# Mathematics Colloquium

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

## Fall 2011

date speaker title host(s)
Sep 9 Manfred Einsiedler (ETH-Zurich) Periodic orbits on homogeneous spaces Fish
Sep 16 Richard Rimanyi (UNC-Chapel Hill) Global singularity theory Maxim
Sep 23 Andrei Caldararu (UW-Madison) The Pfaffian-Grassmannian derived equivalence (local)
Sep 30 Scott Armstrong (UW-Madison) Optimal Lipschitz extensions, the infinity Laplacian, and tug-of-war games (local)
Oct 7 Hala Ghousseini (University of Wisconsin-Madison) Developing Mathematical Knowledge for Teaching in, from, and for Practice Lempp
Oct 14 Alex Kontorovich (Yale) On Zaremba's Conjecture Shamgar
oct 19, Wed Bernd Sturmfels (UC Berkeley) Convex Algebraic Geometry distinguished lecturer Shamgar
oct 20, Thu Bernd Sturmfels (UC Berkeley) Quartic Curves and Their Bitangents distinguished lecturer Shamgar
oct 21 Bernd Sturmfels (UC Berkeley) Multiview Geometry distinguished lecturer Shamgar
Oct 28 Roman Holowinsky (OSU) Equidistribution Problems and L-functions Street
Nov 4 Sijue Wu (U Michigan) Wellposedness of the two and three dimensional full water wave problem Qin Li
Nov 7, Mo, 3pm, SMI 133 Sastry Pantula (NSCU and DMS/NSF) Opportunities in Mathematical and Statistical Sciences at DMS Joint Math/Stat Colloquium
Nov 11 Henri Berestycki (EHESS and University of Chicago) Reaction-diffusion equations and propagation phenomena Wasow lecture
Nov 16, Wed Henry Towsner (U of Conn-Storrs) An Analytic Approach to Uniformity Norms Steffen
Nov 18 Benjamin Recht (UW-Madison, CS Department) The Convex Geometry of Inverse Problems Jordan
Nov 22, Tue, 2:30PM, B205 Zhiwei Yun (MIT) Motives and the inverse Galois problem Tonghai
Nov 28, Mon, 4PM Burglind Joricke (Institut Fourier, Grenoble) Analytic knots, satellites and the 4-ball genus Gong
Nov 29, Tue, 2:30PM, B102 Isaac Goldbring (UCLA) "Nonstandard methods in Lie theory" Lempp
Nov 30, Wed, 4PM Bing Wang (Simons Institute) Uniformization of algebraic varieties Sean
Dec 2 Robert Dudley (University of California, Berkeley) From Gliding Ants to Andean Hummingbirds: The Evolution of Animal Flight Performance Jean-Luc
Dec 5, Mon, 2:25PM, Room 901 Dima Arinkin (UNC-Chapel Hill) Autoduality of Jacobians for singular curves Andrei
Dec 7, Wed, 4PM Toan Nguyen (Brown University) On the stability of Prandtl boundary layers and the inviscid limit of the Navier-Stokes equations Misha Feldman
Dec 9 Xinwen Zhu (Harvard University) Adelic uniformization of moduli of G-bundles Tonghai
Dec 12, Mon, 4PM Jonathan Hauenstein (Texas A&M) Numerical solving of polynomial equations and applications Thiffeault

## Spring 2012

date speaker title host(s)
Jan 26, Thu Peter Constantin (University of Chicago) TBA distinguished lecturer
Jan 27 Peter Constantin (University of Chicago) TBA distinguished lecturer
Feb 3 Scheduled Street
Feb 24 Malabika Pramanik (University of British Columbia) TBA Benguria
March 2 Guang Gong (University of Waterloo) TBA Shamgar
March 16 Charles Doran (University of Alberta) TBA Matt Ballard
March 23 Martin Lorenz (Temple University) TBA Don Passman
March 30 Paolo Aluffi (Florida State University) TBA Maxim
April 6 Spring recess
April 13 Ricardo Cortez (Tulane) TBA Mitchell
April 18 Benedict H. Gross (Harvard) TBA distinguished lecturer
April 19 Benedict H. Gross (Harvard) TBA distinguished lecturer
April 20 Robert Guralnick (University of South California) TBA Shamgar
April 27 Tentatively Scheduled Street
May 4 Mark Andrea de Cataldo (Stony Brook) TBA Maxim
May 11 Tentatively Scheduled Shamgar

