# Colloquia/Spring2014

## Spring 2014

date speaker title host(s)
Mon, Jan 6, 4PM Aaron Lauda (USC) An introduction to diagrammatic categorification Caldararu
Wed, Jan 8, 4PM Karin Melnick (Maryland) Normal forms for local flows on parabolic geometries Kent
Jan 10, 4PM Yen Do (Yale) Convergence of Fourier series and multilinear analysis Denissov
Mon, Jan 13, 4pm Yi Wang (Stanford) Isoperimetric Inequality and Q-curvature Viaclovsky
Wen, Jan 15, 4pm Wei Xiang (University of Oxford) Conservation Laws and Shock Waves Bolotin
Fri, Jan 17, 2:25PM, VV901 Adrianna Gillman (Dartmouth) Fast direct solvers for linear partial differential equations Thiffeault
Thu, Jan 23, 2:25, VV901 Mykhaylo Shkolnikov (Berkeley) Intertwinings, wave equations and growth models Seppalainen
Jan 24 Yaniv Plan (Michigan) Low-dimensionality in mathematical signal processing Thiffeault
Jan 31 Urbashi Mitra (USC) Underwater Networks: A Convergence of Communications, Control and Sensing Gurevich
Feb 7 David Treumann (Boston College) Functoriality, Smith theory, and the Brauer homomorphism Street
Feb 14 Alexander Karp (Columbia Teacher's College) History of Mathematics Education as a Research Field and as Magistra Vitae Kiselev
Feb 21 Svetlana Jitomirskaya (UC-Irvine) Analytic quasiperiodic cocycles Kiselev
Feb 28 Michael Shelley (Courant) Mathematical models of soft active materials Spagnolie
March 7 Steve Zelditch (Northwestern) Shapes and sizes of eigenfunctions Seeger
March 14 Richard Schwartz (Brown) The projective heat map on pentagons Mari-Beffa
March 21 Spring Break No Colloquium
March 26, 7pm, WID Tadashi Tokieda (Cambridge) Toy models Thiffeault (C4 von Neumann Public Lecture)
March 28 Cancelled
April 4 Matthew Kahle (OSU) Recent progress in random topology Dymarz
April 11 Risi Kondor (Chicago) Multiresolution Matrix Factorization Gurevich
April 18 (Wasow Lecture) Christopher Sogge (Johns Hopkins) Focal points and sup-norms of eigenfunctions Seeger
April 25 Charles Doran(University of Alberta) The Mathematics of Supersymmetry: Graphs, Codes, and Super-Curves Song
Tuesday, April 29 (Distinguished Lecture) David Eisenbud(Berkeley) Matrix factorizations old and new, I Erman
Wednesday, April 30 (Distinguished Lecture) David Eisenbud(Berkeley) Matrix factorizations old and new, II Erman
May 2 Lek-Heng Lim (Chicago) Hypermatrices Boston
May 9 Rachel Ward (UT Austin) Sampling theorems for efficient dimensionality reduction and sparse recovery WIMAW

## 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.