
September 8
Paul
Bourgade (Harvard)
Homogenization of the Dyson Brownian motion
I will explain a homogenization result for the Dyson Brownian motion, which gives microscopic statistics from mesoscopic ones. It implies in particular the WignerDysonMehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed.

September 15
Samuel
Watson (MIT)
A conformally invariant metric on CLE_{4}
We will discuss an exploration process, introduced by Wendelin Werner and
Hao Wu, in which conformal loop ensemble (CLE_{κ}) loops grow
uniformly from the boundary of a domain. We relate this process in the case
κ = 4 to the set of level loops of the zeroboundary Gaussian free
field, and we use this point of view to show that the exploration process
is a deterministic function of the CLE loops. We describe how this gives
rise to a conformally invariant metric on CLE_{4}, which we
conjecture can be given a natural geometric interpretation.
Joint work with Scott Sheffield and Hao Wu.

September 22
Allan
Sly (UC Berkeley)
Increasing subsequences on the plane and the Slow Bond Conjecture
For a Poisson process in R^{2} with intensity 1, the distribution of the
maximum number of points on an oriented path from (0,0) to (N,N) has
been studied in detail, culminating in BaikDeiftJohansson's
celebrated TracyWidom fluctuation result. We consider a variant of
the model where one adds, on the diagonal, additional points according
to an independent one dimensional Poisson process with rate λ.
The question of interest here is whether for all positive values of
λ, this results in a change in the law of large numbers for the
the number of points in the maximal path. A closely related question
comes from a variant of Totally Asymmetric Exclusion Process,
introduced by Janowsky and Lebowitz. Consider a TASEP in 1dimension,
where the bond at the origin rings at a slower rate r<1. The question
is whether for all values of r<1, the single slow bond produces a
macroscopic change in the system. We provide affirmative answers to
both questions.
Based on joint work with Riddhipratim Basu and Vladas Sidoravicius

October 6
Philippe Di Francesco (UIUC)
Whittaker functions and fusion product
Whittaker functions have recently reappeared in
the context of random polymers and special McDonald processes.
We revisit their earlier life in representation theory: they
were originally built out of Whittaker vectors, for which we
give a new statistical weighted path formulation, valid for
simple and affine Lie algebras as well as the quantum algebra
U_{q}(sl_{n}). We show how this formulation bypasses the classical
derivation of Todatype differential/difference equations satisfied
by these functions.
We then consider graded tensor products of current algebra modules, and show that their characters obey difference equations,
generalizing the difference Toda equation, allowing for viewing graded
characters as generalized Whittaker functions. This is done
using a constant term expression for the characters via a solution
of the quantum Qsystem, a set of noncommuting integrable recursion
relations attached to the algebra.
Finally, we obtain a new compact expression for graded sl_{n}
characters by constructing a presentation of the quantum Qsystem via
generalized McDonaldRuijsenaars difference operators.
(based on joint works with R. Kedem, and R. Kedem and B. Turmunkh).

October
13 Columbus Day

October 17
Special full day event at Harvard!
The 2014 Charles River Lectures on Probability and Related Topics
The Charles River Lectures on Probability and Related
Topics will be hosted by Harvard. The lectures are jointly
organized by Harvard, MIT and Microsoft Research New
England for the benefit of the greater Boston area
mathematics community. The event features five lectures by
distinguished researchers in the areas of probability and
related topics.
This year's lectures will be
delivered by:
David Brydges (University of British Columbia)
Sourav Chatterjee (Stanford University)
Christophe Garban (CNRS & ENS Lyon, UMPA)
Fabio Toninelli (CNRS & Institut Camille Jordan, Lyon 1)
Srinivasa Varadhan (New York University)

October 20
Maurice Duits (Stockholm University)
CLT's for global and local linear statistics in orthogonal polynomial ensembles
In this talk, I will present some recent joint work with Jonathan
Breuer on the fluctuations of linear statistics in orthogonal
polynomial ensembles. Such ensembles appear naturally in random matrix
theory and integrable probability and the fluctuations of linear
statistics have been studied in various contexts. I will discuss a
new approach for studying such fluctuations based on the recurrence
coefficients or Jacobi operator corresponding to the orthogonal
polynomials. The main results are Central Limit Theorems for linear
statistics on both the global and local scales (or macro and
mesocopic scales) under rather mild conditions on the underlying
measure. If time permits, I will also discuss the extension of the CLT
on the global scale to biorthogonal ensembles.

