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Series: Job Candidate Talk

We will start by describing some general features of quasilinear
dispersive and wave equations. In particular we will discuss a few
important aspects related to the question of global regularity for such
equations.
We will then consider the water waves system for the evolution of a
perfect fluid with a free boundary. In 2 spatial dimensions, under the
influence of gravity, we prove the existence of global irrotational
solutions for suitably small and regular initial data. We also prove
that the asymptotic behavior of solutions as time goes to infinity is
different from linear, unlike the 3 dimensional case.

Series: Job Candidate Talk

The Anderson model on a discrete graph is given by the graph Laplacian perturbed by a random potential. I study spectral properties of this random Schroedinger operator on a random regular graph of fixed degree in the limit where the number of vertices tends to infinity.The choice of model is motivated by its relation to two important and well-studied models of random operators: On the one hand there are similarities to random matrices, for instance to Wigner matrices, whose spectra are known to obey universal laws. On the other hand a random Schroedinger operator on a random regular graph is expected to approximate the Anderson model on the homogeneous tree, a model where both localization (characterized by pure point spectrum) and delocalization (characterized by absolutely continuous spectrum) was established.I will show that the Anderson model on a random regular graph also exhibits distinct spectral regimes of localization and of delocalization. One regime is characterized by exponential decay of eigenvectors. In this regime I analyze the local eigenvalue statistics and prove that the point process generated by the eigenvalues of the random operator converges in distribution to a Poisson process.In contrast to that I will also show that the model exhibits a spectral regime of delocalization where eigenvectors are not exponentially localized.

Series: Job Candidate Talk

Out-of-equilibrium dynamics are a characteristic feature of the
long-time behavior of nonlinear dispersive equations on bounded domains.
This is partly due to the fact that dispersion does not translate into
decay in this setting (in contrast to the case of unbounded domains like
$R^d$). In this talk, we will take the cubic nonlinear Schroedinger
equation as our model, and discuss some aspects of its
out-of-equilibrium dynamics, from energy cascades (i.e. migration of
energy from low to high frequencies) to weak turbulence.

Series: Job Candidate Talk

Discrepancy theory, also referred to as the theory of irregularities of distribution, has been developed into a diverse and fascinating field, with numerous closely related areas, including, numerical integration, Ramsey theory, graph theory, geometry, and theoretical computer science, to name a few. Informally, given a set system S defined over an n-item set X, the combinatorial discrepancy is the minimum, over all two-colorings of X, of the largest deviation from an even split, over all sets in S. Since the celebrated ``six standard deviations suffice'' paper of Spencer in 1985, several long standing open problems in the theory of combinatorial discrepancy have been resolved, including tight discrepancy bounds for halfspaces in d-dimensions [Matousek 1995] and arithmetic progressions [Matousek and Spencer 1996]. In this talk, I will present new discrepancy bounds for set systems of bounded ``primal shatter dimension'', with the property that these bounds are sensitive to the actual set sizes. These bounds are nearly-optimal. Such set systems are abstract, but they can be realized by simply-shaped regions, as halfspaces, balls, and octants in d-dimensions, to name a few. Our analysis exploits the so-called "entropy method" and the technique of "partial coloring", combined with the existence of small "packings".

Series: Job Candidate Talk

First-passage percolation is a model of a random metric on a infinite network. It deals with a collection of points which can be reached within a given time from a fixed starting point, when the network of roads is given, but the passage times of the road are random. It was introduced back in the 60's but most of its fundamental questions are still open. In this talk, we will overview some recent advances in this model focusing on the existence, fluctuation and geometry of its geodesics. Based on joint works with M. Damron and J. Hanson.

