Colloquia/Spring2014
Spring 2015
Go to next semester, Fall 2015.
Abstracts
January 12: Botong Wang (Notre Dame)
Cohomology jump loci of algebraic varieties
In the moduli spaces of vector bundles (or local systems), cohomology jump loci are the algebraic sets where certain cohomology group has prescribed dimension. We will discuss some arithmetic and deformation theoretic aspects of cohomology jump loci. If time permits, we will also talk about some applications in algebraic statistics.
January 14: Jayadev Athreya (UIUC)
Counting points for random (and not-so-random) geometric structures
We describe a philosophy of how certain counting problems can be studied by methods of probability theory and dynamics on appropriate moduli spaces. We focus on two particular cases:
(1) Counting for Right-Angled Billiards: understanding the dynamics on and volumes of moduli spaces of meromorphic quadratic differentials yields interesting universality phenomenon for billiards in polygons with interior angles integer multiples of 90 degrees. This is joint work with A. Eskin and A. Zorich
(2) Counting for almost every quadratic form: understanding the geometry of a random lattice allows yields striking diophantine and counting results for typical (in the sense of measure) quadratic (and other) forms. This is joint work with G. A. Margulis.
January 15: Chi Li (Stony Brook)
On Kahler-Einstein metrics and K-stability
The existence of Kahler-Einstein metrics on Kahler manifolds is a basic problem in complex differential geometry. This problem has connections to other fields: complex algebraic geometry, partial differential equations and several complex variables. I will discuss the existence of Kahler-Einstein metrics on Fano manifolds and its relation to K-stability. I will mainly focus on the analytic part of the theory, discuss how to solve the related complex Monge-Ampere equations and provide concrete examples in both smooth and conical settings. If time permits, I will also say something about the algebraic part of the theory, including the study of K-stability using the Minimal Model Program (joint with Chenyang Xu) and the existence of proper moduli space of smoothable K-polystable Fano varieties (joint with Xiaowei Wang and Chenyang Xu).
January 21: Jun Kitagawa (Toronto)
Regularity theory for generated Jacobian equations: from optimal transport to geometric optics
Equations of Monge-Ampere type arise in numerous contexts, and solutions often exhibit very subtle qualitative and quantitative properties; this is owing to the highly nonlinear nature of the equation, and its degeneracy (in the sense of ellipticity). Motivated by an example from geometric optics, I will talk about the class of Generated Jacobian Equations; recently introduced by Trudinger, this class also encompasses, for example, optimal transport, the Minkowski problem, and the classical Monge-Ampere equation. I will present a new regularity result for weak solutions of these equations, which is new even in the case of equations arising from near-field reflector problems (of interest from a physical and practical point of view). This talk is based on joint works with N. Guillen.
January 23: Nicolas Addington (Duke)
Recent developments in rationality of cubic 4-folds
The question of which cubic 4-folds are rational is one of the foremost open problems in algebraic geometry. I'll start by explaining what this means and why it's interesting; then I'll discuss three approaches to solving it (including one developed in the last year), my own work relating the three approaches to one another, and the troubles that have befallen each approach.
January 26: Minh Binh Tran (CAM)
Nonlinear approximation theory for the homogeneous Boltzmann equation
A challenging problem in solving the Boltzmann equation numerically is that the velocity space is approximated by a finite region. Therefore, most methods are based on a truncation technique and the computational cost is then very high if the velocity domain is large. Moreover, sometimes, non-physical conditions have to be imposed on the equation in order to keep the velocity domain bounded. In this talk, we introduce the first nonlinear approximation theory for the Boltzmann equation. Our nonlinear wavelet approximation is non-truncated and based on a nonlinear, adaptive spectral method associated with a new wavelet filtering technique and a new formulation of the equation. The approximation is proved to converge and perfectly preserve most of the properties of the homogeneous Boltzmann equation. It could also be considered as a general framework for approximating kinetic integral equations.
February 2: Afonso Bandeira (Princeton)
Tightness of convex relaxations for certain inverse problems on graphs
Many maximum likelihood estimation problems are known to be intractable in the worst case. A common approach is to consider convex relaxations of the maximum likelihood estimator (MLE), and relaxations based on semidefinite programming (SDP) are among the most popular. We will focus our attention on a certain class of graph-based inverse problems and show a couple of remarkable phenomena.
