Department of Mathematics

 Winter 2017

We will meet on Tuesdays at 4:00 pm.
Here is a tentative list of speakers (to be extended): Patrick Waters (Temple University), William Felder, Yevgeniy Kovchegov
Registration information: Mth 607, Sec 016 - CRN 31647



Title and abstract

Tuesday, January 10, 4:00 pm
BEXL 321
Patrick Waters
Temple University

"Random matrices and the Stochastic Bessel Operator"

Abstract. The eigenvalues of random matrix "invariant ensembles" can be understood as interacting particle systems, but with temperature restricted to three possible values. Recently "beta ensembles", which extend temperature to all positive values, have been well studied. It has been conjectured that the extremal eigenvalues of a large beta ensemble random matrix with a "hard edge" should be governed by the Stochastic Bessel Operator (SBO). We prove that this conjecture holds when the external field is a polynomial satisfying a convexity condition and β≥1. The law of a smallest SBO eigenvalue gives a two parameter generalization of the famous Tracy-Widom distribution which can be observed in the fluctuations of a spreading coffee stain, the longest increasing subsequence of a random permutation, etc. Joint work with Brian Rider.
Tuesday, February 7, 4:00 pm
BEXL 321
William Felder
Oregon State University

"Distinguished path analysis for continuous-time branching processes: a framework and applications"

Abstract. In this talk we will consider a rubric, laid out by Hardy and Harris, under which many earlier formulations of distinguished path analysis (or "spine techniques") for branching processes are unified.
It has been known for some time that there is a connection between single-particle martingales and certain additive martingales for the corresponding branching processes. This connection is made explicit here, where each is seen to be the projection of a single, more general martingale onto different sub sigma-algebras. We will also see a nice, intuitive formulation of the martingale change of measure that results in the typical alterations along the spine: namely a change in the drift, a change in the offspring distribution ("size biasing"), and a change in the reproductive rate.
The overall formulation is quite elegant, and its power will be demonstrated through consideration of example applications.

Past probability seminars: Fall 2005, Winter 2006, Spring 2006, Fall 2006, Winter 2007, Spring 2007, Fall 2007, Winter 2008, Spring 2008, Fall 2008, Winter 2009, Spring 2009, Fall 2009, Winter 2010, Spring 2010, Fall 2010, Winter 2011, Spring 2011, Fall 2011, Winter 2012, Spring 2012, Fall 2012, Winter 2013, Spring 2013, Fall 2013, Winter 2014, Spring 2014, Fall 2014, Winter 2015, Spring 2015, Fall 2015, Winter 2016, Spring 2016, Fall 2016