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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.)

Series: Job Candidate Talk

Waterborne diseases cause over 3.5 million deaths annually, with cholera
alone responsible for 3-5 million cases/year and over 100,000
deaths/year. Many waterborne diseases exhibit multiple characteristic
timescales or pathways of infection, which can be modeled as direct and
indirect transmission. A major public health issue for waterborne
diseases involves understanding the modes of transmission in order to
improve control and prevention strategies. One question of interest is:
given data for an outbreak, can we determine the role and relative
importance of direct vs. environmental/waterborne routes of
transmission? We examine these issues by exploring the identifiability
and parameter estimation of a differential equation model of waterborne
disease transmission dynamics. We use a novel differential algebra
approach together with several numerical approaches to examine the
theoretical and practical identifiability of a waterborne disease model
and establish if it is possible to determine the transmission rates from
outbreak case data (i.e. whether the transmission rates are
identifiable). Our results show that both direct and environmental
transmission routes are identifiable, though they become practically
unidentifiable with fast water dynamics. Adding measurements of pathogen
shedding or water concentration can improve identifiability and allow
more accurate estimation of waterborne transmission parameters, as well
as the basic reproduction number. Parameter estimation for a recent
outbreak in Angola suggests that both transmission routes are needed to
explain the observed cholera dynamics. I will also discuss some ongoing
applications to the current cholera outbreak in Haiti.

Series: Job Candidate Talk

The cohomology ring of the absolute Galois group Gal(kbar/k) of a field k controls interesting arithmetic properties of k. The Milnor conjecture, proven by Voevodsky, identifies the cohomology ring H^*(Gal(kbar/k), Z/2) with the tensor algebra of k* mod the ideal generated by x otimes 1-x for x in k - {0,1} mod 2, and the Bloch-Kato theorem, also proven by Voevodsky, generalizes the coefficient ring Z/2. In particular, the cohomology ring of Gal(kbar/k) can be expressed in terms of addition and multiplication in the field k, despite the fact that it is difficult even to list specific elements of Gal(kbar/k). The cohomology ring is a coarser invariant than the differential graded algebra of cochains, and one can ask for an analogous description of this finer invariant, controlled by and controlling higher order cohomology operations. We show that order n Massey products of n-1 factors of x and one factor of 1-x vanish, generalizing the relation x otimes 1-x. This is done by embedding P^1 - {0,1,infinity} into its Picard variety and constructing Gal(kbar/k) equivariant maps from pi_1^et applied to this embedding to unipotent matrix groups. This also identifies Massey products of the form <1-x, x, … , x , 1-x> with f cup 1-x, where f is a certain cohomology class which arises in the description of the action of Gal(kbar/k) on pi_1^et(P^1 - {0,1,infinity}). The first part of this talk will not assume knowledge of Galois cohomology or Massey products.

Series: Job Candidate Talk

Bernstein's theorem is a classical result which computes the number of common zeros in (C*)^n of a generic set of n Laurent polynomials in n variables. The theorem of the Newton polygon is a ubiquitous tool in arithmetic geometry which calculates the valuations of the zeros of a polynomial (or convergent power series) over a non-Archimedean field,
along with the number of zeros (counted with multiplicity) with each given valuation. We will explain in what sense both theorems are very special cases of a lifting theorem in tropical intersection theory. The proof of this lifting theorem builds on results of Osserman and Payne, and uses Berkovich analytic spaces and extended tropicalizations of toric
varieties in a crucial way, as well as Raynaud's theory of formal models of
analytic spaces. Most of this talk will be about joint work with Brian Osserman.

Series: Job Candidate Talk

Electronic structure theories, in particular Kohn-Sham density
functional theory, are widely used in computational chemistry and
material sciences nowadays. The computational cost using conventional
algorithms is however expensive which limits the application to
relative small systems. This calls for development of efficient
algorithms to extend the first principle calculations to larger
system. In this talk, we will discuss some recent progress in
efficient algorithms for Kohn-Sham density functional theory. We will
focus on the choice of accurate and efficient discretization for
Kohn-Sham density functional theory.