In some instances of these problems (such as community detection under the stochastic block model) the solution to the SDP matches the ground truth parameters (i.e. achieves exact recovery) for information theoretically optimal regimes. This is established using new nonasymptotic bounds for the spectral norm of random matrices with independent entries.
On other instances of these problems (such as angular synchronization), the MLE itself tends to not coincide with the ground truth (although maintaining favorable statistical properties). Remarkably, these relaxations are often still tight (meaning that the solution of the SDP matches the MLE). For angular synchronization we can understand this behavior by analyzing the solutions of certain randomized Grothendieck problems. However, for many other problems, such as the multireference alignment problem in signal processing, this remains a fascinating open problem.
February 6: Morris Hirsch (UC Berkeley and UW Madison)
Fixed points of Lie transformation group, and zeros of Lie algebras of vector fields
The following questions will be considered:
When a connected Lie group G acts effectively on a manifold M, what general conditions on G, M and the action ensure that the action has a fixed point?
If g is a Lie algebra of vector fields on M, what general conditions on g and M ensure that g has a zero?
Old and new results will be discussed. For example:
Theorem: If G is nilpotent and M is a compact surface of nonzero Euler characteristic, there is a fixed point.
Theorem: Suppose G is supersoluble and M is as above. Then every analytic action of G on M has a fixed point, but this is false for continuous actions, and for groups that are merely solvable.
Theorem: Suppose M is a real or complex manifold that is 2-dimensional over the ground field, and g is a Lie algebra of analytic vector fields on M. Assume some element X in g spans a 1-dimensional ideal. If the zero set K of X is compact and the Poincar'e-Hopf index of X at K is nonzero, then g vanishes at some point of K.
No special knowledge of Lie groups will be assumed.
February 13: Mihai Putinar (UC Santa Barbara)
Quillen’s property of real algebraic varieties
A famous observation discovered by Fejer and Riesz a century ago is the quintessential algebraic component of every spectral decomposition result. It asserts that every non-negative polynomial on the unit circle is a hermitian square. About half a century ago, Quillen proved that a positive polynomial on an odd dimensional sphere is a sum of hermitian squares. Fact independently rediscovered much later by D’Angelo and Catlin, respectively Athavale. The main subject of the talk will be: on which real algebraic sub varieties of [math]\displaystyle{ \mathbb{C}^n }[/math] is Quillen theorem valid? An interlace between real algebraic geometry, quantization techniques and complex hermitian geometry will provide an answer to the above question, and more. Based a recent work with Claus Scheiderer and John D’Angelo.
February 20: David Zureick-Brown (Emory University)
Diophantine and tropical geometry
Diophantine geometry is the study of integral solutions to a polynomial equation. For instance, for integers [math]\displaystyle{ a,b,c \geq 2 }[/math] satisfying [math]\displaystyle{ \tfrac1a + \tfrac1b + \tfrac1c \gt 1 }[/math], Darmon and Granville proved that the individual generalized Fermat equation [math]\displaystyle{ x^a + y^b = z^c }[/math] has only finitely many coprime integer solutions. Conjecturally something stronger is true: for [math]\displaystyle{ a,b,c \geq 3 }[/math] there are no non-trivial solutions.
I'll discuss various other Diophantine problems, with a focus on the underlying intuition and conjectural framework. I will especially focus on the uniformity conjecture, and will explain new ideas from tropical geometry and our recent partial proof of the uniformity conjecture.
Monday February 23: Jayadev Athreya (UIUC)
The Erdos-Szusz-Turan distribution for equivariant point processes
We generalize a problem of Erdos-Szusz-Turan on diophantine approximation to a variety of contexts, and use homogeneous dynamics to compute an associated probability distribution on the integers.
February 27: Allan Greenleaf (University of Rochester)
Erdos-Falconer Configuration problems
In discrete geometry, there is a large collection of problems due to Erdos and various coauthors starting in the 1940s, which have the following general form: Given a large finite set P of N points in d-dimensional Euclidean space, and a geometric configuration (a line segment of a given length, a triangle with given angles or a given area, etc.), is there a lower bound on how many times that configuration must occur among the points of P? Relatedly, is there an upper bound on the number of times any single configuration can occur? One of the most celebrated problems of this type, the Erdos distinct distances problem in the plane, was essentially solved in 2010 by Guth and Katz, but for many problems of this type only partial results are known.
