# Mathematics Colloquium

All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.

## Fall 2013

date speaker title host(s)
Sept 6 Matt Baker (Georgia Institute of Technology) Riemann-Roch for Graphs and Applications Ellenberg
Sept 13 Uri Andrews (University of Wisconsin) A hop, skip, and a jump through the degrees of relative provability
Sept 20 Valerio Toledano Laredo (Northeastern) Flat connections and quantum groups Gurevich
Wed, Sept 25, 2:30PM in B139 Ayelet Lindenstrauss (Indiana University) Taylor Series in Homotopy Theory Meyer
Wed, Sept 25 (LAA lecture) Jim Demmel (Berkeley) Communication-Avoiding Algorithms for Linear Algebra and Beyond Gurevich
Thurs, Sept 26 (LAA lecture, Joint with Applied Algebra Seminar) Jim Demmel (Berkeley) Implementing Communication-Avoiding Algorithms Gurevich
Sept 27 (LAA lecture) Jim Demmel (Berkeley) Communication Lower Bounds and Optimal Algorithms for Programs that Reference Arrays Gurevich
Oct 4 Frank Sottile (Texas A&M) Galois groups of Schubert problems Caldararu
Oct 11 Amie Wilkinson (Chicago) Robust mechanisms for chaos WIMAW (Cladek)
Tues, Oct 15, 4PM (Distinguished Lecture) Alexei Borodin (MIT) Integrable probability I Valko
Wed, Oct 16, 2:30PM (Distinguished Lecture) Alexei Borodin (MIT) Integrable probability II Valko
Oct 18 No colloquium due to the distinguished lecture
Oct 25 Paul Garrett (Minnesota) Boundary-value problems, generalized functions, and zeros of zeta functions Gurevich
Nov 1 Allison Lewko (Columbia University) On sets of large doubling, Lambda(4) sets, and error-correcting codes Stovall
Nov 8 Tim Riley (Cornell) Hydra groups Dymarz
Nov 15 and later Reserved Street
Nov 22 Tianling Jin (University of Chicago) Solutions of some Monge-Ampere equations with degeneracy or singularities. Bolotin
Mon, Nov 25, 4PM Lin Lin (Lawrence Berkeley National Lab) Fast algorithms for electronic structure analysis Jin
Tue, Nov 26, 4PM, B139 Clinton Conley (Cornell) Descriptive set-theoretic graph theory Lempp
Mon, Dec 2, 4PM Simon Marshall (Northwestern) Semiclassical estimates for eigenfunctions on locally symmetric spaces Denissov
Wed, Dec 4, 4PM Steven Sam (Berkeley) Free Resolutions and Symmetry Boston
Fri, Dec 6 Paul Hand (MIT) Simplifications of the lifting approach for quadratic signal recovery problems Thiffeault
Fri, Dec. 6 and Sat Dec. 7 Conference in honor of Dick Askey
Mon, Dec. 9, 4pm, VV B239 Jacob Bedrossian (Courant Institute) Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations Bolotin
Wed, Dec 11, 4PM Lu Wang (Johns Hopkins) Rigidity of Self-shrinkers of Mean Curvature Flow Viaclovsky
Fri, Dec. 13, 2:25pm, VV 901 Chanwoo Kim (Cambridge) Regularity of the Boltzmann equation in convex domains Bolotin
Tues, Dec 17, 4PM Perla Sousi (Cambridge) The effect of drift on the volume of the Wiener sausage Seppalainen
Wed, Dec 18, 4PM Dustin Cartwright (Yale) Tropical Complexes Gurevich

