantine equation is a storied mathematical problem that is easy to state and notoriously difficult to solve. The method of Chabauty--Coleman is one particularly successful technique for ruling out extraneous solutions of a certain class of Diophantine equations. The method is p-adic in nature, and involves producing p-adic analytic functions that vanish on all integer-valued solutions. I will discuss work with Katz and Zureick-Brown on finding uniform bounds on the number of rational points on a curve of fixed genus, defined over a number field, subject to a (conjecturally weak) restriction on its Jacobian. The same technique also makes progress on the uniform Manin-Mumford conjecture on the size of torsion packets on curves of fixed genus.
Manish Mishra : Self-dual cuspidal representations
- Number Theory ( 249 Views )Let F be a non-archimedean local field (such as ℚ_p). The Langlands philosophy says that the arithmetic of F is intimately related to the category R(G) of smooth complex representations of G(F) where G denotes a reductive F-group (for example the general linear group). The building blocks of R(G) are the "supercuspidal" representations of G(F). I will define this term in the talk. The category R(G) comes equipped with an involution - the "contragradient" or the "dual". The supercuspidal representations of G(F) which are self-dual are of considerable interest in the subject. In this talk, I will talk about a joint work with Jeff Adler about the existence of supercuspidals and self-dual supercuspidals. Specifically, we show that G(F) always admits supercuspidal representations. Under some mild hypotheses on G, we determine precisely when G(F) admits self-dual supercuspidal representations. These results are obtained from analogous results for finite reductive groups which I will also talk about.
Simon Brendle : Curvature and topology of manifolds
- String Theory ( 242 Views )The interplay between curvature and topology of Riemannian manifolds is among the most fundamental questions in differential geometry. Over the past century, various different approaches have been developed to attack these types of problems. This includes variational techniques based on geodesics and minimal surfaces, as well as the Ricci flow approach pioneered by Richard Hamilton. In this lecture, I will give an overview of the subject, focusing on the case of positive curvature.
Tony Feng : Steenrod operations and the Artin-Tate pairing
- Number Theory ( 240 Views )In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations.
Joseph Spivey : A How-To Guide to Building Your Very Own Moduli Spaces (they make such great gifts)
- Graduate/Faculty Seminar ( 238 Views )I'll be talking about how to construct the moduli space for genus g Riemann surfaces with r boundary components. I'll draw lots of pictures and focus a lot of attention on genus 1 Riemann surfaces with 1 boundary component. As an application, I'll probably talk about H^1(SL2(Z)) with coefficients in various representations--and the correspondence to modular forms (briefly, and without a whole lot of proofs).
Wuchen Li : Mean-Field Games for Scalable Computation and Diverse Applications
- Applied Math and Analysis ( 235 Views )Mean field games (MFGs) study strategic decision-making in large populations where individual players interact via specific mean-field quantities. They have recently gained enormous popularity as powerful research tools with vast applications. For example, the Nash equilibrium of MFGs forms a pair of PDEs, which connects and extends variational optimal transport problems. This talk will present recent progress in this direction, focusing on computational MFG and engineering applications in robotics path planning, pandemics control, and Bayesian/AI sampling algorithms. This is based on joint work with the MURI team led by Stanley Osher (UCLA).
Marty Golubitsky : Patterns of Synchrony: From Animal Gaits to Binocular Rivalry
- CGTP Group Meeting Seminar ( 233 Views )This talk will discuss previous work on quadrupedal gaits and recent work on a generalized model for binocular rivalry proposed by Hugh Wilson. Both applications show how rigid phase-shift synchrony in periodic solutions of coupled systems of differential equations can help understand high level collective behavior in the nervous system.
Mainak Patel : Temporal Binding Emerges as a Rapid and Accurate Encoding Tool Within a Network Model of the Locust Antennal Lobe
- String Theory ( 224 Views )The locust olfactory system interfaces with the external world through antennal receptor neurons (ORNs), which represent odors in a distributed, combinatorial manner. ORN axons bundle together to form the antennal nerve, which relays sensory information centrally to the antennal lobe (AL). Within the AL, an odor produces a stimulus-specific temporal progression of neuronal spiking, inspiring the hypothesis that the AL encodes odors through dynamically evolving ensembles of active cells. Such a coding strategy, however, requires higher olfactory centers to integrate a prolonged dynamic profile of AL signals prior to stimulus assessment, a process that is likely to be slow and inconsistent with the generation of quick behavioral responses. Our modeling work has led us to propose an alternate hypothesis: the dynamical interplay of fast and slow inhibition within the locust AL induces transient correlations in the spiking activity of an odor-dependent neural subset, giving rise to a temporal binding code and allowing rapid stimulus detection by downstream elements.
