# Seminars & Events for 2015-2016

##### A reverse entropy power inequality for log-concave random vectors

We make two conjectures concerning reverse entropy power inequalities in the log-concave setting, discuss some examples and sketch the proof that the exponent of the entropy of one dimensional projections of a log-concave random vector defines a 1/5-seminorm.

Joint work with Keith Ball and Piotr Nayar.

##### Minimal graphs and harmonic diffeomorphisms.

In 1952, Heinz gave another proof of Bernstein's Theorem ( an entire minimal graph over the euclidean plane is a plane) by showing there is no harmonic diffeomorphism from the disc onto the euclidean plane. Since then the existence theory of harmonic diffeomorphisms between closed surfaces has been developed and useful.

##### Fibered links in S^3

In this talk I will define what it means for a link to be fibered and discuss the construction of fiber surfaces for fibered links through plumbing and twisting. I will give a very brief introduction to sutured manifolds and then present one method of detecting fibered links via sutured manifold decomposition.

##### Divisor functions, function fields and Matrix Integrals

I will examine some very classical questions on the statistics of divisor functions from a modern perspective of function field arithmetic and Random Matrix Theory. As a result one is able to probe new regimes in these problems, hitherto not understood even at a conjectural level.

##### Extremal number of edges in bipartite graphs as a function of the topological connectivity of the matching complex

The topological connectivity of the matching complex of a graph has been found to be a useful tool in solving many kinds of combinatorial problems such as finding rainbow matchings in edge-colored graphs. In this talk I find the extremal cases for this parameter in bipartite graphs in terms of the number of edges and the number of vertices in each side.

##### Motivic cohomology actions and the geometry of eigenvarieties

Venkatesh has recently proposed a fascinating conjecture relating motivic cohomology with automorphic forms and the cohomology of arithmetic groups. I'll describe this conjecture, and discuss its connections with the local geometry of eigenvarieties and nonabelian analogues of the Leopoldt conjecture. This is joint work with Jack Thorne.

##### Taut foliations on graph manifolds

An L-space is a rational homology sphere with simplest possible Heegaard Floer homology. Ozsváth and Szabó have shown that if a closed, connected, orientable three-manifold has a coorientable taut foliation then it is not an L-space. I will explain how to prove the converse to this statement when restricting to graph manifolds.

##### Lagrangian cobordism: what we know and what is it good for

(Joint seminar with Columbia.) I will describe how the notion of Lagrangian cobordism, introduced by Arnold in 1980, offers a systematic perspective on the study of Lagrangian topology.

##### Sharp Trace-Sobolev inequalities of order 4

We establish sharp Sobolev inequalities of order four on Euclidean d-balls for d greater than or equal to four. When d=4, our inequality generalizes the classical second order Lebedev-Milin inequality on Euclidean 2-balls. Our method relies on the use of scattering theory on hyperbolic d-balls.

##### Non-trivial Hamiltonian fibrations via K-theory quantization

We produce examples of non-trivial Hamiltonian fibrations that are not detected by previous methods (the characteristic classes of Reznikov for example), and improve theorems of Reznikov and Spacil on cohomology-surjectivity to the level of classifying spaces. Joint work with Yasha Savelyev.

##### Hölder continuity of solutions to hypoelliptic equations with rough coefficients

The celebrated De Giorgi-Nash theory about Hölder continuity of solutions to elliptic or parabolic equations with rough --i.e. merely measurable-- coefficients in the late 1950s is a cornerstone of modern PDE analysis.

##### An Applied Math Perspective on Climate Science, Turbulence, and Other Complex Systems

You can view slides of the full Lagrange Prize lecture at: http://www.math.nyu.edu/faculty/majda/pdfFiles/2014%20updates/2015/Lagrange_Prize-1.pdf . To read Professor Madja's full biography, click here: http://www.math.nyu.edu/facul

##### The reasonable effectiveness of mathematical deformation theory in physics, especially quantum mechanics and maybe elementary particle symmetries

In 1960 Wigner marveled about ``the unreasonable effectiveness of mathematics in the natural sciences," referring mainly to physics. In that spirit we shall first explain how a posteriori relativity and quantum mechanics can be obtained from previously known theories using the mathematical theory of deformations.

##### The geometric constants in Manin's Conjecture

Manin's Conjecture predicts that the growth of points of bounded height is controlled by certain geometric constants. I will analyze the geometry underlying these constants and discuss applications to Manin's Conjecture. A key tool is the minimal model program.

##### Singular Sets of Geometric PDE's

Given a solution of a nonlinear pde the two primary issues regarding the regularity theory are a priori estimates and the structure of the singular set. We will discuss new techniques in the analysis of these issues, which have been particularly successful in the study of geometrically motivated equations.

##### Ramanujan graph

We discuss the geometry and spectral properties of expander graphs. We'll define Ramanujan graphs and show why they are optimal "expander graphs". Next, we'll explain the construction of explicit Ramanujan graphs by Lubotzky, Phillips, and Sarnak, and relate geometric and spectral properties of LPS Ramanujan graphs to well-known facts in number theory.

##### On Rauzy Induction: Bufetov's Questions

Given an interval exchange transformation (IET) and a sub-interval, there arises a natural visitation matrix that relates the induced IET to the original IET. We show that the original IET, up to topological conjugacy, may be recovered from successive visitation matrices. This answers a question by A. Bufetov and generalizes work by W. A.

##### Augmented trees with high girth

Let G be a graph consisting of a complete binary tree of depth h together with a back edge from each leaf connecting it to one of its ancestors. Suppose further that the girth of G exceeds g. What is the minimum possible depth h=h(g) in such a graph ?

##### A probabilistic approach to the Navier-Stokes equations

In this preliminary talk, we consider the regularity issues of the incompressible Navier-Stokes equations on the basis of probabilistic methods. In the stream function (vector potential) formulation, the Navier-Stokes equations are recast in path-integral forms by "the Feynman-Kac formula". We discuss the regularity of solutions on this basis.

##### Algebraic solutions of differential equations over the projective line minus three points

The Grothendieck–Katz p-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo p has vanishing p-curvatures for almost all p, has finite monodromy. It is known that it suffices to prove the conjecture for differential equations on the projective line minus three points.