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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.

Series: Job Candidate Talk

This talk will be about random lozenge tilings of a class of
planar domains which I like to call "sawtooth domains." The basic
question is: what does a uniformly random lozenge tiling of a large
sawtooth domain look like? At the first order of randomness, a remarkable
form of the law of large numbers emerges: the height function of the tiling
converges to a deterministic "limit shape." My talk is about the next
order of randomness, where one wants to analyze the fluctuations of tiles
around their eventual positions in the limit shape. Quite remarkably, this
analytic problem can be solved in an essentially combinatorial way, using a
desymmetrized version of the double Hurwitz numbers from enumerative
algebraic geometry.

Series: Job Candidate Talk

Under the operation of connected sum, the set of knots in the 3-sphere forms a monoid. Modulo an equivalence relation called concordance, this monoid becomes a group called the knot concordance group. We will consider various algebraic methods -- both classical and modern -- for better understanding the structure of this group.

Series: Job Candidate Talk

In first-passage percolation (FPP), one places random non-negative
weights on the edges of a graph and considers the induced weighted
graph metric. Of particular interest is the case where the graph is
Z^d, the standard d-dimensional cubic lattice, and many of the
questions involve a comparison between the asymptotics of the random
metric and the standard Euclidean one. In this talk, I will survey
some of my recent work on the order of fluctuations of the metric,
focusing on (a) lower bounds for the expected distance and (b) our
recent sublinear bound for the variance for edge-weight distributions
that have 2+log moments, with corresponding concentration results.
This second work addresses a question posed by Benjamini-Kalai-Schramm
in their celebrated 2003 paper, where such a bound was proved for only
Bernoulli weights using hypercontractivity. Our techniques draw
heavily on entropy methods from concentration of measure.

Series: Job Candidate Talk

It is known that certain medium, for example electromagnetic
field and Bose Einstein condensate, has positive speed of sound. It
is observed that if the medium is in its equilibrium state, then an
invading subsonic particle will slow down due to friction; and the
speed of a supersonic particle will slow down to the speed of sound
and the medium will radiate. This is called Cherenkov radiation. It
has been widely discussed in physical literature, and applied in
experiments. In this talk I will
present some rigorous mathematical results. Joint works with Juerg
Froehlich, Israel Michael Sigal, Avy Soffer, Daniel Egli, Arick Shao.