Alison Reddy (University of Illinois Urbana-Champaign)
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Placing students into an appropriate first math course is conducive to their success in the first two years of college, while placement into the wrong course can have devastating effects. Accurate placement requires an accurate and recent assessment, but many students are placed based on standard test scores that are years out of date. Alison Reddy will discuss what makes a good placement program and give data the from the University of Illinois Placement Program which has assessed over 70,000 students over the past 8 years.
Consider Brownian motion on a bounded interval which is redistributed back into the interval whenever hitting the boundary. The redistribution from each boundary point is according to some fixed probability distribution associated with it, and, conditioned on the boundary point ``hit", the redistribution is independent of the past. It is not hard to show that this is an exponentially ergodic Markov process. The convergence rate has an analytic characterization as the spectral gap for an operator that can be viewed as the infinitesimal generator of the process. In some distinguished cases the gap has an explicit numerical expression. The main thrust of the work presented is to provide a probabilistic and intuitive interpretation for the convergence rate through construction of an efficient coupling, that is a coupling in which time for coupling has a tail that coincides with the convergence rate. The coupling is elementary but obtaining efficiency is not straightforward due to the fact that the redistribution ``reshuffles" the system. This is in sharp contrast with reflected BM on an interval for which it is well known that essentially any coupling is efficient. Time permitting, I will also discuss probabilistic treatment of spectral problems for related models.
Efficient Coupling for Diffusion with Redistribution
