# Seminars & Events for 2011-2012

##### A microscopic derivation of Ginzburg-Landau theory of superconductivity

We describe the first rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale.

##### On the Ma-Trudinger-Wang condition

The Ma-Trudinger-Wang condition, first introduced to prove regularity of optimal transport maps for general cost functions, has turned out to be a useful tool for: - Obtaining geometric informations on the underlying manifold. - Making the principal-agent problem theoretically and computationally tractable, allowing to derive uniqueness and stability of the principal's optimum strategy.

##### Noisy interpolation of sparse polynomials, and applications

Let f in $F_q[x]$ be a polynomial of degree $d < q/2$. It is well-known that f can be uniquely recovered from its values at some 2d points even after some small fraction of the values are corrupted. In this talk we will establish a similar result for sparse polynomials.

##### A parametrized version of Gromov's waist of the sphere theorem

Gromov, Memarian, and Karasev--Volovikov proved that any map $f$ from an n-sphere to a k-manifold $(n>=k)$ has a preimage $f^{-1}(z)$ whose epsilon-neighborhoods are at least as large as the epsilon-neighborhoods of the equator $S^{n-k}$, assuming that the degree of f is even in case $n=k$. We present a parametrized generalization.

##### Towards cornered Floer homology

As part of the bordered Floer homology package, Lipshitz, Ozsvath and D. Thurston have associated to a parametrized oriented surface a certain differential graded algebra. I will describe a decomposition theorem for this algebra, corresponding to cutting the surface along a circle.

##### Arithmetic inner product formula

I will introduce an arithmetic version of the classical Rallis' inner product formula for unitary groups, which generalizes the previous works of Kudla, Kudla-Rapoport-Yang and Bruinier-Yang. As Rallis' formula concerns the central L-values of automorphic representations with certain epsilon factor 1.

##### $W^{2,1}$ regularity for the Monge-Ampère equation

The Monge-Ampère equation arises in connections with several problems from geometry and analysis (optimal transport, the Minkowski problem, the affine sphere problem, etc.) The regularity theory for this equation has been widely studied.

##### Morse Theory and Invariants of (almost) Symplectic Manifolds

I will discuss two of my current projects which have different aims but use similar techniques. The first aims to understand the equivariant K-theory of symplectic orbifolds. The second is about the topology of toric origami manifolds. Neither space is a symplectic manifold, but each is almost.

##### Modulation invariant bilinear T(1) theorem

We discuss a T(1) theorem for bilinear singular integral operators with a one-dimensional modulation symmetry.

##### Prolates on the sphere, Extensions and Applications: Slepian functions for geophysical and cosmological signal estimation and spectral analysis

Functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access to, or are only interested in, a study area that is temporally or spatially bounded.

##### Random natural frequencies, active dynamics and coherence stability in populations of coupled rotators

The Kuramoto synchronization model is the reference model for synchronization phenomena in biology (and, to a certain extent, also in other fields). The model is formulated as a dynamical system of interacting plane rotators. Variations of it provide basic models of phenomena beyond synchronization, such as noise induced coherent oscillations.

##### Two gifts from complexity theory: $P$ v. $NP$ and matrix multiplication

I will discuss how the Geometric Complexity Theory of Mulmuley-Sohoni and the problem of determining the complexity of matrix multiplication lead to beautiful questions in algebraic geometry and representation theory.

##### The thermodynamic limit of disordered quantum Coulomb systems

In this talk I will explain how to prove the existence of the thermodynamic limit for a many-body quantum crystal in which the electrons are quantum and the nuclei are classical point particles, with disordered charges and locations around a lattice. This is a collaboration with Xavier Blanc, based on previous work with Christian Hainzl and Jan Philip Solovej.

##### Sharp Thresholds in Statistical Estimation

Sharp thresholds are ubiquitous high-dimensional combinatorial structures. The oldest example is probably the sudden emergence of the giant component in random graphs, first discovered by Erdos an Renyi.

##### Macdonald Processes and Some Applications in Probability and Integrable Systems

Macdonald processes are probability measures on sequences of partitions defined in terms of nonnegative specializations of the Macdonald symmetric functions and two parameters $q$, $t$ in [0,1). Utilizing the Macdonald difference operators we prove several results about observables these processes, including Fredholm determinant formulas for q-Laplace transforms.

##### Interacting Bosons in Random Potentials

In contrast to the cases of single particle Schrödinger operators and ideal Bose gases the effects of random external potentials on many-body systems of interacting particles is still poorly understood.

##### The chromatic number of random Cayley graphs

The study of random Cayley graphs of finite groups is related to the investigation of expanders and to problems in combinatorial number theory and in information theory. I will discuss this topic, focusing on the question of estimating the chromatic number of a random Cayley graph of a given group with a prescribed number of generators.

##### Axioms for Differential Cohomology

A differential cohomology theory (DCT) is a type of refinement of a cohomology theory (restricted to the category of smooth manifolds with corners) that contains information that is not homotopy invariant. A detailed definition of a DCT will be given, as well as axioms for a category larger than that of smooth manifolds with corners.

##### A second main term for counting cubic fields, and biases in arithmetic progressions

We prove the existence of second main term of order $X^{5/6}$ for the function counting cubic fields. This confirms a conjecture of Datskovsky-Wright and Roberts. We also prove a variety of generalizations, including to arithmetic progressions, where we discover a curious bias in the secondary term. Roberts' conjecture has also been proved independently by Bhargava, Shankar, and Tsimerman.

##### Effective dynamics of a non-linear wave equation

We consider the non-linear wave equation on the real line $i u_t-|D|u=|u|^2u$. Its resonant dynamics is given by the Szego equation, which is a completely integrable non-dispersive non-linear equation. We show that the solution of the wave equation can be approximated by that of the resonant dynamics for a long time.