## Abstracts

### Fri, Sept 9: Manfred Einsiedler (ETH-Zurich)

Periodic orbits on homogeneous spaces

We call an orbit xH of a subgroup H<G on a quotient space Gamma \ G periodic if it has finite H-invariant volume. These orbits have intimate connections to a variety of number theoretic problems, e.g. both integer quadratic forms and number fields give rise periodic orbits and these periodic orbits then relate to local-global problems for the quadratic forms or to special values of L-functions. We will discuss whether a sequence of periodic orbits equidistribute in Gamma \ G assuming the orbits become more complicated (which can be measured by a discriminant). If H is a diagonal subgroup (also called torus or Cartan subgroup), this is not always the case but can be true with a bit more averaging. As a theorem of Mozes and Shah show the case where H is generated by unipotents is well understand and is closely related to the work of M. Ratner. We then ask about the rate of approximation, where the situation is much more complex. The talk is based on several papers which are joint work with E.Lindenstrauss, Ph. Michel, and A. Venkatesh resp. with G. Margulis and A. Venkatesh.

### Fri, Sept 16: Richard Rimanyi (UNC)

Global singularity theory

The topology of the spaces A and B may force every map from A to B to have certain singularities. For example, a map from the Klein bottle to 3-space must have double points. A map from the projective plane to the plane must have an odd number of cusp points.

To a singularity one may associate a polynomial (its Thom polynomial) which measures how topology forces this particular singularity. In the lecture we will explore the theory of Thom polynomials and their applications in enumerative geometry. Along the way, we will meet a wide spectrum of mathematical concepts from geometric theorems of the ancient Greeks to the cohomology ring of moduli spaces.

### Fri, Sept 23: Andrei Caldararu (UW-Madison)

The Pfaffian-Grassmannian derived equivalence

String theory relates certain seemingly different manifolds through a relationship called mirror symmetry. Discovered about 25 years ago, this story is still very mysterious from a mathematical point of view. Despite the name, mirror symmetry is not entirely symmetric -- several distinct spaces can be mirrors to a given one. When this happens it is expected that certain invariants of these "double mirrors" match up. For a long time the only known examples of double mirrors arose through a simple construction called a flop, which led to the conjecture that this would be a general phenomenon. In joint work with Lev Borisov we constructed the first counterexample to this, which I shall present. Explicitly, I shall construct two Calabi-Yau threefolds which are not related by flops, but are derived equivalent, and therefore are expected to arise through a double mirror construction. The talk will be accessible to a wide audience, in particular to graduate students. There will even be several pictures!

### Fri, Sept 30: Scott Armstrong (UW-Madison)

Optimal Lipschitz extensions, the infinity Laplacian, and tug-of-war games

Given a nice bounded domain, and a Lipschitz function defined on its boundary, consider the problem of finding an extension of this function to the closure of the domain which has minimal Lipschitz constant. This is the archetypal problem of the calculus of variations "in the sup-norm". There can be many such minimal Lipschitz extensions, but there is there is a unique minimizer once we properly "localize" this Lipschitz minimizing property. This minimizer is characterized by the infinity Laplace equation: the Euler-Lagrange equation for our optimization problem. This PDE is a very highly degenerate nonlinear elliptic equation which does not possess smooth solutions in general. In this talk I will discuss what we know about the infinity Laplace equation, what the important open questions are, and some interesting recent developments. We will even play a probabilistic game called "tug-of-war".

