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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.

Series: Job Candidate Talk

A region of space is cloaked for a class of measurements if observers are not
only unaware of its contents, but also unaware of the presence of the cloak using such
measurements. One approach to cloaking is the change of variables scheme introduced
by Greenleaf, Lassas, and Uhlmann for electrical impedance tomography and by Pendry,
Schurig, and Smith for the Maxwell equations. They used a singular change of variables
which blows up a point into the cloaked region. To avoid this singularity, various
regularized schemes have been proposed. In this talk I present results related to cloaking via
change of variables for the Helmholtz equation using the natural regularized scheme
introduced by Kohn, Shen, Vogelius, and Weintein, where the authors used a transformation
which blows up a small ball instead of a point into the cloaked region. I will discuss the
degree of invisibility for a nite range or the full range of frequencies, and the possibility of
achieving perfect cloaking. At the end of my talk, I will also discuss some results related
to the wave equation in 3d.

Series: Job Candidate Talk

Many mechanical systems have the property that some small perturbations can accumulate over time to lead to large effects. Other perturbations just average out and cancel. It is interesting in applications to find out what systems have these properties and which perturbations average out and which ones grows. A complete answer is far from known but it is known that it is complicated and that, for example, number theory plays a role. In recent times, there has been some progress understanding some mechanisms that lead to instability. One can find landmarks that organize the long term behavior and provide an skeleton for the dynamics. Some of these landmarks provide highways along which the perturbations can accumulate.

Series: Job Candidate Talk

We consider the statistical deconvolution problem where one observes $n$
replications from the model $Y=X+\epsilon$, where $X$ is the unobserved
random signal of interest and where $\epsilon$ is an independent random
error with distribution $\varphi$. Under weak assumptions on the decay of
the Fourier transform of $\varphi$ we derive upper bounds for the
finite-sample sup-norm risk of wavelet deconvolution density estimators
$f_n$ for the density $f$ of $X$, where $f: \mathbb R \to \mathbb R$ is
assumed to be bounded. We then derive lower bounds for the minimax sup-norm
risk over Besov balls in this estimation problem and show that wavelet
deconvolution density estimators attain these bounds. We further show that
linear estimators adapt to the unknown smoothness of $f$ if the Fourier
transform of $\varphi$ decays exponentially, and that a corresponding result
holds true for the hard thresholding wavelet estimator if $\varphi$ decays
polynomially. We also analyze the case where $f$ is a 'supersmooth'/analytic
density. We finally show how our results and recent techniques from
Rademacher processes can be applied to construct global nonasymptotic
confidence bands for the density $f$.

Series: Job Candidate Talk

Tropical geometry can be thought of as geometry over the tropical
semiring, which is the set of real numbers together with the operations max
and +. Just as ordinary linear and polynomial algebra give rise to
convex geometry and algebraic geometry, tropical linear and polynomial
algebra give rise to tropical convex geometry and tropical algebraic
geometry. I will introduce the basic objects and problems in tropical
geometry and discuss some relations with, and applications to,
polyhedral geometry, computational algebra, and algebraic geometry.