UVA Probability Seminar
Wednesdays, 4:30 - 5:30pm
Kerchof 317
 Organizers: Christian Gromoll, Tai Melcher, & Leonid Petrov
 Mathematics Recent Seminars Other Seminars Maps & Directions

# Fall 2014

3 Sep organizational meeting Leo Petrov* (joint with Algebra, meets at 3:30p) Random tilings, representation theory and Macdonald processes I Leo Petrov* (joint with Algebra, meets at 3:30p) Random tilings, representation theory and Macdonald processes II Leo Petrov* (joint with Algebra, meets 2-3p) Random tilings, representation theory and Macdonald processes III Camelia Pop (U Penn) Harnack inequalities for degenerate diffusions Xin Liu (Clemson) Convergence of value functions for controlled subcritical stochastic networks Harry Crane (Rutgers) Exchangeable combinatorial Markov processes Andrei Martínez-Finkelshtein* (University of Almería, visiting Vanderbilt University); joint with Harmonic Analysis & PDE seminar, meets at 4pm Steve Pincus (joint with Statistics) Higher order dangers and precisely constructed taxa in models of randomness Ivan Corwin* (Columbia) meets 4-5p Rob Neel (Lehigh)
* 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 representation-theoretic interpretation (including connections to the character theory of the infinite-dimensional 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 systemstop 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 so-called generalized Kimura diffusion operators. Our main results are a stochastic representation of weak solutions to a degenerate parabolic equation with singular lower-order coefficients, and the proof of the scale-invariant 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 Feynman-Kac 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 Convergence of value functions for controlled subcritical stochastic networks Xin Liu (Clemson) In this talk, we study certain control problems for subcritical multiclass queueing networks. A multiclass queueing network $N$ consists of multiple classes of jobs and multiple servers. We assume that each class of jobs can only be served by one server, while each server could serve multiple classes of jobs. Such network is called subcritical if the traffic intensity at each server is not greater than 1. A server is called critical if its traffic intensity is equal to 1, and it is called strictly subcritical if its traffic intensity is strictly less than 1. Denote by $N_c$ the subnetwork consisting of the critical servers and the classes of jobs served by these servers. It is shown that the value function of the suitably scaled network control problem for the network $N$ converges to that of the diffusion control problem for a modified critical subnetwork that has the same dimension as $N_c$. Our result, in particular, shows how to reduce the dimension of the control problem for subcritical queueing networks, and also suggests a general strategy to construct near optimal controls.top of page Exchangeable combinatorial Markov processes Harry Crane (Rutgers) We discuss some aspects of exchangeable Feller processes on combinatorial state spaces. These processes include many classical combinatorial stochastic processes, such as coalescent and fragmentation processes, as well as processes on more general state spaces. In the special case of partition- and graph-valued processes, we observe several nice properties, including a Lévy-Ito characterization of the jump rates. We also discuss some recent work that extends these properties to processes on general spaces of L-structures from first-order logic.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
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