Seminars and Colloquia by Series

A combinatorial construction of a non-measurable set

Series
Research Horizons Seminar
Time
Wednesday, March 7, 2012 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Robin Thomas – Georgia Tech
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.

Isoperimetric and Functional Inequalities on Discrete Spaces

Series
Research Horizons Seminar
Time
Wednesday, February 22, 2012 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prasad Tetali – Georgia Tech
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.

Train tracks and entropy

Series
Research Horizons Seminar
Time
Wednesday, February 8, 2012 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dan Margalit – Georgia Tech
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.

A Survey of Some Results Related to Roth's Theorem

Series
Research Horizons Seminar
Time
Wednesday, January 25, 2012 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ernie Croot – School of Mathematics, Georgia Tech
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.

FINITE TIME DYNAMICS: the first steps and outlook.

Series
Research Horizons Seminar
Time
Wednesday, January 18, 2012 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Leonid A. Bunimovich – Georgia Tech
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).

Polytope Algebra and Tropical Cycles

Series
Research Horizons Seminar
Time
Wednesday, December 7, 2011 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005.
Speaker
Josephine Yu – Georgia Tech
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.

Chip-firing games on graphs

Series
Research Horizons Seminar
Time
Wednesday, November 30, 2011 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005.
Speaker
Farbod Shokrieh – Georgia Tech.
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.

Discrete Schroedinger Problem.

Series
Research Horizons Seminar
Time
Wednesday, November 16, 2011 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005.
Speaker
Manwah Lilian Wong – Georgia Tech
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.

Viscosity solutions and applications to stochastic optimal control.

Series
Research Horizons Seminar
Time
Wednesday, November 9, 2011 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005.
Speaker
Andrzej Swiech – Georgia Tech.
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.

Getting in shape with eigenvalues

Series
Research Horizons Seminar
Time
Wednesday, November 2, 2011 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Evans Harrell – School of Mathematics, Georgia Tech
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.

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