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Series: Research Horizons Seminar

We survey research spanning more than 20 years on what starts out
to be a very simple problem: Representing a poset as the inclusion order
of circular disks in the plane. More generally, we can speak of spherical
orders, i.e., posets which are inclusion orders of balls in R^d for some d.
Surprising enough, there are finite posets which are not sphere orders.
Quite recently, some elegant results have been obtained for circle orders,
lending more interest to the many open problems that remain.

Series: Research Horizons Seminar

Recently, there has been a lot of interest in estimation of
sparse vectors in high-dimensional spaces and large low rank
matrices based on a finite number of measurements of randomly
picked linear functionals of these vectors/matrices. Such
problems are very basic in several areas (high-dimensional
statistics, compressed sensing, quantum state tomography, etc).
The existing methods are based on fitting the vectors (or the matrices)
to the data using least squares with carefully designed complexity
penalties based on the $\ell_1$-norm in the case of vectors and
on the nuclear norm in the case of matrices. Proving error bounds
for such methods that hold with a guaranteed probability is based on several
tools from high-dimensional probability that will be also discussed.

Series: Research Horizons Seminar

Classical mathematical capillarity theory has as its foundation variational methods introduced by Gauss. There was a heuristic explanation given earlier by Thomas Young, and his explanations did have quantitative scientific content. Due partially to their simplistic nature, the explanations of Young live on today in engineering textbooks, though in certain cases it has been pointed out that they lead to anomolous predictions (which are effectively avoided in the Gaussian variational framework). I will discuss a fundamentally new direction in mathematical capillarity which is motivated by an effort to harmonize the heuristic and rigorous elements of the theory and has other important applications as well.

Series: Research Horizons Seminar

I will present two "real life" applications of Riemannian geometry,one in the field of continuum mechanics, another in the field ofmachine learning (nonlinear dimensionality reduction). I will providequick introductions in order to make the talk accessible to anaudience with no background in either field.

Series: Research Horizons Seminar

This talk will traverse several topics in singularity
theory, algebraic analysis, complex analysis, algebraic geometry, and
statistics. I will outline effective methods to compute the log
canonical threshold, a birational invariant of an algebraic variety,
as well as its potential statistical applications.

Series: Research Horizons Seminar

I will present a construction of a non-measurable set using the fundamental
fact that a graph with no odd cycles is 2-colorable. That will not take very
long, even though I will prove everything from first principles. In the rest
of the time I will discuss the Axiom of Choice and some unprovable
statements. The talk should be accessible to undergraduates.

Series: Research Horizons Seminar

Following exciting developments in the continuous setting of manifolds (and
other geodesic spaces), in joint works with various collaborators, I have
explored discrete analogs of the interconnection between several functional
and isoperimetric inequalities in discrete spaces. Such inequalities include
concentration, transportation, modified versions of the logarithmic Sobolev
inequality, and (most recently) displacement convexity. I will attempt to
motivate and review some of these connections and illustrate with examples.
Time permitting, computational aspects of the underlying functional
constants and other open problems will also be mentioned.

Series: Research Horizons Seminar

To any self-map of a surface we can associate a real number, called the
entropy. This number measures, among other things, the amount of mixing
being effected on the surface. As one example, you can think about a taffy
pulling machine, and ask how efficiently the machine is stretching the
taffy. Using Thurston's notion of a train track, it is actually possible to
compute these entropies, and in fact, this is quite easy in practice. We
will start from the basic definitions and proceed to give an overview of
Thurston's theory. This talk will be accessible to graduate students and
advanced undergraduates.

Series: Research Horizons Seminar

In this talk I will survey some recent results related to
Roth's Theorem on three-term arithmetic progressions. The basic
problem in this area is to determine the largest subset S of the
integers in {1,...,n} containing no triple of the form x, x+d, x+2d.
Roth showed back in the 1950's that the largest such set S has size
o(n), and over the following decades his result has been
considerably improved upon.

Series: Research Horizons Seminar

It is well known that typically equations do not have analytic (expressed by formulas) solutions. Therefore a classical approach to the analysis of dynamical systems (from abstract areas of Math, e.g. the Number theory to Applied Math.) is to study their asymptotic (when an independent variable, "time", tends to infinity) behavior. Recently, quite surprisingly, it was demonstrated a possibility to study rigorously (at least some) interesting finite time properties of dynamical systems. Most of already obtained results are surprising, although rigorously proven. Possible PhD topics range from understanding these (already proven!) surprises and finding (and proving) new ones to numerical investigation of some systems/models in various areas of Math and applications, notably for dynamical analysis of dynamical networks. I'll present some visual examples, formulate some results and explain them (when I know how).