In continuous geometry, there are analogous problems due to Falconer and others. Here, one looks for results that say that if a set A is large enough (in terms of a lower bound on its Hausdorff dimension, say), then the set of configurations of a given type generated by the points of A is large (has positive measure, say). I will describe work on Falconer-type problems using some techniques from harmonic analysis, including estimate for multilinear operators. In some cases, these results can be discretized to obtain at least partial results on Erdos-type problems.
March 6: Larry Guth (MIT)
Introduction to incidence geometry
Incidence geometry is a branch of combinatorics that studies the possible intersection patterns of lines, circles, and other simple shapes. For example, suppose that we have a set of L lines in the plane. An r-rich point is a point that lies in at least r of these lines. For a given L, r, how many r-rich points can we make? This is a typical question in the field, and there are many variations. What if we replace lines with circles? What happens in higher dimensions? We will give an introduction to this field, describing some of the important results, tools, and open problems.
We will discuss two important tools used in the area. One tool is to apply topology to the problem. This tool allows us to prove results in R^2 that are stronger than what happens over finite fields. The second tool is to look for algebraic structure in the problem by studying low-degree polynomials that vanish on the points we are studying. We will also discuss some of the (many) open problems in the field and try to describe the nature of the difficulties in approaching them.
March 13: Cameron Gordon (UT-Austin)
Left-orderability and 3-manifold groups
The fundamental group is a more or less complete invariant of a 3-dimensional manifold. We will discuss how the purely algebraic property of this group being left-orderable is related to two other aspects of 3-dimensional topology, one geometric-topological and the other essentially analytic.
March 20: Aaron Naber (Northwestern)
Regularity and New Directions in Einstein Manifolds
In this talk we give an overview of recent developments and new directions of manifolds which satisfy the Einstein equation Rc=cg, or more generally just manifolds with bounded Ricci curvature |Rc|<C. We will discuss the solution of the codimension four conjecture, which roughly says that Gromov-Hausdorff limits (M^n_i,g_i)->(X,d) of manifolds with bounded Ricci curvature are smooth away from a set of codimension four. In a very different direction, in this lecture we will also explain how Einstein manifolds may be characterized by the behavior of the analysis on path space P(M) of the manifold. That is, we will see a Riemannian manifold is Einstein if and only if certain gradient estimates for functions on P(M) hold. One can view this as an infinite dimensional generalization of the Bakry-Emery estimates.
March 27 11am B239: Ilya Kossovskiy (University of Vienna)
On Poincare's "Probleme local"
In this talk, we describe a result giving a complete solution to the old question of Poincare on the possible dimensions of the automorphism group of a real-analytic hypersurface in two-dimensional complex space. As the main tool, we introduce the so-called CR (Cauchy-Riemann manifolds) - DS (Dynamical Systems) technique. This technique suggests to replace a real hypersurface with certain degeneracies of the CR-structure by an appropriate dynamical system, and then study mappings and symmetries of the initial real hypersurface accordingly. It turns out that symmetries of the singular differential equation associated with the initial real hypersurface are much easier to study than that of the real hypersurface, and in this way we obtain the solution for the problem of Poincare.
This work is joint with Rasul Shafikov.
March 27: Kent Orr (Indiana University)
The Isomorphism Problem for metabelian groups
Perhaps the most fundamental outstanding problem in algorithmic group theory, the Isomorphism Problem for metabelian groups remains a mystery.
I present an introduction to this problem intended to be accessible to graduate students. In collaboration with Gilbert Baumslag and Roman Mikhailov, I present a new approach to this ancient problem which potentially connects to algebraic geometry, cohomology of groups, number theory, Gromov's view of groups as geometric objects, and a fundamental algebraic construction developed for and motivated by the topology of knots and links.
April 10: Jasmine Foo (University of Minnesota)
Evolutionary dynamics in spatially structured populations
In this talk I will present results on a model of spatial evolution on a lattice, motivated by the process of carcinogenesis from healthy epithelial tissue. Cancer often arises through a sequence of genetic alterations. Each of these alterations may confer a fitness advantage to the cell, resulting in a clonal expansion. To model this we will consider a generalization of the biased voter process which incorporates successive mutations modulating fitness, which is interpreted as the bias in the classical process. Under this model we will investigate questions regarding the rate of spread and accumulation of mutations, and study the dynamics of spatial heterogeneity in these evolving populations.
