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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.

Series: Job Candidate Talk

An isoradial graph is one which can be embedded into the plane such that
each face is inscribable in a circle of common radius. We consider the
superposition of an isoradial graph, and its interior dual graph,
approximating a simply-connected domain, and prove that the height function
associated to the dimer configurations is conformally invariant in the
scaling limit, and has the same distribution as a Gaussian Free Field.

Series: Job Candidate Talk

Mixing by fluid flow is important in a variety of situations
in nature and technology. One effect fluid motion can have is to
strongly enhance diffusion. The extent of diffusion enhancement depends
on the properties of the flow. I will give an overview of the area, and
will discuss a sharp criterion describing a class of incompressible
flows that are especially effective mixers. The criterion uses spectral
properties of the dynamical system associated with the flow, and is
derived from a general result on decay rates for dissipative semigroups
of certain structure. The proofs rely on methods developed in studies of
wavepacket spreading in mathematical quantum mechanics.

Series: Job Candidate Talk

Chemical
polymers are long chains of molecules built up from many individual
monomers. Examples are plastics (like polyester and PVC), biopolymers
(like cellulose, DNA, and starch) and rubber. By some estimates over 60%
of research in the chemical industry is related to polymers. The
complex shapes and seemingly random dynamics inherent in polymer chains
make them natural candidates for mathematical modelling. The probability
and statistical physics literature abounds with polymer models, and
while most are simple to understand they are notoriously difficult to
analyze. In
this talk I will describe the general flavor of polymer models and then
speak more in depth on my own recent results for two specific models.
The first is the self-avoiding walk in two dimensions, which has
recently become amenable to study thanks to the invention of the
Schramm-Loewner Evolution. Joint work with Hugo-Duminil Copin shows that
a specific feature of the self-avoiding walk, called the bridge decomposition,
carries over to its conjectured scaling limit, the SLE(8/3) process.
The second model is for directed polymers in dimension 1+1. Recent joint
work with Kostya Khanin and Jeremy Quastel shows that this model can be
fully understood when one considers the polymer in the previously
undetected "intermediate" disorder regime.
This work ultimately leads to the construction of a new type of
diffusion process, similar to but actually very different from Brownian motion.

Series: Job Candidate Talk

In this era of "big data", Mathematics as it applies to
human behavior is becoming a much more relevant and penetrable topic
of research. This holds true even for some of the less desirable
forms of human behavior, such as crime. In this talk, I will discuss
the mathematical modeling of crime on various "scales" and using
many different mathematical techniques, as well as the results of
experiments
that are being performed to test the usefulness and accuracy of these
models.
This will include: models of crime hotspots at the scale of neighborhoods
-- in
the form of systems of PDEs and also statistical models adapted from
literature on
earthquake predictions -- along with the results of the model's application
within the LAPD; a model for gang retaliatory violence on the scale of
social
networks, and its use in the solution of an inverse problem to help solve
gang crimes; and a game-theoretic model of crime and punishment at the
scale of a society, with comparisons of the model to results of lab-based
economic experiments performed by myself and collaborators.