James Bremer : Improved methods for discretizing integral operators
- Presentations ( 218 Views )Integral equation methods are frequently used in the numerical solution of elliptic boundary value problems. After giving a brief overview of the advantages and disadvantages of such methods vis-a-vis more direct techniques like finite element methods, I will discuss two problems which arise in integral equation methods. In both cases, I take a contrarian position. The first is the discretization of integral operators on singular domains (e.g., surfaces with edges and curves with corners). The consensus opinion holds that integral equations given on such domains are exceedingly difficult to discretize and that sophisticated analysis, often specific to a particular boundary value problem, is required. I will explain that, in fact, the efficient solution of a broad class of such problems can be effected using an elementary approach. Exterior scattering problems given on planar domains with tens of thousands of corner points can be solved to 12 digit accuracy on my two year old desktop computer in a matter of hours. The second problem I will discuss is the evaluation of the singular integrals which arise form the discretization of weakly singular integral operators given on surfaces. Exponentially convergent algorithms for evaluating these integrals have been described in the literature and it is widely regarded as a "solved" problem. I will explain why this is not so and describe an approach which yields only algebraic convergence, but nonetheless performs better in practice than standard exponentially convergent methods.
Richard Hain : The Lie Algebra of the Mapping Class Group, Part 2
- Geometry and Topology ( 218 Views )In this talk I will review the construction of the Lie algebra associated to the mapping class group of a (possibly decorated) surface and explain how this generalizes the Lie algebra associated to the pure braid group. I will also explain the analogue of the KZ-equation in the mapping class group case. In the second talk I will discuss filtrations of this Lie algebra associated to curve systems on the surface and their relation to handlebody groups.
Johan Brauer : The Stabilisation of Equilibria in Evolutionary Game Dynamics through Mutation
- Probability ( 208 Views )The multi-population replicator dynamics (RD) can be considered a dynamic approach to the study of multi-player games, where it was shown to be related to Cross-learning, as well as of systems of co-evolving populations. However, not all of its equilibria are Nash equilibria (NE) of the underlying game, and neither convergence to an NE nor convergence in general are guaranteed. Although interior equilibria are guaranteed to be NE, no interior equilibrium can be asymptotically stable in the multi-population RD, resulting, e.g., in cyclic orbits around a single interior NE. We report on our investigation of a new notion of equilibria of RD, called mutation limits, which is based on the inclusion of a naturally arising, simple form of mutation, but is invariant under the specific choice of mutation parameters. We prove the existence of such mutation limits for a large range of games, and consider an interesting subclass, that of attracting mutation limits. Attracting mutation limits are approximated by asymptotically stable equilibria of the (mutation-)perturbed RD, and hence, offer an approximate dynamic solution of the underlying game, especially if the original dynamic has no asymptotically stable equilibria. Therefore, the presence of mutation will indeed stabilise the system in certain cases and make attracting mutation limits near-attainable. Furthermore, the relevance of attracting mutation limits as a game theoretic equilibrium concept is emphasised by the relation of (mutation-)perturbed RD to the Q-learning algorithm in the context of multi-agent reinforcement learning. However, in contrast to the guaranteed existence of mutation limits, attracting mutation limits do not exist in all games, raising the question of their characterization.