April 17: Kay Kirkpatrick (UIUC)
Non-normal asymptotics of the mean-field Heisenberg model
I will discuss spin models of magnets and superconductors, with spins in the circle (XY model) and in the sphere (Heisenberg model), that exhibit interesting phase transitions. I will discuss work with Elizabeth Meckes on the mean-field Heisenberg model and its non-normal behavior at the phase transition. There is much that is still mysterious about these models: I’ll mention work in progress with Tayyab Nawaz and Leslie Ross.
April 24: Marianna Csornyei (University of Chicago)
Tangents of curves and differentiability of functions
One of the classical theorems of Lebesgue tells us that Lipschitz functions on the real line are differentiable almost everywhere. We study possible generalisations of this theorem and some interesting geometric corollaries.
May 1: Bianca Viray (University of Washington)
Cryptography and abelian varieties with extra endomorphisms
Cryptosystems that are based on the discrete logarithm problem require an instantiation of a group of large prime order. For example, for certain primes p, one could use a subgroup of the group of units in F_p, the F_p-points on an elliptic curve, or the F_p-points on the Jacobian of a genus 2 curve. (The group of integers modulo p results in an insecure cryptosystem.) This raises the question of how to construct a curve C of genus 1 or 2 over F_p whose Jacobian has N F_p-points, where N is a large prime. Perhaps surprisingly, this problem is related to curves over the complex numbers whose Jacobian has "extra" endomorphisms, i.e., Jacobians with complex multiplication. This in turn relates to a different counting problem for curves over F_p. We will give an overview of this connection and describe joint work with Lauter which answers this counting problem. This talk will be suitable for a general mathematical audience.
May 8: Marcus Roper (UCLA)
Nonlinear flow in micro- and myco-fluidics
Simulations and asymptotic analysis of particle movement in microfluidic channels is made easier by the fact that the equations of motion for particles moving in a microfluidic pipe are linear and reversible. We discuss two examples of interesting phenomena that that emerge from the breakdown of linearity: (1) Inertial microfluidic devices use high speed flow to focus and segregate particles of different sizes. However, there is no consensus on how focusing forces vary with particle size and flow speed, or even on the number of focusing positions that exist within channels having different shapes. I’ll explain how simulations and experiments may have been misinterpreted and develop a partly numerical, partly asymptotic theory for particle focusing. (2) Fungi form microfluidic networks to transport nutrients, fluid and nuclei. Surprisingly these networks are anticongestive – the greater the density of traffic on a fungal freeway, the faster the traffic travels. I’ll explain how this anticongestive property arises using a kinetic model of the motion of nuclei on a fungal freeway, and speculate about why the fungus might benefit from organizing its flow in this way.
Abstracts
January 6: Aaron Lauda (USC)
An introduction to diagrammatic categorification
Categorification seeks to reveal a hidden layer in mathematical structures. Often the resulting structures can be combinatorially complex objects making them difficult to study. One method of overcoming this difficulty, that has proven very successful, is to encode the categorification into a diagrammatic calculus that makes computations simple and intuitive.
In this talk I will review some of the original considerations that led to the categorification philosophy. We will examine how the diagrammatic perspective has helped to produce new categorifications having profound applications to algebra, representation theory, and low-dimensional topology.
January 8: Karin Melnick (Maryland)
Normal forms for local flows on parabolic geometries
The exponential map in Riemannian geometry conjugates the differential of an isometry at a point with the action of the isometry near the point. It thus provides a linear normal form for all isometries fixing a point. Conformal transformations are not linearizable in general. I will discuss a suite of normal forms theorems in conformal geometry and, more generally, for parabolic geometries, a rich family of geometric structures of which conformal, projective, and CR structures are examples.
January 10, 4PM: Yen Do (Yale)
Convergence of Fourier series and multilinear analysis
Almost everywhere convergence of the Fourier series of square integrable functions was first proved by Lennart Carleson in 1966, and the proof has lead to deep developments in various multilinear settings. In this talk I would like to introduce a brief history of the subject and sketch some recent developments, some of these involve my joint works with collaborators.