### Fri, Oct 7: Hala Ghousseini (University of Wisconsin-Madison)

Developing Mathematical Knowledge for Teaching in, from, and for Practice

Recent research in mathematics education has established that successful teaching requires a specialized kind of professional knowledge known as Mathematical Knowledge for Teaching (MKT). The mathematics education community, however, is beginning to appreciate that to be effective, teachers not only need to know MKT but also be able to use it in interaction with students (Hill & Ball, 2010). Very few examples exist at the level of actual practice of how novice teachers develop such knowledge for use. I will report on my current work on the Learning in, from, and for Practice project to develop, implement, and study what mathematics teacher educators can do to support novice teachers in acquiring and using Mathematical Knowledge for Teaching.

### Fri, Oct 14: Alex Kontorovich (Yale)

On Zaremba's Conjecture

It is folklore that modular multiplication is "random". This concept is useful for many applications, such as generating pseudorandom sequences, or in quasi-Monte Carlo methods for multi-dimensional numerical integration. Zaremba's theorem quantifies the quality of this "randomness" in terms of certain Diophantine properties involving continued fractions. His 40-year old conjecture predicts the ubiquity of moduli for which this Diophantine property is uniform. It is connected to Markoff and Lagrange spectra, as well as to families of "low-lying" divergent geodesics on the modular surface. We prove that a density one set satisfies Zaremba's conjecture, using recent advances such as the circle method and estimates for bilinear forms in the Affine Sieve, as well as a "congruence" analog of the renewal method in the thermodynamical formalism. This is joint work with Jean Bourgain.

### Wed, Oct 19: Bernd Sturmfels (Berkeley)

Convex Algebraic Geometry

This lecture concerns convex bodies with an interesting algebraic structure. A primary focus lies on the geometry of semidefinite optimization. Starting with elementary questions about ellipses in the plane, we move on to discuss the geometry of spectrahedra, orbitopes, and convex hulls of real varieties.

### Thu, Oct 20: Bernd Sturmfels (Berkeley)

Quartic Curves and Their Bitangents

We present a computational study of plane curves of degree four, with primary focus on writing their defining polynomials as sums of squares and as symmetric determinants. Number theorists will enjoy the appearance of the Weyl group $\displaystyle{ E_7 }$ as the Galois group of the 28 bitangents. Based on joint work with Daniel Plaumann and Cynthia Vinzant, this lecture spans a bridge from 19th century algebra to 21st century optimization.

### Fri, Oct 21: Bernd Sturmfels (Berkeley)

Multiview Geometry

The study of two-dimensional images of three-dimensional scenes is foundational for computer vision. We present work with Chris Aholt and Rekha Thomas on the polynomials characterizing images taken by $\displaystyle{ n }$ cameras. Our varieties are threefolds that vary in a family of dimension $\displaystyle{ 11n-15 }$ when the cameras are moving. We use toric geometry and Hilbert schemes to characterize degenerations of camera positions.

### Fri, Oct 28: Roman Holowinsky (OSU)

Equidistribution Problems and L-functions

There are several equidistribution problems of arithmetic nature which have had shared interest between the fields of Ergodic Theory and Number Theory. The relation of such problems to homogeneous flows and the reduction to analysis of special values of automorphic L-functions has resulted in increased collaboration between these two fields of mathematics. We will discuss two such equidistribution problems: the equidistribution of Heegner points for negative quadratic discriminants and the equidistribution of mass of Hecke eigenforms. Equidistribution follows upon establishing subconvexity bounds for the associated L-functions and are fine examples as to why one might be interested in such objects.

### Fri, Nov 4: Sijue Wu (U Michigan)

Wellposedness of the two and three dimensional full water wave problem

We consider the question of global in time existence and uniqueness of solutions of the infinite depth full water wave problem. We show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher orders. For any initial data that is small in its kinetic energy and height, we show that the 2-D full water wave equation is uniquely solvable almost globally in time. And for any initial interface that is small in its steepness and velocity, we show that the 3-D full water wave equation is uniquely solvable globally in time.