Anirban Basak : Ferromagnetic Ising Measures on Large Locally Tree-Like Graphs
- Probability ( 204 Views )We consider the ferromagnetic Ising model on a sequence of graphs $G_n$ converging locally weakly to a rooted random tree. Generalizing [Montanari, Mossel, and Sly (2012)], under an appropriate continuity" property, we show that the Ising measures on these graphs converge locally weakly to a measure, which is obtained by first picking a random tree, and then the symmetric mixture of Ising measures with + and - boundary conditions on that tree. Under the extra assumptions that $G_n$ are edge-expanders, we show that the local weak limit of the Ising measures conditioned on positive magnetization, is the Ising measure with + boundary condition on the limiting tree. The continuity" property holds except possibly for countably many choices of $\beta$, which for limiting trees of minimum degree at least three, are all within certain explicitly specified compact interval. We further show the edge-expander property for (most of) the configuration model graphs corresponding to limiting (multi-type) Galton Watson trees. This talk is based on a joint work with Amir Dembo.
Brian Rider : Log-gases and Tracy-Widom laws
- Probability ( 201 Views )The now ubiquitous Tracy-Widom laws were first discovered in the context of the Gaussian Orthogonal, Unitary, and Symplectic Ensembles (G{O/U/S}E) of random matrix theory. The latter may be viewed as logarithmic gases with quadratic (Gaussian) potential at three special inverses temperatures (beta=1,2,4). A few years back, Jose Ramirez, Balint Virag, and I showed that that one obtains generalizations of the Tracy-Widom laws at all inverse temperatures (beta>0), though still for quadratic potentials. I'll explain how similar ideas (and considerably more labor) extends the result to general potential, general temperature log-gases. This is joint work with Manjunath Krishnapur and Balint Virag.
Lenka Zdeborova : Network Dismantling
- Nonlinear and Complex Systems ( 200 Views )Many systems of interest can be represented by a network of nodes connected by edges. In many circumstances the existence of a giant component is necessary for the network to fulfill its function. Motivated by the need to understand optimal attack strategies, optimal spread of information or immunization policies, we study the network dismantling problem, i.e. the search of a minimal set of nodes whose removal leaves the network broken into components of sub-extensive size. Building on the statistical mechanics perspective we compute the size of the optimal dismantling set for random networks, propose an efficient dismantling algorithm for general networks that outperforms by a large margin existing strategies, and we provide various insights about the problem.
Jim Isenberg : Construcing solutions of the Einstein constraint equations
- Geometry and Topology ( 195 Views )The first step in finding a spacetime solution to the Einstein gravitational field equations via the inital value formulation is to construct initial data which satisfy the Einstein constraint equations. There are three ways of carrying out this construction which have been found to be useful: the conformal and conformal thin sandwich methods, the gluing techniques, and the quasi-spherical approaches. We describe each of these, we discuss their advantages and disadvantages, we outline some of their recent successful applications, and we present some of the outstanding questions remaining to be solved from each of these perspectives.
Suncica Canic : Mathematical modeling for cardiovascular stenting
- Applied Math and Analysis ( 193 Views )The speaker will talk about several projects that are taking place in an interdisciplinary endeavor between the researchers in the Mathematics Department at the University of Houston, the Texas Heart Institute, Baylor College of Medicine, the Mathematics Department at the University of Zagreb, and the Mathematics Department of the University of Lyon 1. The projects are related to non-surgical treatment of aortic abdominal aneurysm and coronary artery disease using endovascular prostheses called stents and stent-grafts. Through a collaboration between mathematicians, cardiovascular specialists and engineers we have developed a novel mathematical model to study blood flow in compliant (viscoelastic) arteries treated with stents and stent-grafts. The mathematical tools used in the derivation of the effective, reduced equations utilize asymptotic analysis and homogenization methods for porous media flows. The existence of a unique solution to the resulting fluid-structure interaction model is obtained by using novel techniques to study systems of mixed, hyperbolic-parabolic type. A numerical method, based on the finite element approach, was developed, and numerical solutions were compared with the experimental measurements. Experimental measurements based on ultrasound and Doppler methods were performed at the Cardiovascular Research Laboratory located at the Texas Heart Institute. Excellent agreement between the experiment and the numerical solution was obtained. This year marks a giant step forward in the development of medical devices and in the development of the partnership between mathematics and medicine: the FDA (the United States Food and Drug Administration) is getting ready to, for the first time, require mathematical modeling and numerical simulations to be used in the development of peripheral vascular devices. The speaker acknowledges research support from the NSF, NIH, and Texas Higher Education Board, and donations from Medtronic Inc. and Kent Elastomer Inc.