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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).

Series: Research Horizons Seminar

A polytope is a convex hull of a finite set of points in a vector space. The set of polytopes in a fixed vector space generate an algebra where addition is formal and multiplication is the Minkowski sum, modulo some relations. The algebra of polytopes were used to solve some variations of Hilbert's third problem about subdivision of polytopes and to give a combinatorial proof of Stanley's g-Theorem that characterizes face numbers of simplicial polytopes. In this talk, we will introduce McMullen's version of polytope algebra and show that it is isomorphic to the algebra of tropical cycles which are balanced weighted polyhedral fans. The tropical cycles can be used to do explicit computations and examples in polytope algebra.

Series: Research Horizons Seminar

I will discuss the theory of chip-firing games, focusing on the interplay between chip-firing games and potential theory on graphs. To motivate the discussion, I will give a new proof of "the pentagon game". I will discuss the concept of reduced divisors and various related algorithmic aspects of the theory. If time permits I will also give some applications, including an "efficient bijective" proof of Kirchhoff's matrix-tree theorem.

Series: Research Horizons Seminar

We will discuss the discrete Schroedinger problem on the integer line and on graphs. Starting from the definition of the discrete Laplacian on the integer line, I will explain why the problem is interesting, how the discrete case relates to the continuous case, and what the open problems are. Recent results by the speaker (joint with Evans Harrell) will be presented.The talk will be accessible to anyone who knows arithmetic and matrix multiplications.

Series: Research Horizons Seminar

I will give a brief introduction to the theory ofviscosity solutions of second order PDE. In particular, I will discussHamilton-Jacobi-Bellman-Isaacs equations and their connections withstochastic optimal control and stochastic differentialgames problems. I will also present extensions of viscositysolutions to integro-PDE.

Series: Research Horizons Seminar

Eigenvalues of linear operators often correspond to physical observables;
for example they determine the energy levels in quantum mechanics and the
frequencies of vibration in acoustics. Properties such as the shape of a
system are encoded in the the set of eigenvalues, known as the "spectrum,"
but in subtle ways. I'll talk about some classic theorems about how
geometry and topology show up in the spectrum of differential operators, and
then I'll present some recent work, with connections to physical models such
as quantum waveguides, wires, and graphs.

Series: Research Horizons Seminar

Vortex methods are an efficient and versatile way to simulate high
Reynolds number flows. We have developed vortex sheet methods for a
variety of flows past deforming bodies, many of which are biologically
inspired. In this talk we will present simulations and asymptotic
analysis of selected problems. The first is a study of oscillated and
freely-swimming flexible foils. We analyze the damped resonances that
determine propulsive performance. The second problem involves multiple
passive flapping ``flags" which interact through their vortex wakes. The
third problem is a study of flexible falling sheets. Here the
flag-flapping instability helps us determine the terminal falling speeds.

Series: Research Horizons Seminar

A multivariate real polynomial p(x) is nonnegative if p(x) is at
least 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, with many applications some of which I will present.
I will explain how the problem of describing nonnegative polynomials fits
into convex algebraic geometry: the study of convex sets with underlying
algebraic structure, that brings together ideas of optimization, convex
geometry and algebraic geometry. I will end by presenting current research
problems in this area.