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Series: Job Candidate Talk

Experimental neuroscience is achieving rapid progress in the ability
to collect neural activity and connectivity data. This holds promise to
directly test many theoretical ideas, and thus advance our understanding
of "how the brain works." How to interpret this data, and what exactly
it can tell us about the structure of neural circuits, is still not
well-understood. A major obstacle is that these data often measure
quantities that are related to more "fundamental" variables by an
unknown nonlinear transformation. We find that combinatorial topology
can be used to obtain meaningful answers to questions about the
structure of neural activity.
In this talk I will first introduce a new method, using tools from
computational topology, for detecting structure in correlation matrices
that is obscured by an unknown nonlinear transformation. I will
illustrate its use by testing the "coding space" hypothesis on neural
data. In the second part of my talk I will attempt to answer a simple
question: given a complete set of binary response patterns of a network,
can we rule out that the network functions as a collection of
disconnected discriminators (perceptrons)? Mathematically this
translates into questions about the combinatorics of hyperplane
arrangements and convex sets.

Series: Job Candidate Talk

Synapses in many cortical areas of the brain are dominated by local, recurrent connections. It has long been suggested, therefore, that cortical networks may serve to restore a noisy or incomplete signal by evolving it towards a stored pattern of activity. These "preferred" activity patterns are constrained by the excitatory connections, and comprise the neural code of the recurrent network. In this talk I will briefly review the permitted and forbidden sets model for cortical networks, first introduced by Hahnloser et. al. (Nature, 2000), in which preferred activity patterns are modeled as "permitted sets" - that is, as subsets of neurons that co-fire at stable fixed points of the network dynamics. I will then present some recent results that provide a geometric handle on the relationship between permitted sets and network connectivity. This allows us to precisely characterize the structure of neural codes that arise from a simple learning rule. In particular, we find "natural codes" that can be learned from few examples, and that closely mimic receptive field codes that have been observed in the brain. Finally, we use our geometric description of permitted sets to prove that these networks can perform error correction and pattern completion for a wide range of connectivities.

Series: Job Candidate Talk

Driving nanomagnets by spin-polarized currents offers exciting
prospects in magnetoelectronics, but the response of the magnet to
such currents remains poorly understood. For a single domain
ferromagnet, I will show that an averaged equation describing the
diffusion of energy on a graph captures the low-damping dynamics of
these systems. In particular, I compute the mean times of thermally
assisted magnetization reversals in the finite temperature system,
giving explicit expressions for the effective energy barriers
conjectured to exist. I will then outline the problem of extending the
analysis to spatially non-uniform magnets, leading to a transition
state theory for infinite dimensional Hamiltonian systems.

Series: Job Candidate Talk

Since Lord Rayleigh conjectured that the disk should minimize the first eigenvalue of the Laplace-Dirichlet operator among all shapes of equal area more than a century ago, extremal eigenvalue problems have been an active research topic. In this talk, I'll demonstrate how extremal eigenvalue problems arise in a variety of contexts, including optics, geometry, and data analysis, and present some recent analytical and computational results in these areas. One of the results I'll discuss is a new graph partitioning method where the optimality criterion is given by the sum of the Dirichlet energies of the partition components. With intuition gained from an analogous continuous problem, we introduce a rearrangement algorithm, which we show to converge in a finite number of iterations to a local minimum of a relaxed objective function. The method compares well to state-of-the-art approaches when applied to clustering problems on graphs constructed from synthetic data, MNIST handwritten digits, and manifold discretizations.

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.