## Seminars and Colloquia by Series

Wednesday, January 25, 2012 - 12:05 , Location: Skiles 005 , Ernie Croot , School of Mathematics, Georgia Tech , Organizer:
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.
Wednesday, January 18, 2012 - 12:05 , Location: Skiles 005 , Leonid A. Bunimovich , Georgia Tech , Organizer:
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).
Wednesday, December 7, 2011 - 12:05 , Location: Skiles 005. , Josephine Yu , Georgia Tech , Organizer:
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.
Wednesday, November 30, 2011 - 12:05 , Location: Skiles 005. , Farbod Shokrieh , Georgia Tech. , Organizer:
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.
Wednesday, November 16, 2011 - 12:05 , Location: Skiles 005. , Manwah Lilian Wong , Georgia Tech , Organizer:
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.
Wednesday, November 9, 2011 - 12:05 , Location: Skiles 005. , Andrzej Swiech , Georgia Tech. , Organizer:
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.
Wednesday, November 2, 2011 - 12:05 , Location: Skiles 005 , Evans Harrell , School of Mathematics, Georgia Tech , Organizer:
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.
Wednesday, October 26, 2011 - 12:05 , Location: Skiles 005 , Silas Alben , School of Mathematics, Georgia Tech , Organizer:
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.
Wednesday, October 12, 2011 - 12:05 , Location: Skiles 005 , Greg Blekherman , Georgia Tech , Organizer:
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.
Wednesday, September 21, 2011 - 12:05 , Location: Skiles 005 , Amit Einav , Georgia Tech , Organizer:
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.