# Seminars & Events for 2014-2015

##### A Lefschetz principle in non-archimedean geometry

PLEASE CLICK ON COLLOQUIUM TITLE FOR COMPLETE ABSTRACT: The "Lefschetz Principle" is the informal idea that for geometric questions about algebraic varieties over fields of characteristic 0, it is often sufficient to assume the ground field is the complex numbers (where analytic tools are available).

##### A toy model in fluid dynamics

Distribution of energy among various Fourier modes and how they fluctuate in a turbulent flow are central questions in fluid dynamics. I will speak about a toy model (introduced by Friedlander Katz Pavlovic) which aims to understand energy transport.

##### Coarse Ricci curvature and the manifold learning problem

We use the framework exploited by Bakry and Emery for the study of logarithmic Sobolev inequalities to consider the statistical problem of estimating the Ricci curvature (and more generally the Bakry-Emery tensor of a manifold with density) of an embedded submanifold of Euclidean space from a point cloud drawn from the submanifold.

##### From Stochastic Modeling to Fractional Modeling - New Tools in Computational Science & Engineering

##### Hodge theory and derived categories of cubic fourfolds

##### Generic K3 categories and Hodge theory

##### Regular operator mappings and multivariate geometric means

We introduce the notion of regular operator mappings of several variables generalising the notion of spectral function. This setting is convenient for studying maps more general than what can be obtained from the functional calculus, and it allows for Jensen type inequalities and multivariate non-commutative perspectives.

##### The Gauss Sphere Problem

The Gauss Sphere Problem is a long-open problem in analytic number theory concerning the number of lattice points in a sphere. In this talk I will discuss a variant of this problem: that of summing a polynomial over the lattice points in a sphere. We will use tools from Fourier analysis, exponential sums and modular forms, to sharpen our estimates step by step.

##### Expansion of Random Graphs - New Proofs, New Results

We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on deep results from combinatorial group theory. It applies to both regular and irregular random graphs. Let G be a random d-regular graph on n vertices.

##### Loop space homology, string homology, and closed geodesics

The homology of free loop space of a manifold enjoys additional structure first identified by Chas and Sullivan. The string multiplication has been studied by Ralph Cohen and John Jones and together with J.~Yan, they have introduced a spectral sequence converging to string homology that is related to the Serre spectral sequence for the free loop space.

##### Iwasawa Main Conjecture for Supersingular Elliptic Curves

We will describe a new strategy to prove the plus-minus main conjecture for elliptic curves having good supersingular reduction at p. It makes use of an ongoing work of Kings-Loeffler-Zerbes on explicit reciprocity laws for Beilinson-Flach elements to reduce to another main conjecture of Greenberg type, which can in turn be proved using Eisenstein congruences on the unitary group U(3,1).

##### Combinatorial tangle Floer homology

In joint work with Vera Vertesi, we extend the functoriality in Heegaard Floer homology by defining a Heegaard Floer invariant for tangles which satisfies a nice gluing formula. We will discuss the construction of this combinatorial invariant for tangles in S^3, D^3, and I x S^2. The special case of S^3gives back a stabilized version of knot Floer homology.

##### On the topology and index of minimal surfaces

We show that for an immersed two-sided minimal surface in R^3, there is a lower bound on the index depending on the genus and number of ends. Using this, we show the nonexistence of an embedded minimal surface

##### A modulated two-soliton with transient turbulent regime for a focusing cubic nonlinear half-wave equation on the real line

In this talk we discuss work in progress regarding a nonlocal focusing cubic half-wave equation on the real line. Evolution problems with nonlocal dispersion naturally arise in physical settings which include models for weak turbulence, continuum limits of lattice systems, and gravitational collapse.

##### Asymptotics beyond all orders: the devil's invention?

"Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever." --- N. H. Abel. The lecture will introduce the concept of an asymptotic series, showing how useful divergent series can be, despite Abel's reservations. We will then discuss Stokes' phenomenon, whereby the coefficients in the series appear to change discontinuously.

##### Statistical Mechanics on Sparse Random Graphs: Mathematical Perspective

Theoretical physics studies of disordered materials lead to challenging mathematical problems with applications to random combinatorial problems and coding theory. The underlying structure is that of many discrete variables that are strongly interacting according to a mean field model determined by a random sparse graph.

##### Symplectic fillings and star surgery

**This is a joint Topology-Symplectic Geometry seminar. Please note special time and location.** Although the existence of a symplectic filling is well-understood for many contact 3-manifolds, complete classifications of all symplectic fillings of a particular contact manifold are more rare.

##### Symplectic fillings and star surgery

**This is a joint Symplectic Geometry-Topology seminar. Please note special time and location.** Although the existence of a symplectic filling is well-understood for many contact 3-manifolds, complete classifications of all symplectic fillings of a particular contact manifold are more rare.

##### On complex geometry of some non-Kahler manifolds

In the talk we discuss a construction of a large family of complex manifolds. This family includes Hopf and Calabi-Eckmann manifolds. Although the question whether a given differentiable manifold admits a complex structure is extremely hard, our construction completely answers it in the case of manifolds with "large" group of U(1)^m-symmetries.

##### A blow-up result for dyadic models of fluid dynamics

Dyadic models in fluid dynamics are toy models for Euler and Navier-Stokes equations. Among many interesting results that can be proved in these models, we will focus on blow-up results; that is, some Sobolev norm can become infinite in finite time. This is joint work with Dong Li.