# Seminars & Events for 2007-2008

##### Undecidability and Hilbert's 10th Problem

Hilbert's famous 10th problem asks whether there is a general algorithm to decide whether a polynomial in many variables has a solution. I will explain how Robinson, Davis, Putnam, and Matijasevic answered the question in the negative by proving a much more interesting theorem: that any listable set can be listed using a polynomial equation.

##### Invariant curves near the boundary of an annulus without a twist hypothesis, following M. R. Herman

Sometime in the nineties, M. R. Herman gave a series of lectures at Columbia on KAM theory. Yasha asked me to speak on one of the results that Herman discussed in his lectures. Here is the result:

##### Averaging Points Two at a Time

In 2006 Brendan McKay asked the following on sci.math.research: We have n points in a disk centered at the centroid of the points. We successively replace the two furthest points from each other by two copies of their average. (After each move we still have n points with the same centroid.

##### On the renormalized volume of quasifuchsian manifolds

The renormalized volume of quasifuchsian hyperbolic 3-manifolds was originally introduced for physical reasons. Takhtajan and Zograf (and others) discovered that it provides a Kähler potential for the Weil-Petersson metric on Teichmüller space. We will give an elementary, differential-geometric account of this result.

##### On a result of Waldspurger in higher rank

An important result of Waldspurger relates the central value of quadratic base change L-functions for GL(2) to period integrals over tori. Subsequently this result was reproved by Jacquet using the relative trace formula. We will explain some progress on extending Waldspurger's result to higher rank via a generalization of Jacquet's approach.

##### Mirror symmetry of Fano toric A-model and Landau-Ginzburg B-model

In this talk, I will introduce the notion of weakly unobstructed Lagrangian submanifolds and balanced Lagrangian submanifolds.

##### The decomposition of global conformal invariants: On a conjecture of Deser and Schwimmer

Global conformal invariants are integrals of geometric scalars which remain invariant under conformal changes of the underlying metric. I will discuss (parts of) my recent proof of a conjecture of Deser and Schwimmer, which states that any such global invariant can be decomposed into standard "building blocks" of three types.

##### Airplane boarding and space-time geometry

It is hard to think of a process that is more boring than boarding an airplane. In the hope of relieving, or at least shortening, some of the pain, airlines have devised various boarding strategies such as back-to-front, window to aisle, boarding by zones or even unassigned seating.

##### The Composite Membrane Problem

We wish to build a body of prescribed shape, and of prescribed mass out of materials of varying density so as to minimize the first Dirichlet eigenvalue with fixed boundary of the body. Existence, uniqueness and regularity of the solution and the resulting free boundary problem will be discussed.

##### Beyond value at risk

Market turmoil and periods of high volatility tend to undermine standard portfolio risk models, which are predicated on the assumption of conditional normality. These models do not account for the extreme events that account for a substantial share of

portfolio loss.

##### Non-Hermitian Anderson model: Lyapunov exponents, eigenvalues, and eigenfunctions

The Non-Hermitian Anderson model was introduce in 1996 by N. Hatano and D. Nelson. Their numerical studies reveled very interesting and unusual spectral properties of this model. The aim of my talk is to explain how the theory of Lyapunov exponents allows one to:

(a) obtain the equations for the curves on which the non-real eigenvalues lie

##### Characters of finite Chevalley groups and categorification

An important branch of representation theory studies representations of reductive groups over finite fields, such as $GL(n,F_q), Sp(2n,F_q)$ etc. A deep theory due mostly to Lusztig and Shoji provides a classification of irreducible representations and a formula for their characters in terms of certain algebro-geometric objects called character sheaves. In a joint work with M.

##### Matriod Theory

Every maximal acyclic subgraph of a given graph has the same number of edges. Every maximal independent subset of a given set of vectors in $\mathbb{R}^n$ has the same size. The common generalization of these lies in matroid theory, which, roughly speaking, is an abstraction of the notion of linear independence.

##### The Finite Field Kakeya Problem

If V is a vector space over the field with q elements, a (finite field) Kakeya set is a subset of V containing a line in every direction. The main problem is to determine how large such a set is forced to be. Recently, Zeev Dvir gave a simple proof of the correct order of magnitude of Kakeya sets by introducing a nice technique from algebraic geometry.

##### Nontrivial coupling at quantum graph vertices obtained through squeezing of Dirichlet networks

The problem discussed in this talk is motivated by efforts to understand approximation of quantum graph Hamiltonians by Laplacians on families of "fat graphs." The emphasis is on new results in the Dirichlet case, however, first we review the background and explain the importance of vertex boundary conditions using a lattice graph example, and mention known result in both the Neumann and Diric

##### Loop products and closed geodesics

The critical points of the energy function on the free loop space $L(M)$ of a compact Riemannian manifold $M$ are the closed geodesics on $M$. Filtration by the length function gives a link between the geometry of closed geodesics and the algebraic structure given by the Chas-Sullivan product on the homology of $L(M)$. Geometry reveals the existence of a related product on the cohomology of $

##### Nonvanishing mod p of Eisenstein series

##### Numerical Methods in Calabi-Yau Compactications of String Theory

After a brief introduction to N=1 compatifications in String Theory, it will become clear why one needs to know explicit solutions to important PDEs, such as the Kaehler-Einstein metrics. This fact motivates the use of numerical methods to approximate solutions to such PDEs.

##### Hamiltonian formulation of general relativity and quasilocal mass

Isometric embeddings of surfaces into the Minkowski space are used as references to derive a quasilocal mass expression from the Hamiltonian formulation of Einstein's equation. This involves an existence and uniqueness theorem of isometric embeddings and a canonical choice of time gauges.

##### Optimal curvature decays on asymptotically locally Euclidean manifolds

We present a method to study curvature decays on asymptotically locally Euclidean manifolds. The method is flexible and can also be applied to elliptic systems of reaction-diffusion type.