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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.

Series: Job Candidate Talk

I will discuss regularity of fully nonlinear elliptic equations
when the usual uniform upper bound on the ellipticity is
replaced by bound on its $L^d$ norm, where $d$ is the dimension of
the ambient space. Our estimates refine the classical theory and require
several new ideas that we believe are of independent interest. As an
application, we prove homogenization for a class of stationary ergodic
strictly elliptic equations.

Series: Job Candidate Talk

Must the Fourier series of an L^2 function converge pointwise almost
everywhere? In the 1960's, Carleson answered this question in the
affirmative, by studying a particular type of maximal singular integral
operator, which has since become known as the Carleson operator. In the
past 40 years, a number of important results have been proved for
generalizations of the original Carleson operator. In this talk we will
describe new joint work with Po Lam Yung that introduces curved
structure to the setting of Carleson operators.

Series: Job Candidate Talk

We study the nonparametric regression model (X1 , Y1 ), ...(Xn , Yn ) , where (Xi )i≥1 is the deterministic design and (Yi )i≥1 is a sequence of real random variables. Assume that the density of Yi is known and can be written as g (., f (Xi )) , which depends on a regression function f at the point Xi . The function f is assumed smooth, i.e. belonging to a Hoelder ball or a Nikol’ski ball. The aim is to estimate the regression function from the observations for two error risks (pointwise and global estimations) and to ﬁnd the optimal estimator (in the sense of rates of convergence) for each density g . We are particularly interested in the study of irregular models, i.e. when the Fisher information does not exist (for example, when the density g is discontinuous like the uniform density). In this case, the rate of convergence can be improved with the use of nolinear estimators like Maximum likelihood or bayesian estimators. We use the locally parametric approach to construct a new local version of bayesian estimators. Under some conditions on the likelihood of the model, we propose an adaptive procedure based on the so-called Lepski’s method (adaptive selection of the bandwidth) which allows us to construct an optimal adaptive bayesian estimator. We apply this theory to several models like multiplicative uniform model, shifted exponential model, alpha model, inhomogeous Poisson model and Gaussian model

Series: Job Candidate Talk

I'll show how on metric measure spaces with Ricci
curvature bounded from below in the sense of Lott-Sturm-Villani there
is a well defined notion of Heat flow, and how the study of the
properties of this flow leads to interesting geometric and analytic
properties of the spaces themselves. A particular attention will be
given to the class of spaces where the Heat flow is linear. (From a
collaboration with Ambrosio and Savare')

Series: Job Candidate Talk

Multiscale numerical methods seek to compute approximate solutions to
physical problems at a reduced computational cost compared to direct
numerical simulations. This talk will cover two methods which have a
fine scale atomistic model that couples to a coarse scale continuum
approximation.
The quasicontinuum method directly couples a continuum approximation
to an atomistic model to create a coherent model for computing
deformed configurations of crystalline lattices at zero temperature.
The details of the interface between these two models greatly affects
the model properties, and we will discuss the interface consistency,
material stability, and error for energy-based and force-based
quasicontinuum variants along with the implications for algorithm
selection.
In the case of crystalline lattices at zero temperature, the
constitutive law between stress and strain is computed using the
Cauchy-Born rule (the lattice deformation is locally linear and equal
to the gradient). For the case of complex fluids, computing the
stress-strain relation using a molecular model is more challenging
since imposing a strain requires forcing the fluid out of equilibrium,
the subject of nonequilibrium molecular dynamics. I will describe the
derivation of a stochastic model for the simulation of a molecular
system at a given strain rate and temperature.

Series: Job Candidate Talk

In his thesis, Margulis computed the asymptotic growth rate for the number of closed geodesics of length less than R on a given closed hyperbolic surface and his argument has been emulated to many other settings. We examine the Teichmüller geodesic flow on the moduli space of a surface, or more generally any stratum of quadratic differentials in the cotangent bundle of moduli space. The flow is known to be mixing, but the spaces are not compact and the flow is not uniformly hyperbolic. We show that the random walk associated to the Teichmüller geodesic ﬂow is biased toward the compact part of the stratum. We then use this to find asymptotic growth rate of for the number of closed loops in the stratum. (This is a joint work with Alex Eskin and Maryam Mirzakhani.)