## 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) Gurevich
Feb 7 David Treumann (Boston College) Street
Feb 14 Alexander Karp (Columbia Teacher's College) History of Mathematics Education as a Research Field and as Magistra Vitae Kiselev
Feb 21
Feb 28 Michael Shelley (Courant) Spagnolie
March 7 Steve Zelditch (Northwestern) Seeger
March 14
March 21 Spring Break No Colloquium
March 28 Michael Lacey (GA Tech) The Two Weight Inequality for the Hilbert Transform Street
April 4 Richard Schwartz (Brown) Mari-Beffa
April 11 Risi Kondor (Chicago) Gurevich
April 18 (Wasow Lecture) Christopher Sogge (Johns Hopkins) Seeger
April 25 Charles Doran(University of Alberta) Song
Monday, April 28 (Distinguished Lecture) David Eisenbud(Berkeley) A mystery concerning algebraic plane curves Maxim
Tuesday, April 29 (Distinguished Lecture) David Eisenbud(Berkeley) Matrix factorizations old and new Maxim
Wednesday, April 30 (Distinguished Lecture) David Eisenbud(Berkeley) Easy solution of polynomial equations over finite fields Maxim
May 2 Lek-Heng Lim (Chicago) Boston
May 9 Rachel Ward (UT Austin) WIMAW

## Abstracts

### Sep 6: Matt Baker (GA Tech)

Riemann-Roch for Graphs and Applications

We will begin by formulating the Riemann-Roch theorem for graphs due to the speaker and Norine. We will then describe some refinements and applications. Refinements include a Riemann-Roch theorem for tropical curves, proved by Gathmann-Kerber and Mikhalkin-Zharkov, and a Riemann-Roch theorem for metrized complexes of curves, proved by Amini and the speaker. Applications include a new proof of the Brill-Noether theorem in algebraic geometry (work of by Cools-Draisma-Payne-Robeva), a "volume-theoretic proof" of Kirchhoff's Matrix-Tree Theorem (work of An, Kuperberg, Shokrieh, and the speaker), and a new Chabauty-Coleman style bound for the number of rational points on an algebraic curve over the rationals (work of Katz and Zureick-Brown).

### Sep 13: Uri Andrews (UW-Madison)

A hop, skip, and a jump through the degrees of relative provability

The topic of this talk arises from two directions. On the one hand, Gödel's incompleteness theorem tell us that given any sufficiently strong, consistent, effectively axiomatizable theory T for first-order arithmetic, there is a statement that is true but not provable in T. On the other hand, over the past seventy years, a number of researchers studying witnessing functions for various combinatorial statements have realized the importance of fast-growing functions and the fact that their totality is often not provable over a given sufficiently strong, consistent, effectively axiomatizable theory T for first-order arithmetic (e.g. the Paris-Harrington and the Kirby-Paris theorems).

I will talk about the structure induced by giving the order (for a fixed T) of relative provability for totality of algorithms. That is, for algorithms describing functions f and g, we say f ≤ g if T along with the totality of g suffices to prove the totality of f. It turns out that this structure is rich, and encodes many facets of the nature of provability over sufficiently strong, consistent, effectively axiomatizable theories for first-order arithmetic. (Work joint with Mingzhong Cai, David Diamondstone, Steffen Lempp, and Joseph S. Miller.)

### Sep 20: Valerio Toledano Laredo (Northeastern)

Flat connections and quantum groups

Quantum groups are natural deformations of the Lie algebra of nxn matrices, and more generally of semisimple Lie algebras. They first arose in the mid eighties in the study of solvable models in statistical mechanics.

I will explain how these algebraic objects can serve as natural receptacles for the (transcendental) monodromy of flat connections arising from representation theory.

These connections exist in rational, trigonometric and elliptic forms, and lead to quantum groups of increasing interest and complexity.

### Wed, Sept 25, 2:30PM Ayelet Lindenstrauss (Indiana University)

Taylor Series in Homotopy Theory

I will discuss Goodwillie's calculus of functors on topological spaces. To mimic the set-up in real analysis, topological spaces are considered small if their nontrivial homotopy groups start only in higher dimensions. They can be considered close only in relation to a map between them, but a map allows us to construct the difference between two spaces, and two spaces are close if the difference between them is small. Spaces can be summed (in different ways) by taking twisted products of them. It is straightforward to construct the analogs of constant, linear, and higher degree homogenous functors, and they can be assembled into "polynomials" and "infinite sums". There are notions of differentiability and higher derivatives, of Taylor towers, and of analytic functions.

