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Series: Job Candidate Talk

We discuss asymptotic-in-time behavior of time-like constant meancurvature hypersurfaces in Minkowski space. These objects model extended relativistic test objects subject to constant normal forces, and appear in the classical field theory foundations of the theory of vibrating strings and membranes. From the point of view of their Cauchy problem, these hypersurfaces evolve according to a geometric system of quasilinear hyperbolic partial differential equations. Inthis talk we will focus on three explicit solutions to the equations:the Minkowski hyperplane, the static catenoid, and the expanding de Sitter space. Their stability properties in the context of the Cauchy problem will be discussed, with emphasis on the geometric origins of the various mechanisms and obstacles that come into play.

Series: Job Candidate Talk

Low-rank structures are common in modern data analysis and signal processing, and they usually
play essential roles in various estimation and detection problems. It is challenging to recover the underlying
low-rank structures reliably from corrupted or undersampled measurements. In this talk, we will introduce
convex and nonconvex optimization methods for low-rank recovery by two examples.
The first example is community detection in network data analysis. In the literature, it has been formulated
as a low-rank recovery problem, and then SDP relaxation methods can be naturally applied. However,
the statistical advantages of convex optimization approaches over other competitive methods, such as spectral
clustering, were not clear. We show in this talk that the methodology of SDP is robust against arbitrary
outlier nodes with strong theoretical guarantees, while standard spectral clustering may fail due to a small
fraction of outliers. We also demonstrate that a degree-corrected version of SDP works well for a real-world
network dataset with a heterogeneous distribution of degrees.
Although SDP methods are provably effective and robust, the computational complexity is usually high
and there is an issue of storage. For the problem of phase retrieval, which has various applications and
can be formulated as a low-rank matrix recovery problem, we introduce an iterative algorithm induced by
nonconvex optimization. We prove that our method converges reliably to the original signal. It requires far
less storage and has much higher rate of convergence compared to convex methods.

Series: Job Candidate Talk

Sir Ronald Ross’ discovery of the transmission mechanism of malaria in 1897 inspired a suite of mathematical models for the transmission of vector-borne disease, known as Ross-Macdonald models. I introduce a common formulation of the Ross-Macdonald model and discuss its extension to address a current topic in malaria control: the introduction of malaria vaccines. Following over two decades of research, vaccine trials for the malaria vaccine RTS,S have been completed, demonstrating an efficacy of roughly 50% in young children. Regions with high malaria prevalence tend to have high levels of naturally acquired immunity (NAI) to severe malaria, leading to large asymptomatic populations. I introduce a malaria model developed to address concerns about how these vaccines will perform in regions with existing NAI, discuss some analytic results and their public health implications, and reframe our question as an optimal control problem.

Series: Job Candidate Talk

Mathematical models of physical phenomena often include parameters that are hard or impossible to measure directly or are subject to
variability, and are thus considered uncertain. Different aspects of modeling
under uncertainty include forward uncertainty propagation, statistical inver-
sion of uncertain parameters, optimal design of experiments, and optimization
under uncertainty. I will focus on recent advances in numerical methods for
infinite-dimensional Bayesian inverse problems and optimal experimental de-
sign. I will also discuss the problem of risk-averse optimization under uncertainty with applications to control of PDEs with uncertain parameters. The
driving applications are systems governed by PDEs with uncertain parameter
fields, such as
ow in the subsurface with an uncertain permeability field, or the
diffusive transport of a contaminant with an uncertain initial condition. Such
problems are computationally challenging due to expensive forward PDE solves
and infinite-dimensional (high-dimensional when discretized) parameter spaces.

