Seminars and Colloquia by Series

On the Steinberg's Conjecture: 3-coloring of planar graphs

Series
SIAM Student Seminar
Time
Friday, April 1, 2011 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 246
Speaker
Peter Whalen – School of Mathematics, Georgia Tech
Steinberg's Conjecture states that any planar graph without cycles of length four or five is three colorable. Borodin, Glebov, Montassier, and Raspaud showed that planar graphs without cycles of length four, five, or seven are three colorable and Borodin and Glebov showed that planar graphs without five cycles or triangles at distance at most two apart are three colorable. We prove a statement similar to both of these results: that any planar graph with no cycles of length four through six or cycles of length seven with incident triangles distance exactly two apart are three colorable. Special thanks to Robin Thomas for substantial contributions in the development of the proof.

A one-dimensional dynamical system with random switching

Series
SIAM Student Seminar
Time
Friday, March 18, 2011 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 246
Speaker
Tobias Hurth – School of Mathematics, Georgia Tech
We will study a simple dynamical system with two driving vector fields on the unit interval. The driving vector fields point to opposite directions, and we will follow the trajectory induced by one vector field for a random, exponentially distributed, amount of time before switching to the regime of the other one. Thanks to the simplicity of the system, we obtain an explicit formula for its invariant density. Basically exploiting analytic properties of this density, we derive versions of the law of large numbers, the central limit theorem and the large deviations principle for our system. If time permits, we will also discuss some ideas on how to prove existence of invariant densities, both in our one-dimensional setting and for more general systems with random switching. The talk will rely to a large extent on my Master's thesis I wrote last year under the guidance and supervision of Yuri Bakhtin.

Invariant Manifolds in Dynamical Systems

Series
SIAM Student Seminar
Time
Friday, December 3, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Nan Lu – School of Mathematics, Georgia Tech
In this talk, I am going to give a elementary introduction of invariant manifold theory in dynamical systems. I will start with the motivation and definition of invariant manifolds. Then I will discuss how to construct various invariant manifolds of maps and flows. Finally, I will discuss some applications. If time is permitted, I will also discuss a little about invariant foliation.

Riemann-Roch Theory for Directed Graphs

Series
SIAM Student Seminar
Time
Friday, November 19, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Spencer Backman – School of Mathematics, Georgia Tech
The talk will begin with an elementary geometric discussion of Riemann-Roch theory for sub-lattices of the integer lattice orthogonal to some positive vector. A pair of necessary and sufficient conditions for such a lattice to have the Riemann-Roch property will be presented. By studying a certain chip firing game on a directed graph related to the lattice spanned by the rows of its Laplacian I will describe a combinatorial method for checking whether a directed graph has the Riemann-Roch property. The talk will conclude with a presentation of arithmetical graphs, which after the application of a simple transformation, may be viewed as a special class of directed graphs. Examples from this class demonstrate that either, both or neither of the Riemann-Roch conditions may be satisfied for a directed graph. This is joint work with Arash Asadi.

A Minimax Problem in Almost Axisymmetric Flows

Series
SIAM Student Seminar
Time
Friday, November 12, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Mark Sedjro – School of Mathematics, Georgia Tech
Almost axisymmetric flows are derived from Boussinesq equations for incompressible fluids. They are supposed to capture special features in tropical cyclones. We establish an unusual minimax equality as the first step towards studying this challenging problem. I will review some basic techniques of the calculus of variations.

Global Stability of Dynamical Networks

Series
SIAM Student Seminar
Time
Friday, November 5, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Ben Webb – School of Mathematics, Georgia Tech
In this talk we consider the collective dynamics of a network of interacting dynamical systems and show that under certain conditions such dynamical networks have a unique global attractor. This involves a combination of techniques from dynamical systems theory as well as newly devised methods in graph theory. However, this talk is intended to be an introduction to both areas of mathematics with a focus on how the combination of the two is yielding new results in graph and dynamical systems theory.

When do random CSPs become hard?

Series
SIAM Student Seminar
Time
Friday, October 29, 2010 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Ricardo Restrepo – School of Mathematics, Georgia Tech
A constraint satisfaction problem (CSP) is an ensemble of boolean clauses, where satisfaction is obtained by an assignment of the variables if every clause is satisfied by such assignment. We will see that when such CSP is arranged following certain random structure, the Fourier expansion of the corresponding clauses allows us to understand certain properties of the solution space, in particular getting a partial understanding of when the 'usual suspects' of the drastical failure of all known satisfiability algorithms, namely long range correlations and clustering, appear. Based in joint work with Prasad Tetali and Andrea Montanari.

On nonparametric multivariate statistical process control charts

Series
SIAM Student Seminar
Time
Friday, October 22, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Giang Do – School of Mathematics, Georgia Tech
Statistical Process Control Charts are key tools in monitoring and controlling production processes to achieve conforming, high quality products. We will conduct a literature review on the Nonparametric Multivariate Statistical Process Control Charts to see what has been done in the area and how the methods have been applied.

The entropy production problem and Villani's conjecture

Series
SIAM Student Seminar
Time
Friday, October 8, 2010 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Amit Einav – School of Mathematics, Georgia Tech
In 1956 Mark Kac published his paper about the Foundation of Kinetic Theory in which he gave a mathematical, probabilistic description of a system of N particles colliding randomly. An interesting result that was found, though not causing any surprise, was the convergence to the stable equilibrium state. The question of the rate of the L2 convergence interested Kac and he conjectured that the spectral gap governing the convergence is uniformly bounded form below as N goes to infinity. While this was proved to be true, and even computed exactly, many situations show that the time scale of the convergence for very natural cases is proportional to N, while we would hope for an exponential decay. A different approach was considered, dealing with a more natural quantity, the entropy. In recent paper some advancement were made about evaluating the rate of change, and in 2003 Villani conjectured that the corresponding 'spectral gap', called the entropy production, is of order of 1/N. In our lecture we'll review the above topics and briefly discuss recently found results showing that the conjecture is essentially true.

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