We will discuss some recent developments in several new directions of
spectral graph theory and mention a number of open problems.

This talk will be an exposition of joint work with Man Wai (Mandy)
Cheung on the effect of the Ricci flow on homogeneous metrics of
positive sectional curvature on flag varieties over the complex,
quaternions and octonians. The speaker’s 1972 paper shows that these
metrics exist only in the case of the variety of flags in the two
dimension projective space over these fields. Here are some of the
results:
All cases can flow from strictly positive curvature to some negative
sectional curvature.
All cases can flow from positive definite Ricci curvature to
indefinite Ricci curvature
The quaternionic and octonianic cases can flow from strictly positive
sectional curvature to indefinite Ricci curvature (in the case of the
quaternions this is a result of Boehm and Wilking).
In the complex case the flow keeps the metrics of strictly positive
curvature in the metrics with positive definite Ricci curvature.

I consider a Markov process inspired by a toy model of a limit order book. "Bid" and "ask" orders arrive in time; the prices are iid uniform on [0,1]. (I'll discuss some extensions.) When a match is possible (bid > ask), the highest bid and lowest ask leave the system. This process turns out to have surprising dynamics, with three limiting behaviours occurring with probability one. At low prices (< 0.21...), bids eventually never leave; at high prices (>0.78...), asks eventually never leave; and in between, the system "ought to" be positive recurrent. I will show how we can derive explicitly the limiting distribution of certain marginals for the middle prices; this makes it possible to extract the numerical values above from a 0-1 Law result.

Lights Out is a single player game on graph G. The game starts with a coloring of the vertices of G with two colors, 0 and 1. At each step, one vertex is toggled which switches the color of that vertex and all of its neighbors. The game is won when all vertices have color 0. This game can be analyzed using linear algebra over a finite field, for example the number of solvable colorings of a graph is 2 to the rank of A + I, where A is the adjacency matrix of the graph, and I is the identity.
We consider the stochastic process arising from toggling a sequence of random vertices. We demonstrate how the process can be viewed as a random walk on an associated state graph. We then find the eigenvalues of the state graph, and use them to bound the rate of convergence and hitting times. We also provide bounds on the average number of steps until this random process reaches the all 0 coloring that are asymptotically tight for many families of graphs.

Alternating direction methods are a commonplace tool for general mathematical programming and optimization. These methods have become particularly important in the field of variational image processing, which frequently requires the minimization of non-differentiable objectives. This paper considers accelerated (i.e., fast) variants of two common alternating direction methods: the Alternating Direction Method of Multipliers (ADMM) and the Alternating Minimization Algorithm (AMA). The proposed acceleration is of the form first proposed by Nesterov for gradient descent methods. In the case that the objective function is strongly convex, global convergence bounds are provided for both classical and accelerated variants of the methods. Numerical examples are presented to demonstrating the superior performance of the fast methods.

If two simple linear algebraic groups have the same F-isomorphism classes of maximal F-tori, are the two groups necessarily isomorphic? When F is a number field, it is an old question attributed to Shimura. We describe the recent solution to this question (which relies on the notion of weak commensurability introduced by Gopal Prasad and Andrei Rapinchuk) and its connection with the question "Can you hear the shape of a drum?" for arithmetic quotients of locally symmetric spaces.

This talk will cover some of the main problems in the field of nonlinear dispersive equations. I will discuss the stability, instability and blow-up for some simpler models such as the cubic Nonlinear Schr\"odinger equations

I will start with a basic introduction to planar algebras. I will
then discuss recent joint work with Scott Morrison (arXiv:1208.3637) and
recent joint work with Stephen Bigelow (arXiv:1208:1564). With
Morrison, we construct a new exotic subfactor planar algebra using
Bigelow's jellyfish algorithm. With Bigelow, we determine exactly
when a planar algebra has a presentation by generators and jellyfish
relations.

I will start with a basic introduction to planar algebras. I will
then discuss recent joint work with Scott Morrison (arXiv:1208.3637) and
recent joint work with Stephen Bigelow (arXiv:1208:1564). With
Morrison, we construct a new exotic subfactor planar algebra using
Bigelow's jellyfish algorithm. With Bigelow, we determine exactly
when a planar algebra has a presentation by generators and jellyfish
relations.

The Margulis Normal Subgroup Theorem states that any normal
subgroup of an irreducible lattice in a center-free higher-rank semisimple
Lie
group is of finite index. Stuck and Zimmer, expanding on Margulis'
approach, showed that any properly ergodic probability-preserving
ergodic action of such a lattice is essentially free.

I will present similar results: my work with Y. Shalom on normal
subgroups of lattices in products of simple locally compact groups and
normal subgroups of commensurators of lattices, and my work with J.
Peterson generalizing this result to stabilizers of ergodic
probability-preserving actions of such groups. As a consequence,
S-arithmetic lattices enjoy the same properties as the arithmetic
lattices (the Stuck-Zimmer result) as do lattices in certain product
groups. In particular, any nontrivial ergodic probability-preserving
action of $\mathrm{PSL}_{n}(\mathbb{Q})$, for $n \geq 3$, is
essentially free.

The key idea in the study of normal subgroups is considering
nonsingular actions which are the extreme opposite of
measure-preserving. Somewhat surprisingly, the key idea in
understanding stabilizers of probability-preserving actions also
involves studying such actions and the bulk of our work is directed
towards properties of these contractive actions.