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Series: Job Candidate Talk

Motivated by rich applications in science and engineering, I am
interested in controlling systems that are characterized by multiple
scales, geometric structures, and randomness. This talk will focus on my
first two steps towards this goal.
The first step is to be able to simulate these systems. We developed
integrators that do not resolve fast scales in these systems but still
capture their effective contributions. These integrators require no
identification of underlying slow variables or processes, and therefore
work for a broad spectrum of systems (including stiff ODEs, SDEs and PDEs).
They also numerically preserve intrinsic geometric structures (e.g.,
symplecticity, invariant distribution, and other conservation laws), and
this leads to improved long time accuracy.
The second step is to understand what noises can do and utilize them. We
quantify noise-induced transitions by optimizing probabilities given by
Freidlin-Wentzell large deviation theory. In gradient systems, transitions
between metastable states were known to cross saddle points. We investigate
nongradient systems, and show transitions may instead cross unstable
periodic orbits. Numerical tools for identifying periodic orbits and for
computing transition paths are proposed. I will also describe how these
results help design control strategies.

Series: Job Candidate Talk

Christian Sadel is a Mathematical Physicists with broad spectrum of competences, who has been working in different areas, Random Matrix Theory (with H. Schulz-Baldes), discrete Schrödinger operators and tree graphs (with A. Klein), cocycle theory (with S. Jitomirskaya & A. Avila), SLE and spectral theory (with B. Virag), application to Mott transports in semiconductors (with J. Bellissard).

When P. Anderson introduced a model for the electronic structure in random disordered systems in 1958, such as randomly doped semiconductors, the surprise was his claim of the possibility of absence of diffusion for the electron motion. Today this phenomenon is called Anderson's localization and corresponds to pure point spectrum with exponentially decaying eigenfunctions for certain random Schrödinger operators (or Anderson models). Mathematically this phenomenon is quite well understood.For dimensions d≥3 and small disorder, the existence of diffusion, i.e. absolutely continuous spectrum, is expected, but mathematically still an open problem. In 1994, A. Klein gave a proof for a.c. spectrum for theinfinite-dimensional, hyperbolic, regular tree. However, generalizations to other hyperbolic trees and so-called "tree-strips" have only been made only in recent years. In my talk I will give an overview of the subject and these recent developments.

Series: Job Candidate Talk

Natural images tend to be compressible, i.e., the amount of information
needed to encode an image is small. This conciseness of information --
in other words, low dimensionality of the signal -- is found throughout a
plethora of applications ranging from MRI to quantum state tomography.
It is natural to ask: can the number of measurements needed to
determine a signal be comparable with the information content? We
explore this question under modern models of low-dimensionality and
measurement acquisition.

Series: Job Candidate Talk

We prove asymptotic stability of shear flows close to the
planar, periodic Couette flow in the 2D incompressible Euler
equations.That is, given an initial perturbation of the Couette flow small in a
suitable regularity class, specifically Gevrey space of class smaller
than 2, the velocity converges strongly in L2 to a shear flow which is also
close to the Couette flow. The vorticity is asymptotically mixed to
small scales by an almost linear evolution and in general enstrophy is lost
in the weak limit. Joint work with Nader Masmoudi. The strong convergence
of the velocity field is sometimes referred to as inviscid damping, due
to the relationship with Landau damping in the Vlasov equations. Recent
work with Nader Masmoudi and Clement Mouhot on Landau damping may also be
discussed.

Series: Job Candidate Talk

Let X be a connected smooth projective variety over Q. If X has a Q point, then X must have local points, i.e. points over the reals and over the p-adic completions Q_p. However, local solubility is often not sufficient. Manin showed that quadratic reciprocity together with higher reciprocity laws can obstruct the existence of a Q point (a global point) even when there exist local points. We will give an overview of this obstruction (in the case of quadratic reciprocity) and then show that for certain surfaces, this reciprocity obstruction can be viewed in a geometric manner. More precisely, we will show that for degree 4 del Pezzo surfaces, Manin's obstruction to the existence of a rational point is equivalent to the surface being fibered into genus 1 curves, each of which fail to be locally solvable.

Series: Job Candidate Talk

We will discuss a general theory of intertwined diffusion processes of any dimension. Intertwined processes arise in many different contexts in probability theory, most notably in the study of random matrices, random polymers and path decompositions of Brownian motion. Recently, they turned out to be also closely related to hyperbolic partial differential equations, symmetric polynomials and the corresponding random growth models. The talk will be devoted to these recent developments which also shed new light on some beautiful old examples of intertwinings. Based on joint works with Vadim Gorin and Soumik Pal.

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".