
Fall 2014
* please note time, date, and/or location change
Abstracts
Random tilings, representation theory and Macdonald processes I 

Leo Petrov* (joint with Algebra, meets at 3:30p)
I will discuss random lozenge tilings of polygons, their representationtheoretic interpretation (including connections to the character theory of the infinitedimensional unitary group), and how this circle of ideas leads to Macdonald processes. The latter give rise to other probabilistic models such as random polymers and similar interacting stochastic particle systems top of page

Harnack inequalities for degenerate diffusions 

Camelia Pop (U Penn)
We study various probabilistic and analytical properties of a class of degenerate diffusion operators arising in Population Genetics, the socalled generalized Kimura diffusion operators. Our main results are a stochastic representation of weak solutions to a degenerate parabolic equation with singular lowerorder coefficients, and the proof of the scaleinvariant Harnack inequality for nonnegative solutions to the Kimura parabolic equation. The stochastic representation of solutions that we establish is a considerable generalization of the classical results on FeynmanKac formulas concerning the assumptions on the degeneracy of the diffusion matrix, the boundedness of the drift coefficients, and on the a priori regularity of the weak solutions. top of page

Higher Order Dangers and Precisely Constructed Taxa in Models of Randomness 

Steve Pincus (joint with Statistics)
The certification, construction and delineation of individual, infinite length `random` sequences has been a longstanding, yet incompletely resolved problem. We address this topic via the study of normal numbers, which have often been viewed as reasonable proxies for randomness, given their limiting equidistribution of subblocks of all lengths. However, limitations arise within this perspective. First, we explicitly construct a normal number that satisfies the Law of the Iterated Logarithm (LIL), yet exhibits pairwise bias towards repeated values, rendering it inappropriate for any collection of random numbers. Accordingly, we deduce that the evaluation of higher order block dynamics, even beyond limiting equidistribution and fluctuational typicality, is imperative in proper evaluation of sequential `randomness`. Second, we develop several criteria motivated by classical theorems for symmetric random walks, which lead to algorithms for generating normal numbers that satisfy a variety of attributes for the series of initial partial sums, including rates of sign changes, patterns of return times to 0, and the extent of fairness of the sequence. Such characteristics are generally unaddressed in most evaluations of `randomness`. More broadly, we can differentiate normal numbers both on the basis of multiple distinct qualitative attributes, as well as quantitatively via a spectrum of rates within each attribute. Furthermore, we exhibit a toolkit of techniques to construct normal sequences that realize diverse a priori specifications, including profound biases. Overall, we elucidate the vast diversity within the category of normal sequences. top of page