What might look like a game of analogies is an extremely useful tool because when one looks at functors that map topological spaces not into the category of topological spaces, but into the category of spectra (the stabilized version of the category of spaces, which will be explained), many of them are, in fact, analytic, so they can be constructed from the homogenous functors of different degrees. And we can use appropriate analogs of calculus theorems to understand them better. I will conclude with some recent work of Randy McCarthy and myself, applying Goodwillie's calculus to algebraic K-theory calculations.

### Sep 25: Jim Demmel (Berkeley)

Communication Avoiding Algorithms for Linear Algebra and Beyond

Algorithm have two costs: arithmetic and communication, i.e. moving data between levels of a memory hierarchy or processors over a network. Communication costs (measured in time or energy per operation) already greatly exceed arithmetic costs, and the gap is growing over time following technological trends. Thus our goal is to design algorithms that minimize communication. We present algorithms that attain provable lower bounds on communication, and show large speedups compared to their conventional counterparts. These algorithms are for direct and iterative linear algebra, for dense and sparse matrices, as well as direct n-body simulations. Several of these algorithms exhibit perfect strong scaling, in both time and energy: run time (resp. energy) for a fixed problem size drops proportionally to the number of processors p (resp. is independent of p). Finally, we describe extensions to algorithms involving arbitrary loop nests and array accesses, assuming only that array subscripts are affine functions of the loop indices.

### Sep 26: Jim Demmel (Berkeley)

Implementing Communication Avoiding Algorithms

Designing algorithms that avoiding communication, attaining lower bounds if possible, is critical for algorithms to minimize runtime and energy on current and future architectures. These new algorithms can have new numerical stability properties, new ways to encode answers, and new data structures, not just depend on loop transformations (we need those too!). We will illustrate with a variety of examples including direct linear algebra (eg new ways to perform pivoting, new deterministic and randomized eigenvalue algorithms), iterative linear algebra (eg new ways to reorganize Krylov subspace methods) and direct n-body algorithms, on architectures ranging from multicore to distributed memory to heterogeneous. The theory describing communication avoiding algorithms can give us a large design space of possible implementations, so we use autotuning to find the fastest one automatically. Finally, on parallel architectures one can frequently not expect to get bitwise identical results from multiple runs, because of dynamic scheduling and floating point nonassociativity; this can be a problem for reasons from debugging to correctness. We discuss some techniques to get reproducible results at modest cost.

### Sep 27: Jim Demmel (Berkeley)

Communication Lower Bounds and Optimal Algorithms for Programs that Reference Arrays

Our goal is to minimize communication, i.e. moving data, since it increasingly dominates the cost of arithmetic in algorithms. Motivated by this, attainable communication lower bounds have been established by many authors for a variety of algorithms including matrix computations.

The lower bound approach used initially by Irony, Tiskin and Toledo for O(n^3) matrix multiplication, and later by Ballard et al for many other linear algebra algorithms, depends on a geometric result by Loomis and Whitney: this result bounds the volume of a 3D set (representing multiply-adds done in the inner loop of the algorithm) using the product of the areas of certain 2D projections of this set (representing the matrix entries available locally, i.e., without communication).

Using a recent generalization of Loomis' and Whitney's result, we generalize this lower bound approach to a much larger class of algorithms, that may have arbitrary numbers of loops and arrays with arbitrary dimensions, as long as the index expressions are affine combinations of loop variables. In other words, the algorithm can do arbitrary operations on any number of variables like A(i1,i2,i2-2*i1,3-4*i3+7*i_4,…). Moreover, the result applies to recursive programs, irregular iteration spaces, sparse matrices, and other data structures as long as the computation can be logically mapped to loops and indexed data structure accesses.

We also discuss when optimal algorithms exist that attain the lower bounds; this leads to new asymptotically faster algorithms for several problems.