### Mo, Nov 7: Sastry Pantula (DMS/NSF, NCSU)

Opportunities in Mathematical and Statistical Sciences at DMS

In this talk, I will give you an overview of the funding and other opportunities at DMS for mathematicians and statisticians. I will also talk about our new program in computational and data-enabled science and engineering in mathematical and statistical sciences (CDS&E-MSS).

### Fri, Nov 11: Henri Berestycki (EHESS and University of Chicago)

Reaction-diffusion equations and propagation phenomena

Starting with the description of reaction-diffusion mechanisms in physics, biology and ecology, I will explain the motivation for this class of non-linear partial differential equations and mention some of the interesting history of these systems. Then, I will review classical results in the homogeneous setting and discuss their relevance. The second part of the lecture will be concerned with recent developments in non-homogeneous settings, in particular for Fisher-KPP type equations. Such problems are encountered in models from ecology. The mathematical theory will be seen to shed light on questions arising in this context.

### Wed, Nov 16: Henry Towsner (U of Conn-Storrs)

An Analytic Approach to Uniformity Norms

The Gowers uniformity norms have proven to be a powerful tool in extremal combinatorics, and a number of "structure theorems" have been given showing that the uniformity norms provide a dichotomy between "structured" objects and "random" objects. While analogous norms (the Gowers-Host-Kra norms) exist in dynamical systems, they do not quite correspond to the uniformity norms in the finite setting. We describe an analytic approach to the uniformity norms in which the "correspondence principle" between the finite setting and the infinite analytic setting remains valid.

### Fri, Nov 18: Ben Recht (UW-Madison)

The Convex Geometry of Inverse Problems

Deducing the state or structure of a system from partial, noisy measurements is a fundamental task throughout the sciences and engineering. The resulting inverse problems are often ill-posed because there are fewer measurements available than the ambient dimension of the model to be estimated. In practice, however, many interesting signals or models contain few degrees of freedom relative to their ambient dimension: a small number of genes may constitute the signature of a disease, very few parameters may specify the correlation structure of a time series, or a sparse collection of geometric constraints may determine a molecular configuration. Discovering, leveraging, or recognizing such low-dimensional structure plays an important role in making inverse problems well-posed.

In this talk, I will propose a unified approach to transform notions of simplicity and latent low-dimensionality into convex penalty functions. This approach builds on the success of generalizing compressed sensing to matrix completion, and greatly extends the catalog of objects and structures that can be recovered from partial information. I will focus on a suite of data analysis algorithms designed to decompose general signals into sums of atoms from a simple---but not necessarily discrete---set. These algorithms are derived in a convex optimization framework that encompasses previous methods based on l1-norm minimization and nuclear norm minimization for recovering sparse vectors and low-rank matrices. I will provide sharp estimates of the number of generic measurements required for exact and robust recovery of a variety of structured models. I will then detail several example applications and describe how to scale the corresponding inference algorithms to massive data sets.

### Tue, Nov 22: Zhiwei Yun (MIT)

"Motives and the inverse Galois problem"

We will use geometric Langlands theory to solve two problems simultaneously. One is Serre's question about whether there exist motives over Q with motivic Galois groups of type E_8 or G_2; the other is whether there are Galois extensions of Q with Galois groups E_8(p) or G_2(p) (the finite simple groups of Lie type). The answers to both questions are YES. No familiarity with either motives or geometric Langlands or E_8 will be assumed.

### Mon, Nov 28: Burglind Joricke (Institut Fourier, Grenoble)

"Analytic knots, satellites and the 4-ball genus"

After introducing classical geometric knot invariants and satellites I will concentrate on knots or links in the unit sphere in $\mathbb C^2$ which bound a complex curve (respectively, a smooth complex curve) in the unit ball. Such a knot or link will be called analytic (respectively, smoothly analytic). For analytic satellite links of smoothly analytic knots there is a sharp lower bound for the 4-ball genus. It is given in terms of the 4-ball genus of the companion and the winding number. No such estimate is true in the general case. There is a natural relation to the theory of holomorphic mappings from open Riemann surfaces into the space of monic polynomials without multiple zeros. I will briefly touch related problems.