Series: Job Candidate Talk

Diabetes is a disease of poor glucose control. Glucose is controlled by two hormones that work in opposite directions: insulin and glucagon. Pancreatic beta-cells release insulin when blood glucose is high, while pancreatic alpha-cells secrete glucagon when blood glucose is low. Both insulin and glucagon secretion are disregulated in people with diabetes. In these people, not enough insulin is secreted in response to elevated glucose levels, while the problem with glucagon secretion is two-fold: too much glucagon is secreted at high glucose levels, while not enough is secreted at low glucose levels. So far, the treatment of diabetes has focused solely on increasing insulin secretion from beta-cells. Therefore, understanding glucose regulated glucagon secretion may lead to new therapies for those with diabetes.There is an ongoing debate as to whether glucose suppresses glucagon secretion directly through an intrinsic mechanism, within the alpha-cell, or indirectly through an extrinsic mechanism. I developed a mathematical model of glucagon secretion in alpha-cells and use it to show that they can control their own secretion. However, experimental evidence shows that factors secreted by pancreatic beta- and delta- cells can also affect glucagon secretion. Therefore, I created the BAD model for pancreatic islets which contains one representative cell of each type and the cellular interactions between them. I use this model to show that these paracrine effects suppress alpha-cell heterogeneity and suggest that delta-cells play a more important role in this than beta-cells.

Series: Job Candidate Talk

The stochastic block model is a random graph model that was originally introduced 30 years ago to model community structure in networks. To generate a random graph from this model, begin with two classes of vertices and then connect each pair of vertices independently at random, with probability p if they are in the same class and probability q otherwise. Some questions come to mind: can we reconstruct the classes if we only observe the graph? What if we only want to partially reconstruct the classes? How different is this model from an Erdos-Renyi graph anyway? The answers to these questions depend on p and q, and we will say exactly how.

Series: Job Candidate Talk

In this talk we will discuss properties of some random polytopes. In particular, we first propose a deviation inequality for the convex hull of i.i.d. random points, uniformly distributed in a convex body. We then discuss statistical properties of this random polytope, in particular, its optimality, when one aims to estimate the support of the corresponding uniform distribution, if it is unknown.We also define a notion of multidimensional quantiles, related to the convex floating bodies, or Tukey depth level sets, for probability measures in a Euclidean space. When i.i.d. random points are available, these multidimensional quantiles can be estimated using their empirical version, similarly to the one-dimensional case, where order statistics estimate the usual quantiles.

Series: Job Candidate Talk

Matrices are one of the most fundamental structures in
mathematics, and it is well known that the behavior of a matrix is dictated
by its eigenvalues. Eigenvalues, however, are notoriously hard to control,
due in part to the lack of techniques available. In this talk, I will
present a new technique that we call the "method of interlacing
polynomials" which has been used recently to give unprecedented bounds on
eigenvalues, and as a result, new insight into a number of old problems.
I will discuss some of these recent breakthroughs, which include the
existence of Ramanujan graphs of all degrees, a resolution to the famous
Kadison-Singer problem, and most recently an incredible result of Anari and
Gharan that has led to an interesting new anomaly in computer science.
This talk will be directed at a general mathematics audience and represents
joint work with Dan Spielman and Nikhil Srivastava.

Series: Job Candidate Talk

High-dimensional statistics is the basis for analyzing large and complex
data sets that are generated by cutting-edge technologies in genetics,
neuroscience, astronomy, and many other fields. However, Lasso, Ridge
Regression, Graphical Lasso, and other standard methods in
high-dimensional statistics depend on tuning parameters that are
difficult to calibrate in practice. In this talk, I present two novel
approaches to overcome this difficulty. My first approach is based on a
novel testing scheme that is inspired by Lepski’s idea for bandwidth
selection in non-parametric statistics. This approach provides tuning
parameter calibration for estimation and prediction with the Lasso and
other standard methods and is to date the only way to ensure high
performance, fast computations, and optimal finite sample guarantees. My
second approach is based on the minimization of an objective function
that avoids tuning parameters altogether. This approach provides
accurate variable selection in regression settings and, additionally,
opens up new possibilities for the estimation of gene regulation
networks, microbial ecosystems, and many other network structures.

Series: Job Candidate Talk

The question of global regularity vs. finite time blow-up remains open for many fluid equations. Even in the cases where global regularity is known, solutions may develop small scales as time progresses. In this talk, I will first discuss an active scalar equation which is an interpolation between the 2D Euler equation and the surface quasi-geostrophic equation. We study the patch dynamics for this equation in the half-plane, and prove that the solutions can develop a finite-time singularity. I will also discuss a passive transport equation whose solutions are known to have global regularity, and our goal is to study how well a given initial density can be mixed if the incompressible flow satisfies some physically relevant quantitative constraints. This talk is based on joint works with A. Kiselev, L. Ryzhik and A. Zlatos.