| Date: | January 18, 2019 |
| Time: | 11:00am |
| Location: | BLOC 628 |
| Speaker: | SHAMGAR GUREVITCH, U Wisconsin-Madison |
| Title: | Harmonic Analysis on GLn over finite fields, and Random Walks |
| Abstract: | There are many formulas that express interesting properties of a group G in terms of sums over its characters.
For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}:
$$trace (\rho(g))/dim (\rho),$$
for an irreducible representation $\rho$ of G and an element g of G.
For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G.
It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}.
This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks.
This is joint work with Roger Howe (TAMU). |
|
| Date: | February 15, 2019 |
| Time: | 11:00am |
| Location: | BLOC 628 |
| Speaker: | Petros Valettas, University of Missouri |
| Title: | Gaussian concentration and convexity |
| Abstract: | We will discuss new forms of concentration and anti-concentration
phenomena explained by convexity rather than isoperimetry. On the side of
applications, this perspective allows for superconcentration and convexity to be
melted together in order to obtain strong small ball and small deviation estimates
in normed spaces. In this framework, the $\ell_\infty$-structure arises naturally as
the (approximate) extremal. The underlying principle dictating these phenomena is
rooted in the sensitivity of the variance.
Based on a joint work with G. Paouris and K. Tikhomirov |
|
| Date: | February 22, 2019 |
| Time: | 11:00am |
| Location: | BLOC 628 |
| Speaker: | Alex Hening, Tufts University |
| Title: | Stochastic persistence and extinction |
| Abstract: | A key question in population biology is understanding the conditions under which the species from an ecosystem persist or go extinct. Theoretical and empirical studies have shown that coexistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of n interacting species that live in a stochastic environment. Our models are described by n dimensional piecewise deterministic Markov processes. These are processes (X(t), r(t)) where the vector X denotes the density of the n species and r(t) is a finite state space process which keeps track of the environment. In any fixed environment the process follows the flow given by a system of ordinary differential equations. The randomness comes from the changes or switches in the environment, which happen at random times. We give sharp conditions under which the the populations persist as well as conditions under which some populations go extinct exponentially fast. As an example we look at the competitive exclusion principle from ecology and show how the random switching can `rescue' species from extinction. The talk is based on joint work with Dang H. Nguyen (University of Alabama). |
|
| Date: | March 22, 2019 |
| Time: | 11:00am |
| Location: | BLOC 628 |
| Speaker: | Eunghyun Lee, Nazarbayev University |
| Title: | Exact formulas in the multi-species ASEP |
| Abstract: | In this talk, we introduce the exact formulas of transition probabilities and the ``block probabilities" of ASEP with multi-species. In particular, we extend Chatterjee and Schutz's result (2010, JSP) on the TASEP with second class particles which gives some determinantal formulas to the ASEP with multi-species. Also, we discuss about similar results for other multi-species models. |
|
| Date: | March 29, 2019 |
| Time: | 11:00am |
| Location: | BLOC 628 |
| Speaker: | Wei-Kuo Chen, University of Minnesota |
| Title: | Phase Transition in the Spiked Gaussian Tensor Models |
| Abstract: | The problem of detecting a deformation in a symmetric Gaussian random tensor is concerned about whether there exists a statistical hypothesis test that can reliably distinguish a low-rank random spike from the noise. In this talk, we will consider the spikes sampled from bounded priors. We will show that there exist critical thresholds for the signal-to-noise ratios, which strictly separate the distinguishability and indistinguishability between the non-spiked and spiked Gaussian random tensors under the total variation distance. Our approach is based on a subtle analysis of the high temperature behavior of the pure p-spin model, arising initially from the field of spin glasses. In particular, the signal-to-noise criticality is identified as the critical temperature, distinguishing the high and low temperature behavior, of the pure p-spin model. |
|
Menu