Students Probability Day V, 14 May 2015


Robert Adler:

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.

Chaim Even-Zohar:

Title: Random knots and their invariants.

Yuval Peled:

Title: On the phase transition in random simplicial complexes

Peleg Michaeli:

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

Ofer Busani:

Title: Continuous Time Random Walk Limits

Paul Balanca:

Title: Uniform multifractal structure of stable trees and super Brownian motion

Mira Shamis:

Title: The Wegner orbital model: localization and Wegner estimate

Yinon Spinka:

Title: Long range order in the 3-state antiferromagnet Potts model

Idan Perl:

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.

Youngwhan Son:

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.

Arie Levit:

Title: Invariant random subgroups and random complexes

Sasha Shamov:

Title: Local random measures over Gaussian fields

Suhamay Saha:

Title: Construction of Asymptotically Optimal Control for a Stochastic Network from a Free Boundary Problem

Shay Moran:

Title: Compression and efficient data representation