McGill University

We study pure Seiberg-Witten theories with $SU(r+1)$ and $Sp(2r)$ gauge groups with no flavors. We study singularity loci of moduli space of the Seiberg-Witten curve. Using exterior derivative and discriminant operators, we can find Argyres-Douglas loci of the SW theory. We also compute BPS charges of the massless dyons of $SU$ and $Sp$ SW theory. In a detailed example of $C_2=Sp(4)$, we find 6 points in the moduli space where we have 2 massless BPS dyons, and 3 of them give Argyres-Douglas loci. We show that BPS charges of the massless dyons jump as we go across Argyres-Douglas loci, giving an explicit example of Argyres-Douglas loci living inside the wall of marginal stability. (Based on work in progress with Keshav Dasgupta)

We will discuss the mixing rate of the standard random walk on the giant

component of the random graph G(n,p). We tie down the mixing rate precisely

for all values of p greater than (1+c)/n for any positive constant c. We need

to develop a new bound on the mixing time of general Markov chains, inspired

by and extending work of Kannan and Lovasz. This is joint work with Nick

Fountoulakis.