### October 4: Frank Sottile (Texas A&M)

Galois groups of Schubert problems

Work of Jordan from 1870 showed how Galois theory can be applied to enumerative geometry. Hermite earlier showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris used this to study Galois groups of many enumerative problems. Vakil gave a geometric-combinatorial criterion that implies a Galois group contains the alternating group. With Brooks and Martin del Campo, we used Vakil's criterion to show that all Schubert problems involving lines have at least alternating Galois group. White and I have given a new proof of this based on 2-transitivity.

My talk will describe this background and sketch a current project to systematically determine Galois groups of all Schubert problems of moderate size on all small classical flag manifolds, investigating at least several million problems. This will use supercomputers employing several overlapping methods, including combinatorial criteria, symbolic computation, and numerical homotopy continuation, and require the development of new algorithms and software.

### October 11: Amie Wilkinson (Chicago)

Robust mechanisms for chaos

What are the underlying mechanisms for robustly chaotic behavior in smooth dynamics?

In addressing this question, I'll focus on the study of diffeomorphisms of a compact manifold, where "chaotic" means "mixing" and and "robustly" means "stable under smooth perturbations." I'll describe recent advances in constructing and using tools called "blenders" to produce stably chaotic behavior with arbitrarily little effort.

### October 15 (Tue) and October 16 (Wed): Alexei Borodin (MIT)

Integrable probability I and II

The goal of the talks is to describe the emerging field of integrable probability, whose goal is to identify and analyze exactly solvable probabilistic models. The models and results are often easy to describe, yet difficult to find, and they carry essential information about broad universality classes of stochastic processes.

### October 25: Paul Garrett (Minnesota)

Boundary-value problems, generalized functions, and zeros of zeta functions

Modern analysis (Beppo Levi, Sobolev, Friedrichs, Schwartz) illuminates work of D. Hejhal and Y. Colin de Verdiere from 30 years ago, clarifying, as in P. Cartier's letter to A. Weil, "how the Riemann Hypothesis was not proven". (Joint with E. Bombieri.)

### November 1: Allison Lewko (Columbia University)

On sets of large doubling, Lambda(4) sets, and error-correcting codes

We investigate the structure of finite sets A of integers such that A+A is large, presenting a counterexample to natural conjectures in the pursuit of an "anti-Freiman" theory in additive combinatorics. We will begin with a brief history of the problem and its connection to the study of Lambda(4) sets in harmonic analysis, and then we will discuss our counterexample and its construction from error-correcting codes. We will conclude by describing some related open problems. This is joint work with Mark Lewko.

### November 8: Tim Riley (Cornell)

Hydra groups

A few years ago Will Dison and I constructed a family of finitely generated groups whose workings include a string-rewriting phenomenon of extraordinary duration which is reminiscent of Hercules' battle with the hydra. I will describe this and the investigations it spurred in hyperbolic geometry, combinatorial group theory, and a problem of how to calculate efficiently with hugely compressed representations of integers.

### November 22: Tianling Jin (University of Chicago)

Solutions of some Monge-Ampere equations with degeneracy or singularities

We will first give a new proof of a celebrated theorem of Jorgens which states that every classical convex solution of det(Hess u)=1 in R^2 has to be a second order polynomial. Our arguments do not use complex analysis, and will be applied to establish such Liouville type theorems for solutions some degenerate Monge-Ampere equations. We will also discuss some results on existence, regularity, classification, and asymptotic behavior of solutions of some Monge-Ampere equations with isolated and line singularities. This is joint work with J. Xiong.

