- You are here:
- GT Home
- Home
- News & Events

Series: ACO Student Seminar

A successful approach to solving linear programming problems exactly has been to solve the problems with increasing levels of fixed precision, checking the final basis in exact arithmetic and then doing additional simplex pivots if necessary. This work is a computational study comparing different techniques for the core element of our exact computation: solving sparse rational systems of linear equations exactly.

Series: Research Horizons Seminar

Dynamics of spatially extended systems is often described by Lattice Dynamical Systems (LDS). LDS were introduced 25 years ago independently by four physicists from four countries. Sometimes LDS themselves are quite relevant models of real phenomena. Besides, very often discretizations of partial differential equations lead to LDS. LDS consist of local dynamical systems sitting in the nodes of a lattice which interact between themselves. Mathematical studies of LDS started in 1988 and introduced a thermodynamic formalism for these spatially extended dynamical systems. They allowed to give exact definitions of such previously vague phenomena as space-time chaos and coherent structures and prove their existence in LDS. The basic notions and results in this area will be discussed. It is a preparatory talk for the next day colloquium where Dynamical Networks, i.e. the systems with arbitrary graphs of interactions, will be discussed.

Series: Geometry Topology Seminar

The Dehn function of a finitely presented group measures the difficulty in filling loops in the presentation complex of the group. Higher dimensional Dehn functions are a natural generalization: the n-dimensional Dehn function of a group captures the difficulty of filling n-spheres with (n+1)-balls in suitably defined complexes associated with the group. A fundamental question in the area is that of determining which functions arise as Dehn functions. I will give an overview of known results and describe recent progress in the 2-dimensional case. This is joint work with Josh Barnard and Noel Brady.

Series: Analysis Seminar

Variable transformations are used to enhance the normally poor performance of trapezoidal rule approximations of finite-range integrals I[f]=\int^1_0f(x)dx. Letting x=\psi(t), where \psi(t) is an increasing function for 0 < t < 1 and \psi(0)=0 and \psi(1)=1, the trapezoidal rule is applied to the transformed integral I[f]=\int^1_0f(\psi(t))\psi'(t)dt. By choosing \psi(t) appropriately, approximations of very high accuracy can be obtained for I[f] via this approach. In this talk, we survey the various transformations that exist in the literature. In view of recent generalizations of the classical Euler-Maclaurin expansion, we show how some of these transformations can be tuned to optimize the numerical results. If time permits, we will also discuss some recent asymptotic expansions for Gauss-Legendre integration rules in the presence of endpoint singularities and show how their performance can be optimized by tuning variable transformations. The variable transformation approach presents a very flexible device that enables one to write his/her own high-accuracy numerical integration code in a simple way without the need to look up tables of abscissas and weights for special Gaussian integration formulas.

Monday, September 15, 2008 - 13:00 ,
Location: Skiles 255 ,
Peijun Li ,
Department of Mathematics, Purdue University ,
Organizer: Haomin Zhou

Near-field optics has developed dramatically in recent years due to the possibility of breaking the diffraction limit and obtaining subwavelength resolution. Broadly speaking, near-field optics concerns phenomena involving evanescent electromagnetic waves, to which the super-resolving capability of near-field optics may be attributed. In order to theoretically understand the physical mechanism of this capability, it is desirable to accurately solve the underlying scattering problem in near-field optics. We propose an accurate global model for one of the important experimental modes of near-field optics, photon scanning tunneling microscopy, and develop a coupling of finite element and boundary integral method for its numerical solution. Numerical experiments will be presented to illustrate the effectiveness of the proposed method and to show the features of wave propagation in photon scanning tunneling microscope. The proposed model and developed method have no limitations on optical or geometrical parameters of probe and sample, they can be used for realistic simulations of various near-field microscope configurations.

Series: Combinatorics Seminar

Let K^r_{r+1} denote the complete r-graph on r+1 vertices. The Turan density of K^r_{r+1} is the largest number t such that there are infinitely many K^r_{r+1}-free r-graphs with edge density t-o(1). Determining t(K^r_{r+1}) for r > 2 is a famous open problem of Turan. The best upper bound for even r, t(K^r_{r+1})\leq 1-1/r, was given by de Caen and Sidorenko. In a joint work with Linyuan Lu, we slightly improve it. For example, we show that t(K^r_{r+1})\leq 1 - 1/r - 1/(2r^3) for r=4 mod 6. One of our lemmas also leads to an exact result for hypergraphs. Given r > 2, let p be the smallest prime factor of r-1. Every r-graph on n > r(p-1) vertices such that every r+1 vertices contain 0 or r edges must be empty or a complete star.

Friday, September 12, 2008 - 14:00 ,
Location: Skiles 269 ,
Stavros Garoufalidis ,
School of Mathematics, Georgia Tech ,
Organizer: Stavros Garoufalidis

We will discuss, with examples, the Jones polynomial of the two simplest knots (the trefoil and the figure eight) and its loop expansion.

Series: Stochastics Seminar

Under certain conditions, we obtain exact asymptotic expressions for the stationary distribution \pi of a Markov chain. In this talk, we will consider Markov chains on {0,1,...}^2. We are particularly interested in deriving asymptotic expressions when the fluid limit of the most probable paths from the origin to the rare event are nonlinear. For example, we will derive asymptotic expressions for a large deviation along the x-axis (e.g., \pi(\ell, y) for fixed y) when the most probable paths to (\ell,y) initially climb the y-axis before turning southwest and drifting towards (\ell,y).

Series: ACO Student Seminar

In order to estimate the spread of potential pandemic diseases and the efficiency of various containment policies, it is helpful to have an accurate model of the structure of human contact networks. The literature contains several explicit and implicit models, but none behave like actual network data with respect to the spread of disease. We discuss the difficulty of modeling real human networks, motivate the study of some open practical questions about network structure, and suggest some possible avenues of attack based on some related research.

Series: Research Horizons Seminar

A plasma is a gas of ionized particles. For a dilute plasma of very high temperature, the collisions can be ignored. Such situations occur, for example, in nuclear fusion devices and space plasmas. The Vlasov-Poisson and Vlasov-Maxwell equations are kinetic models for such collisionless plasmas. The Vlasov-Poisson equation is also used for galaxy evolution. I will describe some mathematical results on these models, including well-posedness and stability issues.