# Colloquia

# Mathematics Colloquium

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

## 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 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
satisfying , Darmon and Granville proved that the individual generalized Fermat equation *x*^{a} + *y*^{b} = *z*^{c} has only finitely many coprime integer solutions. Conjecturally something stronger is true: for 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.