### Monday, Nov 25: Lin Lin (Lawrence Berkeley National Lab)

Fast algorithms for electronic structure analysis

Kohn-Sham density functional theory (KSDFT) is the most widely used electronic structure theory for molecules and condensed matter systems. For a system with N electrons, the standard method for solving KSDFT requires solving N eigenvectors for an O(N) * O(N) Kohn-Sham Hamiltonian matrix. The computational cost for such procedure is expensive and scales as O(N^3). We have developed pole expansion plus selected inversion (PEXSI) method, in which KSDFT is solved by evaluating the selected elements of the inverse of a series of sparse symmetric matrices, and the overall algorithm scales at most O(N^2) for all materials including insulators, semiconductors and metals. The PEXSI method can be used with orthogonal or nonorthogonal basis set, and the physical quantities including electron density, energy, atomic force, density of states, and local density of states are calculated accurately without using the eigenvalues and eigenvectors. The recently developed massively parallel PEXSI method has been implemented in SIESTA, one of the most popular electronic structure software using atomic orbital basis set. The resulting method can allow accurate treatment of electronic structure in a unprecedented scale. We demonstrate the application of the method for solving graphene-like structures with more than 20,000 atoms, and the method can be efficiently parallelized 10,000 - 100,000 processors on Department of Energy (DOE) high performance machines.

### November 26 (Tuesday): Clinton Conley (Cornell)

Descriptive set-theoretic graph theory

Familiar graph-theoretic problems (for example, vertex coloring) exhibit a stark change of character when measurability constraints are placed on the structures and functions involved. While discussing some ramifications in descriptive set theory, we also pay special attention to interactions with probability (concerning random colorings of Cayley graphs) and ergodic theory (characterizing various dynamical properties of groups). The talk will include joint work with Alexander Kechris, Andrew Marks, Benjamin Miller, and Robin Tucker-Drob.

### December 2 (Monday): Simon Marshall (Northwestern)

Semiclassical estimates for eigenfunctions on locally symmetric spaces

Let M be a compact Riemannian manifold, and f an L^2-normalised Laplace eigenfunction on M. If p > 2, a theorem of Sogge tells us how large the L^p norm of f can be in terms of its Laplace eigenvalue. For instance, when p is infinity this is asking how large the peaks of f can be. I will present an analogue of Sogge's theorem for eigenfunctions of the full ring of invariant differential operators on a locally symmetric space, and discuss some links between this result and number theory.

### December 4 (Wednesday): Steven Sam (Berkeley)

Free Resolutions and Symmetry

This talk is about the use of symmetry in the study of modules and free resolutions in commutative algebra and algebraic geometry, and specifically how it clarifies, organizes, and rigidifies calculations, and how it enables us to find finiteness in situations where it a priori does not seem to exist. I will begin the talk with an example coming from classical invariant theory and determinantal ideals using just some basic notions from linear algebra. Then I will explain some of my own work which builds on this setting in several directions. Finally, I'll discuss a recent program on twisted commutative algebras, developed jointly with Andrew Snowden, which formalizes the synthesis of representation theory and commutative algebra and leads to new finiteness results in seemingly infinite settings.

### December 6: Paul Hand (MIT)

Simplifications of the Lifting Approach for Quadratic Signal Recovery Problems

Many signal recovery problems are quadratic in nature, such as phase retrieval and sparse principal component analysis. Such problems in R^n can be convexified by introducing n^2 variables corresponding to each quadratic combination of unknowns. This approach often gives rise to an n x n matrix recovery problem that is convex and has provable recovery guarantees. Because the dimensionality has been squared, it is an important task to find simplifications that make computation more tractable. We will discuss two examples where the lifting approach can be simplified while retaining recovery guarantees. These examples will be the phase retrieval problem and a special case of sparse principal component analysis.

### December 9 (Monday): Jacob Bedrossian (Courant Institute)

Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations

We prove asymptotic stability of shear flows close to the planar, periodic Couette flow in the 2D incompressible Euler equations. That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically mixed to small scales by an almost linear evolution and in general enstrophy is lost in the weak limit. The strong convergence of the velocity field is sometimes referred to as inviscid damping, due to the relationship with Landau damping in the Vlasov equations. Joint work with Nader Masmoudi.

