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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.

Series: Research Horizons Seminar

Sharp trace inequalities play a major role in the world of
Mathematics. Not only do they give a connection between 'boundary values' of
the trace and 'interior values' of the function, but also the truest form of
it, many times with a complete classification of when equality is attained.
The result presented here, motivated by such inequality proved by Jose'
Escobar, is a new trace inequality, connecting between the fractional
laplacian of a function and its restriction to the intersection of the
hyperplanes x_(n)=0, x_(n-1)=0, ..., x_(n-j+1)=0 where 1<=j<=n. We will show
that the inequality is sharp and discussed the natural space for it, along
with the functions who attain equality in it.
The above result is based on a joint work with Prof. Michael Loss.

Series: Research Horizons Seminar

Starting in the 30's Physicists were concerned
with the problem of motion of dislocations or the problem
of deposition of materials over a periodic structure.
This leads naturally to a variational problem (minimizing the energy).
One wants to know very delicate properties of the minimizers, which
was a problem that Morse was studying at the same time.
The systematic mathematical study of these problems started in the
80's with the work of Aubry and Mather who developed the basis to
deal with very subtle problems. The mathematics that have become
useful include dynamical systems, partial differential equations,
calculus of variations and numerical analysis. Physical intuition also
helps.
I plan to explain some of the basic questions and, perhaps illustrate some
of the results.

Series: Research Horizons Seminar

Hosts: Amey Kaloti and Ricardo Restrepo

Fourier series provide a way of writing almost any signal as a superposition of pure tones, or musical notes. But this representation is not local, and does not reflect the way that music is actually generated by instruments playing individual notes at different times. We will discuss time-frequency representations, which are a type of local Fourier representation of signals. This gives us a mathematical model for representing music. While the model is crude for music, it is in fact apowerful mathematical representation that has appeared widely throughout mathematics (e.g., partial differential equations), physics (e.g., quantum mechanics), and engineering (e.g., time-varying filtering). We ask one very basic question: are the notes in this representation linearly independent? This seemingly trivial question leads to surprising mathematical difficulties.