# Seminars & Events for 2016-2017

##### The free-boundary Brakke flow

A surface has geometric free-boundary if it meets some barrier hypersurface orthogonally, like a bubble on a bathtub. We extend Brakke's weak notion of mean curvature flow to have a free-boundary condition, which allows the surface to ``pop'' upon tangential contact with the barrier.

##### Tunnell's Work on the Congruent Number Problem

The congruent number problem is a classical diophantine problem which asks to determine which integers are the areas of right triangles with rational sides. We explain Tunnell's theorem on congruent numbers using elliptic curves and modular forms.

##### Three graph classes: mock threshold, cute, and nice graphs

A graph class, defined in one way, can be characterized in several other ways. Forbidden induced subgraphs, intersection of subobjects, relationship among invariants, and vertex ordering are some of the most common ways. A graph is mock threshold if every induced subgraph of it has a vertex with degree or codegree at most 1.

##### Topology and Combinatorics of 'unavoidable' complexes and the family tree of the Van Kampen-Flores theorem

Special classes of simplicial complexes (chessboard, `unavoidable’, threshold, `simple games’, etc.) play a central role in applications of algebraic topology in discrete geometry and combinatorics.

##### Exponential self-similar mixing by incompressible flows

I will address the problem of the optimal mixing of a passive scalar under the action of an incompressible flow in two space dimensions. The scalar solves the continuity equation with a divergence-free velocity field which satisfies a bound in the Sobolev space $W^{s,p}$, where $s \geq 0$ and $1\leq p\leq \infty$.

##### Co-oriented Taut Foliations

I will describe a new construction of (codimension one) co-oriented taut foliations (CTFs) of 3-manifolds. It follows from this construction that if K is either an alternating knot or a Montesinos knot, then K is not an L-space knot if and only if every nontrivial Dehn filling on K yields a 3-manifold that contains a CTF. This work is joint with Charles Delman.

##### The Arithmetic of Noncongruence Subgroups of SL(2,Z)

After beginning by giving a brief overview of how one can think of noncongruence modular curves as moduli spaces of elliptic curves with G-structures, we will discuss how these moduli interpretations fits into the greater body of knowledge concerning noncongruence subgroups, in particular focusing on the Unbounded Denominators Conjecture for their modular forms.

##### Moduli of Riemann surface and Bers conjecture

This is a continuation of the October 24 talk. It was Koebe who first proved that closed Riemann surface can be uniformized by Schottky groups.

##### Lagrangian Whitney sphere links

Let n>1. Given two maps of an n-dimensional sphere into Euclidean 2n-space with disjoint images, there is a Z/2 valued linking number given by the homotopy class of the corresponding Gauss map. We prove, under some restrictions on n, that this vanishes when the components are immersed Lagrangian spheres each with exactly one double point of high Maslov index.

##### Clique-based semidefinite relaxation of the quadratic assignment problem

The matching problem between two adjacency matrices, A and B, can be formulated as the NP-hard quadratic assignment problem (QAP). While the QAP admits a semidefinite (SDP) relaxation that is often tight in practice, this SDP scales badly as it involves a matrix variable of size n^2 by n^2.

##### Non-linear stability of Kerr-de Sitter black holes

In joint work with András Vasy, we recently proved the stability of the Kerr-de Sitter family of black holes as solutions of the initial value problem for the Einstein vacuum equations with positive cosmological constant, for small angular momenta but without any symmetry assumptions on the initial data.

##### Statistics, Machine Learning, and Understanding the 2016 Election

Although 2016 is a highly unusual political year, elections and public opinion follow predictable statistical properties. I will review how the Presidential, Senate, and House races can be tracked and forecast from freely available polling data. Missing data can be filled in using a Google-Wide Association Study (GoogleWAS).

##### Tropical curve counting in superabundant geometries

I will discuss a general framework using Artin fans -- certain logarithmic algebraic stacks -- in which to understand the relationship between logarithmic stable maps and tropical curve counting. These objects provide a flexible tool to study correspondences between algebraic and tropical curves. In particular, we obtain new realization theorems for tropical curves in superabundant settings.

##### Hermite interpolation and approximation in manifolds

In this talk we study the Hermite interpolation and approximation problem. It aims at producing a function together with its derivatives, which interpolate or approximate given discrete point-vector data. The classical Hermite method interpolates data in linear spaces using polynomial functions.

##### Universal asymptotic behavior of discrete particle systems beyond integrable cases

The talk is about a class of systems of 2d statistical mechanics, such as random tilings, noncolliding walks, log-gases and random matrix-type distributions. Specific members in this class are integrable, which means that available exact formulas allow delicate asymptotic analysis leading to the Gaussian Free Field, sine-process, Tracy-Widom distributions.

##### What We Know and Don’t Know about the Space of Solutions of the Einstein Constraint Equations

Ten years ago, Robert Bartnick and I wrote a review article summarizing what was known at the time about the Einstein constraint equations and their solutions. In that article, we noted that while much was known about solutions of the constraints which have constant mean curvature (CMC) or are nearly CMC, very little was known about solutions which are far from CMC.

##### The stability of Kerr-de Sitter black holes

In this lecture I will discuss Kerr-de Sitter black holes, which are rotating black holes in a universe with a positive cosmological constant, i.e. they are explicit solutions (in 3+1 dimensions) of Einstein's equations of general relativity. They are parameterized by their mass and angular momentum.

##### Sharp Sobolev Inequalities and Applications

Interest in extremal cases of Sobolev embeddings is not just intrinsic but also arises from some useful geometric information they can provide. We'll first discuss tools used to obtain these inequalities before turning to applications on manifolds such as rigidity results.

##### Stochastic 3D Navier-Stokes Equations with Uniformly Large Initial Vorticity

##### The union-closed sets conjecture

The union-closed sets conjecture is an elementary problem posed by Frankl in 1979. It goes like this: for any finite family of finite sets, closed under taking union, there exists an element that belongs to at least half of the sets in the family. This problem attracted recent interest for being a polymath project.