### Wednesday, Dec 11: Lu Wang (Johns Hopkins)

Rigidity of Self-shrinkers of Mean Curvature Flow

The study of mean curvature flow not only is fundamental in geometry, topology and analysis, but also has important applications in applied mathematics, for instance, image processing. One of the most important problems in mean curvature flow is to understand the possible singularities of the flow and self-shrinkers, i.e., self-shrinking solutions of the flow, provide the singularity models.

In this talk, I will describe the rigidity of asymptotic structures of self-shrinkers. First, I show the uniqueness of properly embedded self-shrinkers asymptotic to any given regular cone. Next, I give a partial affirmative answer to a conjecture of Ilmanen under an infinite order asymptotic assumption, which asserts that the only two-dimensional properly embedded self-shrinker asymptotic to a cylinder along some end is itself the cylinder. The feature of our results is that no completeness of self-shrinkers is required.

The key ingredients in the proof are a novel reduction of unique continuation for elliptic operators to backwards uniqueness for parabolic operators and the Carleman type techniques. If time permits, I will discuss some applications of our approach to shrinking solitons of Ricci flow.

### Friday, Dec 13: Chanwoo Kim (Cambridge)

Regularity of the Boltzmann equation in convex domains

A basic question about regularity of Boltzmann solutions in the presence of physical boundary conditions has been open due to characteristic nature of the boundary as well as the non-local mixing of the collision operator. Consider the Boltzmann equation in a strictly convex domain with the specular, bounce-back and diffuse boundary condition. With the aid of a distance function toward the grazing set, we construct weighted classical $\displaystyle{ C^{1} }$ solutions away from the grazing set for all boundary conditions. For the diffuse boundary condition, we construct $\displaystyle{ W^{1,p} }$ solutions for 1< p<2 and weighted $\displaystyle{ W^{1,p} }$ solutions for $\displaystyle{ 2\leq p\leq \infty }$ as well. On the other hand, we show second derivatives do not exist up to the boundary in general by constructing counterexamples for all boundary conditions. This is a joint work with Guo, Tonon, Trescases.

### December 17: Perla Sousi (Cambridge)

The effect of drift on the volume of the Wiener sausage

The Wiener sausage at time t is the algebraic sum of a Brownian path on [0,t] and a ball. Does the expected volume of the Wiener sausage increase when we add drift? How do you compare the expected volume of the usual Wiener sausage to one defined as the algebraic sum of the Brownian path and a square (in 2D) or a cube (in higher dimensions)? We will answer these questions using their relation to the detection problem for Poisson Brownian motions, and rearrangement inequalities on the sphere (with Y. Peres). We will also discuss generalisations of this to Levy processes (with A. Drewitz and R. Sun) as well as an adversarial detection problem and its connections to Kakeya sets (with Babichenko, Peres, Peretz and Winkler).

### December 18: Dustin Cartwright (Yale)

Tropical Complexes

Tropical geometry is a way of understanding algebraic varieties by the limiting behavior of their degenerations. Through tropicalization, algebraic operations are replaced with combinatorial constructions and piecewise linear functions. I will introduce tropical complexes, which a way of understanding the geometry of algebraic varieties through combinatorics. Tropical complexes are Delta-complexes together with additional integral data, for which one has parallels and concrete comparisons with the behavior of algebraic varieties.

### 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.

### Wen, 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.

### 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.

### March 28: Michael Lacey (GA Tech)

The Two Weight Inequality for the Hilbert Transform

The individual two weight inequality for the Hilbert transform asks for a real variable characterization of those pairs of weights (u,v) for which the Hilbert transform H maps L^2(u) to L^2(v). This question arises naturally in different settings, most famously in work of Sarason. Answering in the positive a deep conjecture of Nazarov-Treil-Volberg, the mapping property of the Hilbert transform is characterized by a triple of conditions, the first being a two-weight Poisson A2 on the pair of weights, with a pair of so-called testing inequalities, uniform over all intervals. This is the first result of this type for a singular integral operator. (Joint work with Sawyer, C.-Y. Shen and Uriate-Tuero)

## Past talks

Last year's schedule: Colloquia 2012-2013