# Seminars & Events for 2015-2016

##### Real Gromov-Witten theory in all genera

We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the quintic threefold. Our approach to the orientability problem is based entirely on the topology of real bundle pairs over symmetric surfaces.

##### Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic maps - Part 1

This series of talks is an extension of the colloquium talk where we outline a proof of recent results concerning the structure of stationary and minimizing harmonic maps. Specifically, we will show that the singular stratum S^k(f) are k-rectifiable for a stationary map, and for a minimizing map that the singular set S(f) has finite n-3 measure. We will also show sharp sobolev estimates for s

##### Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic maps (Part 2)

This series of talks is an extension of the colloquium talk where we outline a proof of recent results concerning the structure of stationary and minimizing harmonic maps. Specifically, we will show that the singular stratum S^k(f) are k-rectifiable for a stationary map, and for a minimizing map that the singular set S(f) has finite n-3 measure. We will also show sharp sobolev estimates for s

##### $\ell^p(\mathbf Z^d)$ boundedness for discrete operators of Radon types: maximal and variational estimates

**PLEASE NOTE ROOM CHANGE FROM LAST TERM: SEMINAR WILL NOW BE HELD IN FINE 110.** In recent times - particularly the last two decades - discrete analogues in harmonic analysis have gone through a period of considerable changes and developments. This is due in part to Bourgain's pointwise ergodic theorem for the squares on $L^p(X, \mu)$ for any $p>1$.

##### Comparison Lemmas and Convex-Optimization-Based Signal Recovery

In the past couple of decades, non-smooth convex optimization has emerged as a powerful tool for the recovery of structured signals (sparse, low rank, finite constellation, etc.) from (possibly) noisy measurements in a variety of applications in statistics, signal processing, machine learning, communications, etc.

##### Local volumes and equisingularity theory

The epsilon multiplicity of a module is a natural generalization of the Hilbert-Samuel multiplicity of an ideal. It can be expressed as the volume of a suitable Cartier divisor associated with the module. In this talk, I will present a result that determines the change of the epsilon multiplicity of a module across flat families.

##### Harmonicity and Invariance on the Slice

The subject of this talk is the slice of the discrete cube - i.e. the uniform distribution over all binary vector of a certain weight, or probabilistically the product measure on the cube conditioned on having a specific sum.

##### Embeddings into Euclidean space and related obstructions

We will discuss several results about bi-Lipschitz embeddings of metric spaces into Hilbert space, motivated by related classic theorems in Banach spaces. Emphasis will be given on techniques for proving non-embeddability and Ramsey-type theorems in this context.

##### Polygonal Outer Billiards

I will introduce the polygonal outer billiards problem, presenting a bit of history, open problems, and connections to other dynamical systems. I will focus on the case of regular polygons.

##### The Taylor model in magnetohydrodynamics

**Please note double seminar on this date. **We shall discuss a model introduced by J.B. Taylor in 1963, that comes from a formal asymptotic limit of MHD (magnetohydrodynamics) equations with rotation. This asymptotic model, relevant to the Earth's dynamo problem, should in principle allow for easier numerics.

##### Casson towers and slice knots

A link is slice if it is the boundary of a disjoint union of flat discs in the 4-ball. The link slicing problem is closely related to the surgery and s-cobordism programme for classifying 4-manifolds. A Casson tower is a 4-manifold with boundary built from iteratively attempting to find an embedded disc in a 4-manifold using immersed discs.

##### Adjoint Selmer groups for polarized automorphic Galois representations

Given the p-adic Galois representation associated to a regular algebraic polarized cuspidal automorphic representation, one naturally obtains a pure weight zero representation called its adjoint representation. Because it has weight zero, a conjecture of Bloch and Kato says that the only de Rham extension of the trivial representation by this adjoint representation is the split extension.

##### Clique-stable set separation

Consider the following communication complexity problem: Alice is given a clique K, Bob is given a stable set S, and they have to decide via a non-deterministic protocol whether K intersects S or not. A ``certificate'' for the non-intersection is a bipartition of the vertices such that K is included in one side, and S is included in the other side.

##### The Hele-Shaw and Muskat problem

**Please note double seminar on this date. **In this talk we are going to review some recent results concerning the evolution of a free boundary under Darcy's flow. This is known as the Muskat problem or the Hele-Shaw cell problem with gravity. In particular, we will present a new method based on the study of the bulk rather than the appropriate integral equation.

##### Embedded Willmore tori in three-manifolds with small area constraint

While there are lots of contributions on Willmore surfaces in the three-dimensional Euclidean space, the literature on curved manifolds is still relatively limited. One of the main aspects of the Willmore problem is the loss of compactness under conformal transformations.

##### 3d mirror symmetry and symplectic duality

**Please note special location (Fine 224) and time (3:00-4:00)****. **In recent work of Braden, Licata, Proudfoot, and Webster, a "symplectic duality" was described between pairs of module categories O(M), O(M') associated to certain pairs of complex symplectic manifolds (M, M').

##### The two membranes problem

We will consider the two membranes obstacle problem for two different operators, possibly non-local. In the case when the two operators have different orders, we discuss how to obtain $C^{1,\gamma}$ regularity of the

##### A restriction estimate using polynomial partitioning

The restriction conjecture is an open problem in Fourier analysis first raised by Stein in the late 60’s. It is about the L^p estimates obeyed by an oscillatory integral operator. I will explain a recent approach to this problem, which gives a slightly better estimate in three dimensions.

##### Applications of interlacing polynomials in graph theory

In this talk, I will discuss some recent applications of interlacing polynomials in graph theory. I will begin by discussing the more classical results concerning graph polynomials, including the real rootedness of the matching polynomial and the extension of Chudnovsky and Seymour to the independence polynomial of claw-free graphs. I will then discuss results using the ``method of interlacing

##### On Minkowski bases for Newton-Okounkov bodies

The Newton-Okounkov bodies of linear series on an n-dimensional projective variety is a compact convex body in real n-space which carries information about the linear series. However, in general it is hard to determine in practice. We show that under certain conditions there exist simple "building blocks" for al Newton-Okounkov bodies of a given variety, a so called Minkowski basis.