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Series: Combinatorics Seminar

Hajos' conjecture is false, and it seems that graphs without a
subdivision of a big complete graph do not behave as well as
those without a minor of a big complete graph.
In fact, the graph minor theorem (a proof of Wagner's
conjecture) is not true if we replace the minor relation by the
subdivision relation. I.e, For every infinite sequence
G_1,G_2, ... of graphs, there exist distinct integers
i < j such that G_i is a minor of G_j, but if we replace
''minor" by ''subdivision", this is no longer true.
This is partially because we do not really know what the graphs
without a subdivision of a big complete graph look like.
In this talk, we shall discuss this issue. In particular,
assuming some moderate connectivity condition, we can say
something, which we will present in this talk.
Topics also include coloring graphs without a subdivision of a
large complete graph, and some algorithmic aspects. Some of the
results are joint work with Theo Muller.

Series: SIAM Student Seminar

The Fundamental Theorem of Algebra implies that a complex valued nxn matrix has n eigenvalues (including multiplicities). In this talk we introduce a general method for reducing the size of a square matrix while preserving this spectrum. This can then be used to improve on the classic eigenvalue estimates of Gershgorin, Brauer, and Brualdi. As this process has a natural graph theoretic interpretation this talk should be accessible to most anyone with a basic understanding of matrices and graphs. These results are based on joint work with Dr. Bunimovich.

Series: Job Candidate Talk

In this talk, I will first discuss several chemotaxis models includingthe classical Keller-Segel model.Chemotaxis is the phenomenon in which cells, bacteria, and other single-cell or multicellular organisms direct their movements according to certain chemicals (chemoattractants) in their environment. The mathematical models of chemotaxis are usually described by highly nonlinear time dependent systems of PDEs. Therefore, accurate and efficient numerical methods are very important for the validation and analysis of these systems. Furthermore, a common property of all existing chemotaxis systems is their ability to model a concentration phenomenon that mathematically results in solutions rapidly growing in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular, spiky behavior. In either case, capturing such solutions numerically is a challenging problem. In our work we propose a family of stable (even at times near blow up) and highly accurate numerical methods, based on interior penalty discontinuous Galerkin schemes (IPDG) for the Keller-Segel chemotaxis model with parabolic-parabolic coupling. This model is the basic step in the modeling of many real biological processes and it is described by a system of a convection-diffusion equation for the cell density, coupled with a reaction-diffusion equation for the chemoattractant concentration.We prove theoretical hp error estimates for the proposed discontinuous Galerkin schemes. Our proof is valid for pre-blow-up times since we assume boundedness of the exact solution.Numerical experiments to demonstrate the stability and accuracy of the proposed methods for chemotaxis models and comparison with other methods will be presented. Ongoing research projects will be discussed as well.

Thursday, January 21, 2010 - 11:05 ,
Location: Skiles 269 ,
Bruce Reed ,
McGill University ,
Organizer: Robin Thomas

The term Probabilistic Method refers to the proof of
deterministic statements using probabilistic tools. Two of the most famous
examples arise in number theory. these are: the first non-analytic proof
of the prime number theorem given by Erdos in the 1940s, and the recent
proof of the Hardy-Littlewood Conjecture (that there are arbitrarily long
arithmetic progressions of primes) by Green and Tao.
The method has also been succesfully applied in the field of graph
colouring. We survey some of the results thereby obtained.
The talk is targeted at a general audience. We will first define graph
colouring, explain the type of graph colouring problems which tend to
attract interest, and then explain the probabilistic tools which are
used
to solve them, and why we would expect the type of tools that are used to
be effective for solving the types of problems typically studied.

Series: Analysis Seminar

Müntz polynomials arise from consideration of Müntz's Theorem, which is a beautiful generalization of Weierstrass's Theorem. We prove a new surprisingly simple representation for the Müntz orthogonal polynomials on the interval of orthogonality, and in particular obtain new formulas for some of the classical orthogonal polynomials (e.g. Legendre, Jacobi, Laguerre). This allows us to determine the strong asymptotics on the interval, and the zero spacing behavior follows. This is the first time that such asymptotics have been obtained for general Müntz exponents. We also look at the asymptotic behavior outside the interval, and the asymptotic properties of the associated Christoffel functions.

Wednesday, January 20, 2010 - 11:00 ,
Location: SKiles 269 ,
Lee Childers ,
Georgia Tech, School of Applied Physiology ,
lee@gatech.edu ,
Organizer:

Cycling represents an integration of man and machine. Optimizing this integration through changes in rider position or bicycle component selection may enhance performance of the total bicycle/rider system. Increasing bicycle/rider performance via mathematical modeling was accomplished during the US Olympic Superbike program in preparation for the 1996 Atlanta Olympic Games. The purpose of this presentation is to provide an overview on the science of cycling with an emphasis on biomechanics using the track pursuit as an example. The presentation will discuss integration and interaction between the bicycle and human physiological systems, how performance may be measured in a laboratory as well as factors affecting performance with an emphasis on biomechanics. Then reviewing how people pedal a bicycle with attention focused on forces at the pedal and the effect of position variables on performance. Concluding with how scientists working on the US Olympic Superbike program incorporated biomechanics and aerodynamic test data into a mathematical model to optimize team pursuit performance during the 1996 Atlanta Olympic Games.

Series: Algebra Seminar

Weyl proved that if an N-dimensional real vector v has linearly independent coordinates over Q, then its integer multiples v, 2v, 3v, .... are uniformly distributed modulo 1. Stated multiplicatively (via the exponential map), this can be viewed as a Haar-equidistribution result for the cyclic group generated by a point on the N-dimensional complex unit torus. I will discuss an analogue of this result over a non-Archimedean field K, in which the equidistribution takes place on the N-dimensional Berkovich projective space over K. The proof uses a general criterion for non-Archimedean equidistribution, along with a theorem of Mordell-Lang type for the group variety G_m^N over the residue field of K, which is due to Laurent.

Series: PDE Seminar

I will discuss the intermediate and long time dynamics
of solutions of the nonlinear Schroedinger - Gross Pitaevskii equation,
governing nonlinear dispersive waves in a spatially
non-homogeneous background.
In particular, we present results (with B. Ilan)
on solitons with frequencies near a spectral band edge associated
with periodic potential, and results (with Z. Gang) on large
time energy distribution in systems with multiple bound states.
Finally, we discuss how such results can inform strategies
for control of soliton-like states in optical and quantum systems.

Series: Job Candidate Talk

The Edrei-Thoma theorem characterizes totally positive functions, and plays an important role in character theory of the infinite symmetric group. The Loewner-Whitney theorem characterizes totally positive elements of the general linear group, and is fundamental for Lusztig's theory of total positivity in reductive groups. In this work we derive a common generalization of the two theorems. The talk is based on joint work with Thomas Lam.

Friday, January 15, 2010 - 14:00 ,
Location: Skiles 269 ,
Mohammad Ghomi ,
Georgia Tech ,
Organizer:

We study the topology of the space bd K^n of complete convex
hypersurfaces of R^n which are homeomorphic to R^{n-1}. In particular,
using Minkowski sums, we construct a deformation retraction of bd K^n
onto the Grassmannian space of hyperplanes. So every hypersurface in bd
K^n may be flattened in a canonical way. Further, the total curvature
of each hypersurface evolves continuously and monotonically under this
deformation. We also show that, modulo proper rotations, the subspaces
of bd K^n consisting of smooth, strictly convex, or positively curved
hypersurfaces are each contractible, which settles a question of H.
Rosenberg.