Mon, January 13: Yi Wang (Stanford)
Isoperimetric Inequality and Q-curvature
A well-known question in differential geometry is to prove the isoperimetric inequality under intrinsic curvature conditions. In dimension 2, the isoperimetric inequality is controlled by the integral of the positive part of the Gaussian curvature. In my recent work, I prove that on simply connected conformally flat manifolds of higher dimensions, the role of the Gaussian curvature can be replaced by the Branson's Q-curvature. The isoperimetric inequality is valid if the integral of the Q-curvature is below a sharp threshold. Moreover, the isoperimetric constant depends only on the integrals of the Q-curvature. The proof relies on the theory of $A_p$ weights in harmonic analysis.
January 15: Wei Xiang (University of Oxford)
Conservation Laws and Shock Waves
The study of continuum physics gave birth to the theory of quasilinear systems in divergence form, commonly called conservation laws. In this talk, conservation laws, the Euler equations, and the definition of the corresponding weak solutions will be introduced first. Then a short history of the studying of conservation laws and shock waves will be given. Finally I would like to present two of our current research projects. One is on the mathematical analysis of shock diffraction by convex cornered wedges, and the other one is on the validation of weakly nonlinear geometric optics for entropy solutions of nonlinear hyperbolic systems of conservation laws.
Fri, Jan 17, 2:25PM, VV901 Adrianna Gillman (Dartmouth) Fast direct solvers for linear partial differential equations
Fri, Jan 17: Adrianna Gillman (Dartmouth)
Fast direct solvers for linear partial differential equations
The cost of solving a large linear system often determines what can and cannot be modeled computationally in many areas of science and engineering. Unlike Gaussian elimination which scales cubically with the respect to the number of unknowns, fast direct solvers construct an inverse of a linear in system with a cost that scales linearly or nearly linearly. The fast direct solvers presented in this talk are designed for the linear systems arising from the discretization of linear partial differential equations. These methods are more robust, versatile and stable than iterative schemes. Since an inverse is computed, additional right-hand sides can be processed rapidly. The talk will give the audience a brief introduction to the core ideas, an overview of recent advancements, and it will conclude with a sampling of challenging application examples including the scattering of waves.
Thur, Jan 23: Mykhaylo Shkolnikov (Berkeley)
Intertwinings, wave equations and growth models
We will discuss a general theory of intertwined diffusion processes of any dimension. Intertwined processes arise in many different contexts in probability theory, most notably in the study of random matrices, random polymers and path decompositions of Brownian motion. Recently, they turned out to be also closely related to hyperbolic partial differential equations, symmetric polynomials and the corresponding random growth models. The talk will be devoted to these recent developments which also shed new light on some beautiful old examples of intertwinings. Based on joint works with Vadim Gorin and Soumik Pal.
Jan 24: Yaniv Plan (Michigan)
Low-dimensionality in mathematical signal processing
Natural images tend to be compressible, i.e., the amount of information needed to encode an image is small. This conciseness of information -- in other words, low dimensionality of the signal -- is found throughout a plethora of applications ranging from MRI to quantum state tomography. It is natural to ask: can the number of measurements needed to determine a signal be comparable with the information content? We explore this question under modern models of low-dimensionality and measurement acquisition.
Thur, Jan 30: Urbashi Mitra (USC)
Underwater Networks: A Convergence of Communications, Control and Sensing
The oceans cover 71% of the earth’s surface and represent one of the least explored frontiers, yet the oceans are integral to climate regulation, nutrient production, oil retrieval and transportation. Future scientific and technological efforts to achieve better understanding of oceans and water-related applications will rely heavily on our ability to communicate reliably between instruments, vehicles (manned and unmanned), human operators, platforms and sensors of all types. Underwater acoustic communication techniques have not reached the same maturity as those for terrestrial radio communications and present some unique opportunities for new developments in information and communication theories. Key features of underwater acoustic communication channels are examined: slow speed of propagation, significant delay spreads, sparse multi-path, time-variation and range-dependent available bandwidth. Another unique feature of underwater networks is that the cost of communication, sensing and control are often comparable resulting in new tradeoffs between these activities. We examine some new results (with implications wider than underwater systems) in channel identifiability, communicating over channels with state and cooperative game theory motivated by the underwater network application.
Feb 7: David Treumann (Boston College)
Functoriality, Smith theory, and the Brauer homomorphism
Smith theory is a technique for relating the mod p homologies of X and of the fixed points of X by an automorphism of order p. I will discuss how, in the setting of locally symmetric spaces, it provides an easy method (no trace formula) for lifting mod p automorphic forms from G^{sigma} to G, where G is an arithmetic group and sigma is an automorphism of G of order p. This lift is compatible with Hecke actions via an analog of the Brauer homomorphism from modular representation theory, and is often compatible with a homomorphism of L-groups on the Galois side. The talk is based on joint work with Akshay Venkatesh. I hope understanding the talk will require less number theory background than understanding the abstract.
