# Seminars & Events for 2013-2014

##### The Van der Corput lemma and equations in Physics

I will prove the Van der Corput lemma and then describe how it applies to partial differential equations arising in Physics. It will provide intuition for a number of qualitative properties of non-relativistic/relativistic massive particles and massless particles. For example, a single picture will show that relativistic massive particles have rest mass and cannot reach the speed of light.

##### Low-lying Fundamental Geodesics

It is classical that an element of the class group of a real quadratic field corresponds to a closed geodesic on the modular surface, but not every closed geodesic arises this way; we call those that do "fundamental." Given a fixed compact subset W of (the unit tangent bundle of) the modular surface, we say a closed geodesic is "low-lying" if it is contained in W; in particular, it does not ent

##### Polynomial bounds for the grid-minor theorem

One of the key results in Robertson and Seymour's seminal work on graph minors is the grid-minor theorem, that every graph of sufficiently large treewidth contains any fixed grid as a minor. This theorem has found many applications in graph theory and algorithms. Let f(k) be maximum such that every graph of treewidth at least k contains a grid minor of size f(k).

##### On the well-posedness of an interface damped free boundary fluid-structure model

We address a fluid-structure system which consists of the incompressible Navier-Stokes equations and a damped linear wave equation defined on two dynamic domains. The equations are coupled through transmission boundary conditions and additional boundary stabilization effects imposed on the free moving interface separating the two domains.

##### Cylindrical contact homology as a well-defined homology?

In this talk I will explain how the heuristic arguments sketched in literature since 1999 fail to define a homology theory. These issues will be made clear with concrete examples and we will explore what stronger conditions are necessary to develop a theory without the use of virtual chains or polyfolds in 3 dimensions.

##### The hyperbolic Ax-Lindemann conjecture

**Please note special day, time and location. **The hyperbolic Ax Lindemann conjecture is a functional transcendental statement which describes the closure of "algebraic flows" on Shimura varieties. We will describe the proof of this conjecture and its consequences for the André-Oort conjecture. This is a joint work with Bruno Klingler and Andrei Yafaev.

##### Quantum groups and quantum cohomology

Quantum groups, which originated in mathematical physics in the study of solvable 1+1 dimensional models, are noncocommutative Hopf algebra deformations of the universal enveloping of a Lie algebra. PLEASE CLICK ON LECTURE TITLE FOR COMPLETE ABSTRACT.

##### Local well-posedness for the equation of minimal hypersurface in Minkowski space

A timelike minimal hypersurface in Minkowski space satisfies a quasilinear wave equation. I will explain how the minimal hypersurface equation exhibits a null structure and how to utilize the null structure in order to lower the regularity requirements on the initial data for the Cauchy problem.

##### Fast Direct Solvers for Elliptic PDEs

That the linear systems arising upon the discretization of elliptic PDEs can be solved very efficiently is well-known, and many successful iterative solvers with linear complexity have been constructed (multigrid, Krylov methods, etc).

##### Physical Principles Underlying the Fractional Quantum Hall Effect

PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT. I review an approach to the theory of the Quantum Hall Effect (QHE) somewhat analogous to Landau's theory of phase transitions

##### The geometric genus of normal surface singularities

We discuss several topological characterizations of the geometric genus of a complex normal surface singularity under certain topological and analytic restrictions. The `classical' cases include the rational and elliptic singularities.

##### Universal spaces for birational invariants

Anabelian geometry techniques allow the construction of explicit universal spaces which capture birational properties of algebraic varieties. I will describe this theory and its applications (joint with F. Bogomolov).

##### Curry-Howard Correspondence

The Curry-Howard correspondence relates formal proofs in logic and computer programs in a typed functional programming language. It is the underlying theory behind Coq, one of the major systems today for verifying correctness of proofs using computers. The theory is further extended by Vladimir Voevodsky in the univalent foundations program.

##### Virtual domination of $3$-manifolds

For any closed oriented hyperbolic $3$-manifold $M$, and any closed oriented $3$-manifold $N$, we will show that $M$ admits a finite cover $M'$, such that there exists a degree-$2$ map $f: M' \rightarrow N$.

##### Hypergraphs of bounded disjointness

A k-uniform hypergraph is said to be intersecting if no pair of edges is disjoint. The maximal size of an intersecting k-uniform hypergraph with a given groundset is given by the beautiful and well-known theorem of Erdos, Ko and Rado. A k-uniform hypergraph is s-almost intersecting if every edge is disjoint from exactly s other edges.

##### Remarks on the cohomology of the Lubin-Tate tower

##### The 2D Magnetohydrodynamic Equations with Partial Dissipation

The magnetohydrodynamic (MHD) equations model electrically conducting fluids such as plasmas and are important in understanding many natural phenomena such as solar flares and the formation of Northern Lights. Mathematically the MHD equations can be difficult to analyze due to the nonlinear coupling between the induction equation and the Navier-Stokes equations with the Lorentz force.

##### On Floer cohomology and non-archimedian geometry

Ideas of Kontsevich-Soibelman and Fukaya indicate that there is a natural rigid analytic space (the mirror) associated to a symplectic manifold equipped with a Lagrangian torus fibration. I will explain a construction which associates to a Lagrangian submanifold a sheaf on this space, and explain how this should be the mirror functor.

##### Quantum groups and quantum cohomology - second lecture

This lecture is #2 in a series of 10 lectures: Quantum groups, which originated in mathematical physics in the study of solvable 1+1 dimensional models, are noncocommutative Hopf algebra deformations of the universal enveloping of a Lie algebra.The adjective "quantum" in quantum cohomology has a very different meaning and origin.

##### New Stable Periodic Solutions to the n-Body Problem

Recently, some new surprisingly-elegant stable periodic solutions to the 4-body problem were discovered. I will review the current state-of-the-art for finding periodic solutions to the n-body problem and I will describethe interesting new ones that have been found. I will also discuss the key question of the stability of these orbits.