## Seminars and Colloquia by Series

Friday, April 29, 2011 - 13:00 , Location: Skiles 246 , Jie Ma , School of Mathematics, Georgia Tech , Organizer:
Fix k vertices in a graph G, say a_1,...,a_k, if there exists a cycle that visits these vertices with this specified order, we say such a cycle is (a_1,a_2,...,a_k)-ordered. It is shown by Thomas and Wollan that any 10k-connected graph is k-linked, therefore any 10k-connected graph has an (a_1,a_2,...,a_k)-ordered for any a_1,...,a_k. However, it is possible that we can improve this bound when k is small. It is shown by W. Goddard that any 4-connected maximal planar graph has an (a_1,...,a_4)-ordered cycle for any choice of 4 vertices. We will present a complete characterization of 4-ordered cycle in planar graphs. Namely, for any four vertices a,b,c,d in planar graph G, if there is no (a,b,c,d)-ordered cycle in G, then one of the follows holds: (1) there is a cut S separating {a,c} from {b,d} with |S|\leq 3; (2) roughly speaking, a,b,d,c "stay" in a face of G with this order.
Friday, April 22, 2011 - 13:00 , Location: Skiles 246 , Amey Kaloti , School of Mathematics, Georgia Tech , Organizer:

Hosted also by Ben Webb

We will try to define what Khovanov homology for a link in a S^3 is. We will then try to give a proof figuring out unknotting number of certain kinds of knots in S^3.
Friday, April 8, 2011 - 13:00 , Location: Skiles 246 , Nathan Parrish , School of Electrical and Computer Engineering, Georgia Tech , Organizer:
The discussion will focus on some recent advances in improving performance of rendering 3D scenes. First, a Monte Carlo method based upon the Metropolis algorithm is described. Then a method of using spherical harmonics to generate vectors and matrices which allow efficient high-quality rendering in real time will be described. Finally, a discussion will be made of possible future areas for improving the efficiency of such algorithms.
Friday, April 1, 2011 - 13:00 , Location: Skiles 246 , Peter Whalen , School of Mathematics, Georgia Tech , Organizer:
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.
Friday, March 18, 2011 - 13:00 , Location: Skiles 246 , Tobias Hurth , School of Mathematics, Georgia Tech , Organizer:
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.
Friday, December 3, 2010 - 13:00 , Location: Skiles 255 , Nan Lu , School of Mathematics, Georgia Tech , Organizer:
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.
Friday, November 19, 2010 - 13:00 , Location: Skiles 255 , Spencer Backman , School of Mathematics, Georgia Tech , Organizer:
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.
Friday, November 12, 2010 - 13:00 , Location: Skiles 255 , Mark Sedjro , School of Mathematics, Georgia Tech , Organizer:
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.
Friday, November 5, 2010 - 13:00 , Location: Skiles 255 , Ben Webb , School of Mathematics, Georgia Tech , Organizer:
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.
Friday, October 29, 2010 - 13:05 , Location: Skiles 255 , Ricardo Restrepo , School of Mathematics, Georgia Tech , Organizer:
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.