Ramon van Handel (Princeton University) (Two lectures)

                                                                                A New Approach to Nonasymptotic Random Matrix Theory  

Abstract: Classical random matrix theory is largely concerned with the asymptotic 
properties very special random matrix models, such as matrices with i.i.d. entries, 
invariant ensembles, and other mean-field models. On the other hand, matrix concentration 
inequalities, which are widely used in applied mathematics to obtain nonasymptotic 
bounds on very general random matrices, can only provide crude and often suboptimal 
information on the spectrum. Very recently, however, a new approach to nonasymptotic 
random matrix theory has resulted in a drastically improved understanding of arbitrarily 
structured random matrices. This theory opens the door to applications that were beyond 
the reach of previous methods. My aim is to introduce some of the basic ingredients of 
this theory, and to illustrate by means of concrete examples some ways in which it can be used. .
 

Pablo Ferrari (University of Buenos Aires) (Two lectures)

      Talk 1: Hidden Temperature in the KMP Model
    

Abstract: In the KMP model a positive energy $Z_i$ is associated with each site $i\in\{0,\frac1n,
\dots,\frac{n-1}{n},1\}$. When a Poisson clock rings at the bond $ij$ with energies $Z_i,Z_j$, 
those values are substituted by $U(Z_i+Z_j)$ and $(1-U)(Z_i+Z_j)$, respectively, where $U$ is a 
uniform random variable in $(0,1)$. The energy at the boundary vertices  $b\in\{0,1\}$ is always 
an exponential random variable with mean $T_b$, the fixed boundary condition. We show that the 
stationary measure for the resulting Markov process $Z(t)$ is the distribution of a vector $Z$ 
with coordinates $Z_i=T_iX_i$, where $X_{k/n}$ are iid exponential$(1)$ random variables, the 
law of $T$ is the invariant measure for an opinion model with the same boundary conditions, and 
$X,T$ are independent. The result confirms a conjecture based on the large deviations of the model. 
The discrete derivative of the opinion model behaves as a neural spiking process, which is also 
analysed. The hydrostatic limit shows that the empirical measure associated to the stationary 
distribution converges to the linear interpolation of $T_0$ and $T_{1}$. Joint work with Anna de 
Masi and Davide Gabrielli L'Aquila. 

      Talk 2: Soliton Decomposition of the Zigzag Random Walk 

Abstract: The box-ball system is a one-dimensional transport cellular automaton that 
exhibits soliton behavior. The slot decomposition of a ball configuration reveals 
soliton components travelling ballistically. This linearization yields generalized 
hydrodynamic theorems. Furthermore, it provides methods for constructing invariant 
measures through the slot diagrams of the excursions. In this talk, we will see that 
the slot diagram of a continuous soliton-weighted random excursion of the zig-zag 
process is a nonhomogeneous Poisson process in the positive quadrant, and that the 
slot decomposition of the corresponding translation-invariant measure is a Poisson 
process in the upper semiplane with intensity $dx q(y) dy$ , where q is an integrable 
explicit function, proportional to the law of the maximum of the excursion. Based on 
works with Inés Armendáriz, Pablo Blanc, Davide Gabrielli, Chi Nguyen, Leo Rolla 
and Minmin Wang.

Sandra Cerrai (University of Maryland)	

	The Smoluchowski-Kramers Diffusion Approximation for Constrained Stochastic Wave Equations

Abstract: cWe study the validity of the Smoluchowski-Kramers diffusion approximation for a class
of stochastic damped wave equations constrained to live on the unitary sphere of the space of 
square-integrable functions.  We demonstrate that, in the small mass limit, the solution converges 
to the solution of a stochastic parabolic equation subject to the same constraint. We further show 
that an extra noise-induced drift emerges, which in fact does not account for the Stratonovich-to-Itô 
correction term.

 
Sumit Mukerjee (Columbia University)

   Permutation Limits

Abstract: Permutation limit theory arises by viewing a permutation as a probability measure 
on the unit square, and is motivated by dense graph limit theory. Using the theory of 
permutation limits (permutons), we can compute limiting properties of various permutation 
statistics for random permutations, such as number of fixed points, number of small cycles, 
pattern counts, and degree distribution of permutation graphs. We can also derive LDPs for 
random permutations. Our results apply to many non uniform distributions on permutations, 
including the celebrated Mallows model, and mu-random permutations. This is based on joint 
work with Bhaswar Bhattacharya, Jacopo Borga, Sayan Das and Peter Winkler.

Arnab Sen (University of Minnesota)

       Some Results on Levy Spin Glasses