Series: Job Candidate Talk

Kronecker coefficients lie at the intersection of representation theory, algebraic combinatorics and, most recently, complexity theory. They count the multiplicities of irreducible representations in the tensor product of two other irreducible representations of the symmetric group. While their study was initiated almost 75 years, remarkably little is known about them. One of the major problems of algebraic combinatorics is to find an explicit positive combinatorial formula for these coefficients. Recently, this problem found a new meaning in the field of Geometric Complexity Theory, initiated by Mulmuley and Sohoni, where certain conjectures on the complexity of computing and deciding positivity of Kronecker coefficients are part of a program to prove the "P vs NP" problem. In this talk we will give an overview of this topic and we will describe several problems with some results on different aspects of the Kronecker coefficients. We will explore Saxl conjecture stating that the tensor square of certain irreducible representation of S_n contains every irreducible representation, and present a criterion for determining when a Kronecker coefficient is nonzero. In a more combinatorial direction, we will show how to prove certain unimodality results using Kronecker coefficients, including the classical Sylvester's theorem on the unimodality of q-binomial coefficients (as polynomials in q). We will also present some results on complexity in light of Mulmuley's conjectures. The presented results are based on joint work with Igor Pak and Ernesto Vallejo.

Series: Job Candidate Talk

Many fundamental theorems in extremal graph theory can be expressed as linear inequalities between homomorphism densities. Lovasz and, in a slightly different formulation, Razborov asked whether it is true that every such inequality follows from a finite number of applications of the Cauchy-Schwarz inequality. In this talk we will show that the answer to this question is negative. Further, we will show that the problem of determining the validity of a linear inequality between homomorphism densities is undecidable. Hence such inequalities are inherently difficult in their full generality. These results are joint work with Hamed Hatami. On the other hand, the Cauchy-Schwarz inequality (a.k.a. the semidefinite method) represents a powerful tool for obtaining _particular_ results in asymptotic extremal graph theory. Razborov's flag algebras provide a formalization of this method and have been used in over twenty papers in the last four years. We will describe an application of flag algebras to Turan’s brickyard problem: the problem of determining the crossing number of the complete bipartite graph K_{m,n}. This result is based joint work with Yori Zwols.

Series: Job Candidate Talk

In synthetic aperture radar (SAR) imaging, two important applications are formation of high resolution images and motion estimation of moving targets on the ground. In scenes with both stationary targets and moving targets, two problems arise. Moving targets appear in the computed image as a blurred extended target in the wrong location. Also, the presence of many stationary targets in the vicinity of the moving targets prevents existing algorithms for monostatic SAR from estimating the motion of the moving targets. In this talk I will discuss a data pre-processing strategy I developed to address the challenge of motion estimation in complex scenes. The approach involves decomposing the SAR data into components that correspond to the stationary targets and the moving targets, respectively. Once the decomposition is computed, existing algorithms can be applied to compute images of the stationary targets alone. Similarly, the velocity estimation and imaging of the moving targets can then be carried out separately.The approach for data decomposition adapts a recent development from compressed sensing and convex optimization ideas, namely robust principle component analysis (robust PCA), to the SAR problem. Classicalresults of Szego on the distribution of eigenvalues for Toeplitz matrices and more recent results on g-Toeplitz and g-Hankel matrices are key for the analysis. Numerical simulations will be presented.

Series: Job Candidate Talk

Electrical stimulation of cardiac cells causes an action
potential wave to propagate through myocardial tissue, resulting in
muscular contraction and pumping blood through the body. Approximately two
thirds of unexpected, sudden cardiac deaths, presumably due to ventricular
arrhythmias, occur without recognition of cardiac disease. While
conduction failure has been linked to arrhythmia, the major players in
conduction have yet to be well established. Additionally, recent
experimental studies have shown that ephaptic coupling, or field effects,
occurring in microdomains may be another method of communication between
cardiac cells, bringing into question the classic understanding that
action potential propagation occurs primarily through gap junctions. In
this talk, I will introduce the mechanisms behind cardiac conduction, give
an overview of previously studied models, and present and discuss results
from a new model for the electrical activity in cardiac cells with
simplifications that afford more efficient numerical simulation, yet
capture complex cellular geometry and spatial inhomogeneities that are
critical to ephaptic coupling.

Series: Job Candidate Talk

We will discuss amenability of the topological full group of a minimal Cantor system. Together with the results of H. Matui this provides examples of finitely generated simple amenable groups. Joint with N. Monod.