Feb 14: Alexander Karp (Columbia Teacher's College)
History of Mathematics Education as a Research Field and as Magistra Vitae
The presentation will be based on the experience of putting together and editing the Handbook on the History of Mathematics Education, which will be published by Springer in the near future. This volume, which was prepared by a large group of researchers from different countries, contains the first systematic account of the history of the development of mathematics education in the whole world (and not just in some particular country or region). The editing of such a book gave rise to thoughts about the methodology of research in this field, and also about what constitutes an object of such research. These are the thoughts that the presenter intends to share with his audience. From them, it is natural to pass to an analysis of the current situation and how it might develop.
Feb 21: Svetlana Jitomirskaya (UC-Irvine)
Analytic quasiperiodic cocycles
Analytic quasiperiodic matrix cocycles is a simple dynamical system, where analytic and dynamical properties are related in an unexpected and remarkable way. We will focus on this relation, leading to a new approach to the proof of joint continuity of Lyapunov exponents in frequency and cocycle, at irrational frequency, first proved for SL(2,C) cocycles in Bourgain-Jitom., 2002. The approach is powerful enough to handle singular and multidimensional cocycles, thus establishing the above continuity in full generality. This has important consequences including a dense open version of Bochi-Viana theorem in this setting, with a completely different underlying mechanism of the proof. A large part of the talk is a report on a joint work with A. Avila and C. Sadel.
February 28: Michael Shelley (Courant)
Mathematical models of soft active materials
Soft materials that have an "active" microstructure are important examples of so-called active matter. Examples include suspensions of motile microorganisms or particles, "active gels" made up of actin and myosin, and suspensions of microtubules cross-linked by motile motor-proteins. These nonequilibrium materials can have unique mechanical properties and organization, show spontaneous activity-driven flows, and are part of self-assembled structures such as the cellular cortex and mitotic spindle. I will discuss the nature and modeling of these materials, focusing on fluids driven by "active stresses" generated by swimming, motor-protein activity, and surface tension gradients. Amusingly, the latter reveals a new class of fluid flow singularities and an unexpected connection to the Keller-Segel equation.
March 7: Steve Zelditch (Northwestern)
Shapes and sizes of eigenfunction
Eigenfunctions of the Laplacian (or Schroedinger operators) arise as stationary states in quantum mechanics. They are not apriori geometric objects but we would like to relate the nodal (zero) sets and Lp norms of eigenfunctions to the geometry of geometrics. I will explain what is known (and unknown) and norms and nodal sets of eigenfunctions. No prior knowledge of quantum mechanics is assumed.
March 14: Richard Schwartz (Brown)
The projective heat map on pentagons
In this talk I'll describe several maps defined on the space of polygons. These maps are described in terms of simple straight-line constructions, and are therefore natural with respect to projective geometry. One of them, the pentagram map, is now known to be a discrete completely integrable system. I'll concentrate on a variant of the pentagram map, which behaves somewhat like heat flow on convex polygons but which does crazy things to non-convex polygons. I'll sketch a computer-assisted analysis of what happens for pentagons. I'll illustrate the talk with computer demos.
April 4: Matthew Kahle (OSU)
"Recent progress in random topology"
The study of random topological spaces: manifolds, simplicial complexes, knots, groups, has received a lot of attention in recent years. This talk will mostly focus on random simplicial complexes, and especially on a certain kind of topological phase transition, where the probability that that a certain homology group is trivial passes from 0 to 1 within a narrow window. The archetypal result in this area is the Erdős–Rényi theorem, which characterizes the threshold edge probability where the random graph becomes connected.
One recent breakthrough has been in the application of "Garland's method", which allows one to prove homology-vanishing theorems by showing that certain Laplacians have large spectral gaps. This reduces problems in random topology to understanding eigenvalues of certain random matrices, and the method has been surprisingly successful.
This talk is intended for a broad mathematical audience, and I will not assume any particular prerequisites in probability or topology. Part of this is joint work with Christopher Hoffman and Elliot Paquette.
