Students Probability Day V, 14 May 2015
Title: Phase Transitions and Topology
Abstract: Phase transitions are phenomena occurring in large random systems, as water cools and turns to ice, is heated and turns to vapor, or as iron loses or gains magnetism. Although not usually recognized as such, there is topology in these transitions, generally studied only at the topologically simple level of connectivity.
Recently, there has been growing interest in far more sophisticated phase transitions, studying large random topological structures undergoing transitions in their homological structure. The results in this area, to a large extent initially motivated by questions from topological data analysis, are building the foundations of a new area of research, that one might call Random Topology.
The aim of the lecture will be to describe these developments, mainly via a number of examples, ranging from the asymptotics of the Betti numbers of large simplicial complexes to a version of a random Morse theory. While the details of the examples are not simple, my aim will be to present everything in an audience friendly, widely accessible fashion.
Sunder Ram Krishnan:
Title: Asymptotic critical radius of a randomly embedded manifold.
Title: Random knots and their invariants.
Title: On the phase transition in random simplicial complexes
Title: graph theoretic properties of the trace of a random walk on random graphs
Patric Karl Gloede:
Title: Well-posedness of skew product martingale problems
Title: Continuous Time Random Walk Limits
Title: Uniform multifractal structure of stable trees and super Brownian motion
Title: The Wegner orbital model: localization and Wegner estimate
Title: Long range order in the 3-state antiferromagnet Potts model
Title: Harmonic functions and polynomials on discrete groups
Abstract: Alexopoulos proved that on a finitely generated nilpotent group, the restriction of a harmonic function of polynomial growth to a torsion-free nilpotent subgroup of finite index is always a polynomial. We give a simpler, more direct proof for this result using a more elementary definition for polynomials. We also show that the Laplacian operator is onto, and calculate the precise dimension of the space of harmonic functions of given polynomial growth, refining previous results.
Title: Birkhoff sum fluctuations of substitution dynamical systems
Abstract: In this talk we will discuss deviation of Birkhoff sums for substitution
dynamical systems with an incidence matrix having eigenvalues of
modulus 1. Especially we will describe central limit theorem for fixed
points of substitution. This is a joint work with E. Paquette.
Title: Invariant random subgroups and random complexes
Title: Local random measures over Gaussian fields
Title: Construction of Asymptotically Optimal Control for a Stochastic Network from a Free Boundary Problem
Title: Compression and efficient data representation