Georgia Institute of Technology College of Sciences

The multi-server queue with non-homogeneous Poisson arrivals and customer abandonment is a fundamental dynamic rate queueing model for large scale service systems such as call centers and hospitals. Scaling the arrival rates and number of servers gives us fluid and diffusion limits. The diffusion limit suggests a Gaussian approximation to the stochastic behavior. The fluid mean and diffusion variance can form a two-dimensional dynamical system that approximates the actual transient mean and variance for the queueing process. Recent work showed that a better approximation for mean and variance can be computed from a related two-dimensional dynamical system. In this spirit, we introduce a new three-dimensional dynamical system that estimates the mean, variance, and third cumulant moment. This surpasses the previous two approaches by fitting the number in the queue to a quadratic function of a Gaussian random variable. This is based on a paper published in QUESTA and is joint work with Jamol Pender of Cornell University.

This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse. More tales following up on the talk I gave at GaTech in Nov., 2013. It is not assumed listeners heard that earlier talk.

The probabilistic method, pioneered by P. Erdös, has been key in many proofs from asymptotic geometric analysis. This method allows one to take advantage of numerous tools and concepts from probability theory to prove theorems which are not necessarily a-priori related to probability. The objective of this talk is to demonstrate several recent results which take advantage of stochastic calculus to prove results of a geometric nature. We will mainly focus on a specific construction of a moment-generating process, which can be thought of as a stochastic version of the logarithmic Laplace transform. The method we introduce allows us to attain a different viewpoint on the method of semigroup proofs, namely a path-wise point of view. We will first discuss an application of this method to concentration inequalities on high dimensional convex sets. Then, we will briefly discuss an application to two new functional inequalities on Gaussian space; an L1 version of hypercontractivity of the convolution operator related to a conjecture of Talagrand (joint with J. Lee) and a robustness estimate for the Gaussian noise-stability inequality of C.Borell (improving a result of Mossel and Neeman).

The evolution, through spatially periodic linear dispersion, of rough initial data leads to surprising quantized structures at rational times, and fractal, non-differentiable profiles at irrational times. The Talbot effect, named after an optical experiment by one of the founders of photography, was first observed in optics and quantum mechanics, and leads to intriguing connections with exponential sums arising in number theory. Ramifications of these phenomena and recent progress on the analysis, numerics, and extensions to nonlinear wave models will be discussed.