April 11: Risi Kondor (Chicago)
Multiresolution Matrix Factorization
Matrices that appear in modern data analysis and machine learning problems often exhibit complex hierarchical structure, which goes beyond what can be uncovered by traditional linear algebra tools, such as eigendecomposition. In this talk I describe a new notion of matrix factorization inspired by multiresolution analysis that can capture structure in matrices at multiple different scales. The resulting Multiresolution Matrix Factorizations (MMFs) not only provide a wavelet basis for sparse approximation, but can also be used for matrix compression and as a prior for matrix completion. The work presented in this talk is joint with Nedelina Teneva and Vikas Garg.
April 18: Christopher Sogge (Johns Hopkins)
Focal points and sup-norms of eigenfunctions
If (M,g) is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes saturating sup-norm estimates. The condition is that there exists a self-focal point x_0\in M for the geodesic flow at which the associated Perron-Frobenius operator U: L^2(S_{x_0}^*M) \to L^2(S_{x_0}^*M) has a nontrivial invariant function. The proof is based on von Neumann's ergodic theorem and stationary phase. This is joint work with Steve Zelditch.
April 25: Charles Doran (University of Alberta)
The Mathematics of Supersymmetry: Graphs, Codes, and Super-Curves
In physics, supersymmetry is a pairing between bosons and fermions appearing in theories of subatomic particles. One may study supersymmetry mathematically by using Adinkras, which are graphs with vertices representing the particles in a supersymmetric theory and edges corresponding to the supersymmetry pairings. In combinatorial terms, Adinkras are N-regular, edge N-colored bipartite graphs with signs assigned to the edges and heights assigned to the vertices, subject to certain conditions. We will see how to capture some of the structure of an Adinkra using binary linear error-correcting codes, and all of it using a very special case of a geometric construction due to Grothendieck. The talk is designed to be accessible to an undergraduate audience.
April 29 and 30: David Eisenbud (University of California, Berkeley and MSRI)
Matrix factorizations old and new
You cannot factor f=xy-z^2 nontrivially as a product of power series, but you can factor f times a 2x2 identity matrix as the product of the matrices
| x | z | and | y | -z | ||
| z | y | -z | x. |
It turns out that any power series of order at least has a "matrix factorization" in this sense, and that this is the key to understanding the simplest infinite free resolutions, as I proved in the 1980s. Such matrix factorizations have since proven useful in many contexts. Recently Irena Peeva and I have discovered what I believe is the natural extension of this idea to systems of polynomials called complete intersections. I'll explain some of the old theory and sketch the new development.
The first talk will be aimed at a general audience and the second talk will cover some of the recent advances.
May 2: Lek-Heng Lim (Chicago)
Hypermatrices.
This talk is intended for those who, like the speaker, have at some point
wondered whether there is a theory of three- or higher-dimensional
matrices that parallels matrix theory. We will explain why a d-dimensional
hypermatrix is related to but not quite the same as an order-d tensor.
We discuss how notions like rank, norm, determinant, eigen and singular values may be generalized to hypermatrices. We will see that, far from being artificial constructs, these notions have appeared naturally in a wide range of applications and can be enormously useful. We will examine several examples, highlighting three from the speaker's recent work: (i) rank of 3-hypermatrices and blind source separation in signal processing, (ii) positive definiteness of 6-hypermatrices and self-concordance in convex optimization, (iii) nuclear norm of 3-hypermatrices and bipartite separability in quantum computing.
(i) is joint work with Pierre Comon and (iii) is joint work with Shmuel Friedland
May 9: Rachel Ward (UT Austin)
Sampling theorems for efficient dimensionality reduction and sparse recovery.
Embedding high-dimensional data sets into subspaces of much lower dimension is important for reducing storage cost and speeding up computation in several applications, including numerical linear algebra, manifold learning, and theoretical computer science. Moreover, central to the relatively new field of compressive sensing, if the original data set is known to be sparsely representable in a given basis, then it is possible to efficiently 'invert’ a random dimension-reducing map to recover the high-dimensional data via e.g. l1-minimization. We will survey recent results in these areas, and then show how near-equivalences between fundamental concepts such as restricted isometries and Johnson-Lindenstrauss embeddings can be used to leverage results in one domain and apply to another. Finally, we discuss how these and other recent results for structured random matrices can be used to derive sampling strategies in various settings, from low-rank matrix completion to function interpolation.