Series: Geometry Topology Seminar
The hyperbolic volume and the colored Jones polynomial are two of the most powerful invariants in knot theory. In this talk we aim to extend these invariants to arbitrary graphs embedded in 3-space. This provides new tools for studying questions about graph embedding and it also sheds some new light on the volume conjecture. According to this conjecture, the Jones polynomial and the volume of a knot are intimately related. In some special cases we will prove that this still holds true in the case of graphs.
Friday, September 5, 2008 - 15:00 , Location: Skiles 255 , Ernie Croot , School of Mathematics, Georgia Tech , Organizer: Prasad Tetali
Let A be a set of n real numbers. A central problem in additive combinatorics, due to Erdos and Szemeredi, is that of showing that either the sumset A+A or the product set A.A, must have close to n^2 elements. G. Elekes, in a short and brilliant paper, showed that one can give quite good bounds for this problem by invoking the Szemeredi-Trotter incidence theorem (applied to the grid (A+A) x (A.A)). Perhaps motivated by this result, J. Solymosi posed the following problem (actually, Solymosi's original problem is slightly different from the formulation I am about to give). Show that for every real c > 0, there exists 0 < d < 1, such that the following holds for all grids A x B with |A| = |B| = n sufficiently large: If one has a family of n^c lines in general position (no three meet at a point, no two parallel), at least one of them must fail to be n^(1-d)-rich -- i.e. at least one of then meets in the grid in fewer than n^(1-d) points. In this talk I will discuss a closely related result that I and Evan Borenstein have proved, and will perhaps discuss how we think we can use it to polish off this conjecture of Solymosi.
Thursday, September 4, 2008 - 15:00 , Location: Skiles 269 , Heinrich Matzinger , School of Mathematics, Georgia Tech , Organizer: Heinrich Matzinger
A common subsequence of two sequences X and Y is a sequence which is a subsequence of X as well as a subsequence of Y. A Longest Common Subsequence (LCS) of X and Y is a common subsequence with maximal length. Longest Common subsequences can be represented as alignments with gaps where the aligned letter pairs corresponds to the letters in the LCS. We consider two independent i.i.d. binary texts X and Y of length n. We show that the behavior of the the alignment corresponding to the LCS is very different depending on the number of colors. With 2-colors, long blocks tend to be aligned with no gaps, whilst for four or more colors the opposite is true. Let Ln denote the length of the LCS of X and Y. In general the order of the variance of Ln is not known. We explain how a biased affect of a finite pattern can influence the order of the fluctuation of Ln.
Wednesday, September 3, 2008 - 12:00 , Location: Skiles 255 , Robin Thomas , School of Mathematics, Georgia Tech , Organizer:
I will explain and prove a beautiful and useful theorem of Alon and Tarsi that uses multivariate polynomials to guarantee, under suitable hypotheses, the existence of a coloring of a graph. The proof method, sometimes called a Combinatorial Nullstellensatz, has other applications in graph theory, combinatorics and number theory.
Wednesday, September 3, 2008 - 11:00 , Location: Skiles 255 , Annalisa Bracco , School of Earth & Atmospheric Sciences, Georgia Tech , Organizer: Christine Heitsch
In the ocean, coherent vortices account for a large portion of the ocean turbulent kinetic energy and their presence significantly affects the dynamics and the statistical properties of ocean flows, with important consequences on transport processes. Mesoscale vortices also affect the population dynamics of phyto- and zooplankton, and are associated with secondary currents responsible for localized vertical fluxes of nutrients. The fact that the nutrient fluxes have a fine spatial and temporal detail, generated by the eddy field, has important consequences on primary productivity and the horizontal velocity field induced by the eddies has been suggested to play an important role in determining plankton patchiness. Owing to their trapping properties, vortices can also act as shelters for temporarily less-favoured planktonic species. In this contribution, I will review some of the transport properties associated with coherent vortices and their impact on the dynamics of planktoni ecosystems, focusing on the simplified conceptual model provided by two-dimensional turbulence.
Series: PDE Seminar
Shear flow instability is a classical problem in hydrodynamics. In particular, it is important for understanding the transition from laminar to turbulent flow. First, I will describe some results on shear flow instability in the setting of inviscid flows in a rigid wall. Then the effects of a free surface (or water waves) and viscosity will be discussed.
Friday, August 29, 2008 - 15:00 , Location: Skiles 255 , Kevin Costello , School of Mathematics, Georgia Tech , Organizer: Prasad Tetali
Let f be a polynomial or multilinear form in a large number of variables. A basic question we can ask about f is how dispersed it becomes as the number of variables increases. To be more concrete: If we randomly (and independently) set each entry to be either 1 or -1, what is the largest concentration of the output of f on any single value, assuming all (or most) of the coefficients of f are nonzero? Can we somehow describe the structure of those forms which are close to having maximal concentration? If f is a linear polynomial, this is a question originally examined by Littlewood and Offord and answered by Erdos: The maximal concentration occurs when all the nonzero coefficients of f are equal. Here we will consider the case where f is a bilinear or quadratic form.
Thursday, August 28, 2008 - 15:00 , Location: Skiles 269 , Mikhail Lifshits , School of Mathematics, Georgia Tech , Organizer: Heinrich Matzinger
We consider a random field of tensor product type X and investigate the quality of approximation (both in the average and in the probabilistic sense) to X by the processes of rank n minimizing the quadratic approximation error. Most interesting results are obtained for the case when the dimension of parameter set tends to infinity. Call "cardinality" the minimal n providing a given level of approximation accuracy. By applying Central Limit Theorem to (deterministic) array of covariance eigenvalues, we show that, for any fixed level of relative error, this cardinality increases exponentially (a phenomenon often called "intractability" or "dimension curse") and find the explosion coefficient. We also show that the behavior of the probabilistic and average cardinalities is essentially the same in the large domain of parameters.
Series: Graph Theory Seminar
The problem of generating random integral tables from the set of all nonnegative integral tables with fixed marginals is of importance in statistics. The Diaconis-Sturmfels algorithm for this problem performs a random walk on the set of such tables. The moves in the walk are referred to as Markov bases and correspond to generators of a certain toric ideal. When only one and two-way marginals are considered, one can naturally associate a graph to the model. In this talk, I will present a characterization of all graphs for which the corresponding toric ideal can be generated in degree four, answering a question of Develin and Sullivant. I will also discuss some related open questions and demonstrate applications of the Four Color theorem and results on clean triangulations of surfaces, providing partial answers to these questions. Based on joint work with Daniel Kral and Ondrej Pangrac.
Wednesday, August 27, 2008 - 12:00 , Location: Skiles 255 , Tom Trotter, Teena Carroll, Luca Dieci , School of Mathematics, Georgia Tech , Organizer:
* Dr. Trotter: perspective of the hiring committee with an emphasis on research universities. * Dr. Carroll: perspective of the applicant with an emphasis on liberal arts universities. * Dr. Dieci: other advice, including non-academic routes.