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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.

Series: Job Candidate Talk

Despite its long history, the theory of ellipticpartial differential equations in non-smooth media is abundant with openproblems. We will discuss the main achievements in the theory, recentdevelopments, surprising paradoxes related to the behavior of solutions nearthe boundary, and some fundamental questions which still remain open.

Series: Job Candidate Talk

Let H = A+UBU* where A and B are two N-by-N Hermitian matrices and U is
a random unitary transformation. When N is large, the point measure of
eigenvalues of H fluctuates near a probability measure which depends
only on eigenvalues of A and B. In this talk, I will discuss this limiting
measure and explain a result about convergence to the limit in a local regime.

Series: Job Candidate Talk

The immune system is a complex, multi-layered biological system, making it difficult to characterize dynamically. Perhaps, we can better understand the system’s construction by isolating critical, functional motifs. From this perspective, we will investigate two simple, yet ubiquitous motifs:state transitions and feedback regulation.Numerous immune cells exhibit transitions from inactive to activated states. We focus on the T cell response and develop a model of activation, expansion, and contraction. Our study suggests that state transitions enable T cells to detect change and respond effectively to changes in antigen levels, rather than simply the presence or absence of antigen. A key component of the system that gives rise to this change detector is initial activation of naive T cells. The activation step creates a barrier that separates the slow dynamics of naive T cells from the fast dynamics of effector T cells, allowing the T cell population to compare short-term changes in antigen levels to long-term levels. As a result, the T cell population responds to sudden shifts in antigen levels, even if the antigen were already present prior to the change. This feature provides a mechanism for T cells to react to rapidly expandingsources of antigen, such as viruses, while maintaining tolerance to constant or slowly fluctuating sources of stimulation, such as healthy tissue during growth.For our second functional motif, we investigate the potential role of negative feedback in regulating a primary T cell response. Several theories exist concerning the regulation of primary T cell responses, the most prevalent being that T cells follow developmental programs. We propose an alternative hypothesis that the response is governed by a feedback loop between conventional and adaptive regulatory T cells. By developing a mathematical model, we show that the regulated response is robust to a variety of parameters and propose that T cell responses may be governed by a simple feedback loop rather than by autonomous cellular programs.

Series: Job Candidate Talk

The Jones polynomial is a link invariant that can be understood in
terms of the representation theory of the quantum group associated to sl2. This
description facilitated a vast generalization of the Jones polynomial to other
quantum link and tangle invariants called Reshetikhin-Turaev invariants. These
invariants, which arise from representations of quantum groups associated to
simple Lie algebras, subsequently led to the definition of quantum 3-manifold
invariants. In this talk we categorify quantum groups using a simple diagrammatic
calculus that requires no previous knowledge of quantum groups. These
diagrammatically categorified quantum groups not only lead to a representation
theoretic explanation of Khovanov homology but also inspired Webster's recent
work categorifying all Reshetikhin-Turaev invariants of tangles.

Series: Job Candidate Talk

A multivariate real polynomial $p$ is nonnegative if $p(x) \geq 0$ for all $x \in R^n$. I will review the history and motivation behind the problem of representing nonnegative polynomials as sums of squares. Such representations are of interest for both theoretical and practical computational reasons. I will present two approaches to studying the differences between nonnegative polynomials and sums of squares. Using techniques from convex geometry we can conclude that if the degree is fixed and the number of variables grows, then asymptotically there are significantly more nonnegative polynomials than sums of squares. For the smallest cases where there exist nonnegative polynomials that are not sums of squares, I will present a complete classification of the differences between these sets based on algebraic geometry techniques.

Series: Job Candidate Talk

High throughput genetic sequencing arrays with thousands of
measurements
per sample and a great amount of related censored clinical data have
increased demanding need for better measurement specific model
selection.
In this paper we establish strong oracle properties of non-concave
penalized methods for non-polynomial (NP) dimensional data with
censoring in the framework of Cox's proportional hazards model.
A class of folded-concave penalties are employed and both LASSO and
SCAD are discussed specifically. We unveil the question under which
dimensionality and correlation
restrictions can an oracle estimator be constructed and grasped. It is
demonstrated that non-concave penalties lead to significant reduction
of the "irrepresentable condition" needed for LASSO model selection
consistency.
The large deviation result for martingales, bearing interests of its
own, is developed for characterizing the strong oracle property.
Moreover, the non-concave regularized estimator, is shown to achieve
asymptotically the information bound of the oracle estimator. A
coordinate-wise algorithm is developed for finding the grid of
solution paths for penalized hazard regression problems, and its
performance is evaluated on simulated and gene association study
examples.