# Seminars & Events for 2015-2016

##### Probabilistic interpretation of conservation laws and optimal transport in one dimension

We consider partial differential equations that describe the conservation of one or several quantities, possibly taking an additional dissipation mechanism into account, set on the real line. Such models are for instance relevant in gas dynamics or in the study of road traffic.

##### Composition laws

We will first look at Gauß's classical composition law for binary integral quadratic forms and some of its applications. Then, we describe a few special cases of Bhargava's higher composition laws.

##### Quantum invariants in the (oriented) knot Floer cube of resolutions

Using filtrations on the knot Floer cube of resolutions, I will define a triply graded invariant that categorifies the HOMFLY-PT polynomial and has a spectral sequence converging to HFK. Along the way I will discuss relationships between this construction and deformations of sl(n) homology.

##### Space-time resonances and high-frequency instabilities in the two-fluid Euler-Maxwell system

We show that space-time resonances induce high-frequency instabilities in the two-fluid Euler-Maxwell system. This implies in particular that the Zakharov approximation to Euler-Maxwell is stable if and only if the group velocity vanishes in the Schrödinger equation satisfied by the envelope of the WKB electrical field.

##### A proportionality of scalar curvatures on Hermitian manifolds and Schrödinger operators

**Please note special day, time and location.** On a Kähler manifold there is a clear connection between the complex geometry and underlying Riemannian geometry. However, in the non-Kähler setting, such a link is not so obvious.

##### Unlikely Intersections For Two-Parameter Families of Polynomials

Inspired by work of Masser and Zannier for torsion specializations of points on the Legendre elliptic curve, Baker and DeMarco proved that if v,w are two points in C, then there are at most finitely many t in C such that v and w are both preperiodic for the polynomial x^2 + t, unless of course v equals plus or minus w. Here we prove a two-dimensional version of this result, namely that if v, w

##### On graphs decomposable into induced matchings of linear size

A ``Ruzsa-Szemeredi graph'' is a graph on n vertices whose edge set can be partitioned into induced matchings of size cn. The study of these graphs goes back more than 35 years and has connections with number theory, combinatorics, complexity theory and information theory. In this talk we will discuss the history and some recent developments in this area.

##### An overview of Benjamini-Schramm convergence in group theory and dynamics

When studying an infinite geometric object or graph it is natural to want a "good" finite or bounded model for the sake of computations. But what does "good" mean here? This notion is formalized by Benjamini-Schramm convergence: "good" means that locally the finite object looks like the infinite one, except for a small density of singularities.

##### Dehn twists exact sequences through Lagrangian cobordism

In this talk we first introduce a new "singularity-free" approach to the proof of Seidel's long exact sequence, including the fixed-point version. This conveniently generalizes to Dehn twists along Lagrangian submanifolds which are rank one symmetric spaces and their covers, including RP^n and CP^n, matching a mirror prediction due to Huybrechts and Thomas.

##### Motion by curvature in the subriemannian Heisenberg group

In this talk we will present some properties of the motion by curvature in subriemannian setting. This describes the motion of a surface when each point is moving in the normal direction with speed proportional to the mean curvature.

##### Minimal surfaces of finite total curvature in Hˆ2xR

**Please note special time (4:15). ** In this talk we will present some recent developments in the theory of finite total curvature minimal surfaces in Hˆ2xR and we will show a characterization for such surfaces in terms of their behavior at infinity. This is a joint work with Hauswirth and Rodriguez.

##### Celebration of the Life and Work of John F. Nash, Jr.

**Lectures on Nash’s work:**(All talks will be in McDonnell A02)

##### On the global dynamics of three dimensional imcompressible magnetohydrodynamics

We construct and study global solutions for the 3-dimensional imcompressible MHD systems with arbitrary small viscosity.

##### The master equation and the convergence problem in mean-field games

We discuss the convergence, as $N$ tends to infinity, of a system of $N$ coupled Hamilton-Jacobi equations, called the Nash system. This system arises in differential game theory. We describe the limit problem in terms of the so-called ``master equation", a kind of second order partial differential equation stated on the space of probability measures.

##### Random regular digraphs: singularity and spectrum

**Please note special day (Tuesday).** We consider two random matrix ensembles associated to large random regular digraphs: (1) the 0/1 adjacency matrix, and (2) the adjacency matrix with iid bounded edge weights. Motivated by universality conjectures, we show that the spectral distribution for the latter ensemble is asymptotically described by the circular law, assuming the gra

##### Schramm -- Loewner Evolution and Liouville Quantum Multifractality

We describe some recent advances in the study of the fundamental coupling of a canonical model of random paths, the Schramm--Loewner Evolution (SLE), to a canonical model of random surfaces, Liouville Quantum Gravity (LQG). The latter is expected to be the conformally invariant continuum limit of various models of random planar maps.

##### Toric degenerations and symplectic geometry of projective varieties

I will explain some recent general results about symplectic geometry of projective varieties using toric degenerations (motivated by commutative algebra and the theory of Newton-Okounkov bodies). The main result is the following: Let X be a smooth n-dimensional complex projective variety equipped with an integral Kahler form.

##### The maximum of the characteristic polynomial of random unitary matrices

A recent conjecture of Fyodorov, Hiary & Keating (FHK) states that the maxima of the characteristic polynomial of random unitary matrices behave like the maxima of a specific class of Gaussian fields, the so-called log-correlated Gaussian fields. These include important examples such as branching Brownian motion and the 2D Gaussian free field.

##### Geometric identities on moduli spaces and their applications

We will discuss the Bridgeman-Kahn and Mirzakhani-McShane identities on moduli spaces of bordered Riemann surfaces. As applications, we will look at Bridgeman’s proof of the classical Abel identity and Mirzakhani’s work on Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces.