# Seminars & Events for 2014-2015

##### Level raising mod 2 and arbitrary 2-Selmer ranks

We prove a level raising mod p=2 theorem for elliptic curves over Q, generalizing theorems of Ribet and Diamond-Taylor. As an application, we show that the 2-Selmer rank can be arbitrary in level raising families. We will begin by explaining our motivation from W. Zhang's approach to the p-part of the BSD conjecture.

##### Interface Singularities for the Euler Equations

The fluid interface ``splash'' singularity was introduced by Castro, C\'{o}rdoba, Fefferman, Gancedo, \& G\'{o}mez-Serrano. A splash singularity occurs when a fluid interface remains locally smooth but self-intersects in finite time. In this talk, I will very briefly discuss how we construct splash singularities for the one-phase 3-D Euler and Navier-Stokes equations. I will then disc

##### Geometry of the space of probability measures

The space of probability measures, on a compact Riemannian manifold, carries the Wasserstein metric coming from optimal transport. Otto found a remarkable formal Riemannian metric on this infinite-dimensional space. It is a challenge to make rigorous sense of the ensuing formal calculations, within the framework of metric geometry.

##### Belief Propagation Algorithms: From Matching Problems to Network Discovery in Cancer Genomics

* This is a PACM Distinguished Lecture.* We review a certain class of algorithms, belief propagation algorithms, inspired by the study of phase transitions in computationally difficult problems. We show how these algorithms can be used both in the mathematical analysis of relatively simple problems like matching, and in the heuristic analysis of more complex problems.

##### Gauged linear $\sigma$-model and gauged Witten equation

This is a joint work with Gang Tian. I will talk about the analytical properties of the classical equation of motion in gauged linear $\sigma$-model, which we call the gauged Witten equation. This is a generalization of the Witten equation in Landau-Ginzburg A-model (Fan-Jarvis-Ruan, Witten) and the symplectic vortex equation (Mundet, Cieliebak-Gaio-Salamon).

##### From Newton's dynamics to the heat equation

**This is a joint Analysis - Analysis of Fluids and Related Topics seminar.** The goal of this lecture is to show how the brownian motion can be derived rigorously from a deterministic system of hard spheres in the limit where the number of particles $N$ tends to infinity, and their diameter simultaneously converges to 0.

##### TBA - Saint-Raymond

**This is a joint Analysis of Fluids and Related Topics - Analysis seminar.**

##### Pluriclosed flow and generalized Kahler geometry

In joint work with G.

##### Interface dynamics for incompressible flows

The goal of these lectures is to present the main ideas and arguments of recent results concerning global solutions and finite time singularities for interface dynamics. In particular those contour dynamics that are given by basic fluid mechanics systems; Euler´s equation, Darcy´s law and the Quasi-geostrophic equation.

##### Heat kernel on affine buildings

Let $\mathscr{X}$ be a thick affine building of rank $r+1$. We consider a finite range isotropic random walk on vertices of $\mathscr{X}$.

##### The Euclidean Distance Degree

The nearest point map of a real algebraic variety with respect to Euclidean distance is

an algebraic function. The Euclidean distance degree is the number of critical points for

this optimization problem. We focus on varieties seen in engineering applications, and

we discuss tools for exact computation. Our guiding example is the Eckart-Young Theorem

##### Finding Rational Curves

Rational curves are one of the most basic objects to look at on an algebraic variety, but they have a large impact on the geometric structure of the variety in which they sit. In this talk we will talk about what are rational curves, why they are useful, and how they are found.

##### Stochastic higher spin vertex models on the line

We show how transfer matrices of higher spin vertex models (generalizing the six-vertex model) can be conjugated into stochastic matrices describing interacting particle systems. Bethe ansatz produces eigenfunctions and we prove their completeness on the line. This, along with a self duality of the transfer matrices, provides a means to study the long time behavior of these stochastic systems.

##### Interface dynamics for incompressible flows

The goal of these lectures is to present the main ideas and arguments of recent results concerning global solutions and finite time singularities for interface dynamics. In particular those contour dynamics that are given by basic fluid mechanics systems; Euler´s equation, Darcy´s law and the Quasi-geostrophic equation.

##### Knot invariants via the cotangent bundle

In recent years, symplectic geometry has emerged as a key tool in the study of low-dimensional topology. One approach, championed by Arnol'd, is to study the topology of a smooth manifold through the symplectic geometry of its cotangent bundle, building on the familiar concept of phase space from classical mechanics.

##### The polynomial method

**In 2008, Zeev Dvir gave a surprisingly short proof of the Kakeya conjecture over finite fields: a finite subset of F_q^n containing a line in every direction has cardinality at least c_n q^n. The "polynomial method" introduced by Dvir has led to a wave of activity in applications of algebraic and arithmetic geometry to extremal problems in combinatorial geometry, including a theme sem**

##### Quantum unique ergodicity and arithmetic Fuchsian group

Quantum unique ergodicity conjecture discusses the limiting behavior of eigenfunctions of Laplacian on compact negatively curved manifolds. Results so far have connected the research areas of number theory, spectral theory and ergodic theory.

##### TBA - Parzanchevski

##### Selmer groups, automorphic periods, and Bloch-Kato Conjecture

**Double-header seminars: actual time to be determined. **The Bloch-Kato Conjecture, which generalizes the B-SD Conjecture to higher dimensional varieties, predicts a relation between certain Selmer group and L-function. The famous works of Gross-Zagier and Kolyvagin give results for elliptic curves when the analytic rank is at most 1.

##### On the stability of Prandtl boundary layer expansions of Navier-Stokes in the inviscid limit

I will present two recent results concerning the stability of boundary layer asymptotic expansions of solutions of Navier-Stokes with small viscosity. First, we show that the linearization around an arbitrary stationary shear flow (other than the Couette flow) admits an unstable eigenfunction with small wave number, when viscosity is sufficiently small.