# Seminars & Events for 2008-2009

##### Large deviations of the current and phase transitions

Using the framework of the hydrodynamic limits, we will discuss the large deviations of a particle current through a diffusive system. The deviations can lead to dynamical phase transitions. In the case of asymmetric dynamics we will explain how the large deviation functional of the current provides a physical interpretation to the non-entropic solutions of Burgers equation.

##### Bordered Floer homology: bimodules and computations

We will review the structure of bordered Floer homology, including how it depends on the parametrization of the boundary. We will then discuss how to compute it, and consequently another algorithm for computing HF-hat. This is work in progress with Peter Ozsvath and Dylan Thurston.

##### A rigid irregular connection on the projective line

From the trace formula and the global Langlands correspondence one can infer the existence of a particular rigid l-adic local system on the projective line with tame ramification at 0 and wild ramification, of the mildest possible kind, at infinity, for any simple algebraic group.

##### The number of 3-SAT functions

We are interested in the number, say $G(k,n)$, of k-SAT functions of $n$ variables (a k-SAT function being a Boolean function representable by a k-SAT formula in, say, conjunctive normal form).

We show that $G(3,n)$ is asymptotic to $2^{n + {n \choose 3}}$, a strong form of a conjecture of Bollobas, Brightwell and Leader.

##### 11-dimensional supergravity and Dirichlet problem for forms on asymptotically hyperbolic spaces

##### Entire functions and gap theorems

Several classical problems of Analysis can be translated into a universal language based on Hilbert spaces of entire functions and kernels of Toeplitz operators. Problems that can be treated this way include completeness/minimality problems for systems of exponentials or special functions in $L^2$ and spectral problems for second order differential operators.

##### Coupling Einstein's equations to Dirac spinors can prevent the big bang/crunch singularity in the Friedmann model

We consider a spatially homogeneous and isotropic system of Dirac particles coupled to classical gravity. We recover, on the one hand, the dust and radiation dominated closed Friedmann-Robertson-Walker space-times. On the other hand, we find particular solutions where the oscillations of the Dirac spinors prevent the formation of the big bang or big crunch singularity.

##### Constructing moduli spaces of objects with infinite automorphisms

Moduli problems parameterizing objects with infinite automorphisms (eg. semi-stable vector bundles) often do not admit coarse moduli schemes but may admit moduli schemes identifying certain non-isomorphic objects. I will introduce techniques to study such moduli stacks and address the question of how such moduli schemes can be intrinsically constructed.

##### Mathai-Quillen's Thom form and Atiyah-Hirzebruch's Riemann-Roch theorem

After Hirzebruch's generalization of the classical Riemann-Roch formula, Grothendieck extended this result to a relative version. Then Atiyah and Hirzebruch gave a "differentiable analogue" of Grothendieck's theorem, which is called Atiyah-Hirzebruch's Riemann-Roch theorem.

##### On mixing properties of locally Hamiltonian flows on surfaces

We consider area-preserving flows on surfaces which are locally given by smooth Hamiltonians. It turns out that the presence or absence of mixing depends on the type of fixed points. We proved in our PhD thesis that the presence of centers is generically enough to create mixing. Recently we showed that if such flows have only saddles, they are generically not mixing, but weakly mixing.

##### Linear Programming Relaxations for the TSP

The most successful solution approaches to the traveling salesman problem are based on lower bounds obtained from linear programming relaxations. We will discuss a number of open problems concerning this approach, with emphasis on topics that could improve the practical performance of LP-based algorithms.

##### Existence and rigidity of pseudo-Anosov flows transverse to R-covered foliations

Pseudo-Anosov flows are extremely common in three manifolds and they are very useful. How many pseudo-Anosov flows are there in a manifold up to topological conjugacy? We analyse this question in the context of flows transverse to a given foliation F.

##### Eigenvarieties and $p$-adic families of finite slope automorphic representations

##### Stochastic finite element approximations of elliptic problems of higher stochastic order

In this talk we will address numerical methods for two stochastic elliptic models, where the coefficients are perturbed by colored noise or white noise. An overview of the development of numerical methods will be given for the first model. We will focus on a stochastic Galerkin finite element method for the second model.

##### A Second Boundary Value Problem for special Lagrangian submanifolds

Given any two uniformly convex regions in Euclidean space, we show that there exists a unique diffeomorphism between them, such that the graph of the diffeomorphism is a special Lagrangian submanifold in the product space. This is joint work with Simon Brendle.

##### Surface Correspondence via Discrete Uniformization

Many applied-science fields like medical imaging, computer graphics and biology use meshes to model surfaces. It is a challenging problem to determine whether, how and to what extent such surfaces correspond to each other, e.g. to see whether they are differently parametrized views of one object, or whether they indicate movement of part of an object with respect to its other parts.

##### $h$—Principle and fluid dynamics

Joint Princeton University and Institute for Advanced Study Analysis Seminar

##### New results for reaction-diffusion equations arising from reversible chemistry

Entropy/entropy dissipation methods have been used with success lately in the study of the large time behavior of kinetic equations, nonlinear diffusions, etc., and have led to the development of the concept of hypocoercivity.

##### An Asymptotic Expansion for the Dimer $\Lambda_d$

The dimer problem is to count the number of ways a $d$-dimensional "chessboard" can be completely covered by non-overlapping dimers (dominoes), each dimer covering two nearest neighbor boxes. The number is approximately $exp^{\Lambda_d*V}$ as the volume $V$ goes to infinity.

##### Vector bundles with sections

Classical Brill-Noether theory studies, for given $g, r, d$, the space of line bundles of degree $d$ with $r+1$ global sections on a curve of genus $g$. We will review the main results in this theory, and the role of degeneration techniques in proving them, and then we will discuss the situation for higher-rank vector bundles, where even the most basic questions remain wide open.