### Tue, Nov 29: Isaac Goldbring (UCLA)

"Nonstandard methods in Lie theory"

Nonstandard analysis is a way of rigorously using "ideal" elements, such as infinitely small and infinitely large elements, in mathematics. In this talk, I will survey the use of nonstandard methods in Lie theory. I will focus on two applications in particular: the positive solution to Hilbert's fifth problem (which establishes that locally euclidean groups are Lie groups) and nonstandard hulls of infinite-dimensional Lie groups and algebras. I will also briefly discuss the recent work of Breuillard, Green, and Tao (extending work of Hrushovski) concerning the classification of approximate groups, which utilizes nonstandard methods and the local version of Hilbert's fifth problem in an integral way. I will assume no prior knowledge of nonstandard analysis or Lie theory.

### Wed, Nov 30: Bing Wang (Simons Center for Geometry and Physics)

Uniformization of algebraic varieties

For algebraic varieties of general type with mild singularities, we show the Bogmolov-Yau inequality holds. If equality is attained, then this variety is a global quotient of complex hyperbolic space away from a subvariety.

### Mon, Dec 5: Dima Arinkin (UNC-Chapel Hill)

"Autoduality of Jacobians for singular curves"

Let C be a (smooth projective algebraic) curve. It is well known that the Jacobian J of C is a principally polarized abelian variety. In other words, J is self-dual in the sense that J is identified with the space of topologically trivial line bundles on itself.

Suppose now that C is singular. The Jacobian of C parametrizes topologically trivial line bundles on C; it is an algebraic group which is no longer compact. By considering torsion-free sheaves instead of line bundles, one obtains a natural singular compactification J' of J.

In this talk, I consider (projective) curves C with planar singularities. The main result is that J' is self-dual: J' is identified with a space of torsion-free sheaves on itself. This autoduality naturally fits into the framework of the geometric Langlands conjecture; I hope to sketch this relation in my talk.

### Wed, Dec 7: Toan Nguyen (Brown University)

"On the stability of Prandtl boundary layers and the inviscid limit of the Navier-Stokes equations"

In fluid dynamics, one of the most classical issues is to understand the dynamics of viscous fluid flows past solid bodies (e.g., aircrafts, ships, etc...), especially in the regime of very high Reynolds numbers (or small viscosity). Boundary layers are typically formed in a thin layer near the boundary. In this talk, I shall present various ill-posedness results on the classical Prandtl boundary-layer equation, and discuss the relevance of boundary-layer expansions and the vanishing viscosity limit problem of the Navier-Stokes equations. I will also discuss viscosity effects in destabilizing stable inviscid flows.

### Fri, Dec 9: Xinwen Zhu (Harvard University)

"Adelic uniformization of moduli of G-bundles"

It is well-known from Weil that the isomorphism classes of rank n vector bundles on an algebraic curve can be written as the set of certain double cosets of GL(n,A), where A is the adeles of the curve. I will introduce such presentation in the world of algebraic geometry and discuss two of its applications: the first is the Tamagawa number formula in the function field case (proved by Gaitsgory-Lurie), which is a formula for the volume of the automorphic space; and thesecond is the Verlinde formula in positive characteristic, which is a formula for the dimensions of global sections of certain line bundles on the moduli spaces.

### Mon, Dec 12: Jonathan Hauenstein (Texas A&M)

"Numerical solving of polynomial equations and applications"

Systems of polynomial equations arise in many areas of mathematics, science, economics, and engineering with their solutions, for example, describing equilibria of chemical reactions and economics models, and the design of specialized robots. These applications have motivated the development of numerical methods used for solving polynomial systems, collectively called Numerical Algebraic Geometry. This talk will explore fundamental numerical algebraic geometric algorithms for solving systems of polynomial equations and the application of these algorithms to problems in arising engineering and mathematical biology.