October 27
Ofer
Zeitouni (Weizmann Institute)
On roots of random polynomials
I will discuss some recent results concerning the
distribution of roots of random polynomials, focusing on
results concerning roots of systems of polynomials. In
particular, I will discuss the probability that a random
Littlewood polynomial has multiple roots. If time
permits, I will also discuss a nonstandard large
deviation result for random polynomials with positive
real coefficients.
(Based on joint works with G. Kozma, with R. Peled and
A. Sen, and with S. Ghosh)

November 3
Madhu Sudan (Microsoft Research New England)
Communication with Imperfectly Shared Randomness
In the 1980's Yao introduced the model of communication complexity where Alice, who knows x in {0,1}^{n}, and Bob, who knows y in {0,1}^{n}, wish to communicate with each other, exchanging fewest possible number of bits, to determine some function f(x,y). For many natural functions f this communication complexity is much smaller than the trivial n bits suggesting that communication with a goal in mind may be much less expensive than otherwise. This message gets amplified even more if Alice and Bob share some random string r chosen independently of x and y: in this setting even more functions f can be determined quickly (with high probability).
In this talk I will introduce a relaxation of this model where Alice and Bob do not share the random string perfectly, but rather Alice knows r and Bob knows some string s that is correlated with r. I will describe some recent results showing that any communication protocol with k bits of communication between Alice and Bob with perfect sharing of randomness, continues to have a moderately lowcomplexity, 2^{k} bit, protocol with shared correlation. Furthermore, I will show that this result is tight in that there exist problems where this exponential jump is necessary. The technical core of these results rely on the understanding of the influence of variables in the analysis of Boolean functions and recently developed probabilistic tools such as the "invariance principle" of Mossel, O'Donnell and Oleszkiewicz. One hope of the talk is to explain what these tools are, and why computer scientists find them useful.
Based on joint work with Clément Canonne (Columbia), Venkatesan Guruswami (CMU) and Raghu Meka.

Thursday November 13, Room E18466A, at 2:30
Yuval Peres
(Microsoft Research)
Rigidity and tolerance for perturbed lattices
Consider a perturbed lattice {v+Y_{v}} obtained
by adding IID ddimensional Gaussian variables {Y_{v}} to the lattice
points in Z^{d}. Suppose that one point, say Y_{0},
is removed from this perturbed lattice; is it possible for an
observer, who sees just the remaining points, to detect that a point
is missing? In one and two dimensions, the answer is positive: the
two point processes (before and after Y_{0} is removed) can be
distinguished using smooth statistics, analogously to work of Sodin
and Tsirelson (2004) on zeros of Gaussian analytic
functions. (cf. Holroyd and Soo (2011) ). Further rigidity results
for these zeros were obtained with Ghosh (2012), based on an idea of
Nazarov and Sodin. The situation in higher dimensions is more
delicate; our solution depends on a gametheoretic idea, in one
direction, and on the unpredictable paths constructed by Benjamini,
Pemantle and the speaker (1998), in the other. I will also describe a
related point process where removal of one point can be detected but
not the removal of (any!) two points.
(Joint work with Allan Sly, UC Berkeley).

November 17
Russell Lyons (Indiana University)
Random walks on groups and the KaimanovichVershik conjecture
Let G be an infinite group with a finite symmetric
generating set S. The corresponding Cayley graph on G has an edge
between x,y in G if y is in xS. KaimanovichVershik (1983), building
on fundamental results of Furstenberg, Derriennic and Avez, showed
that G admits nonconstant bounded harmonic functions iff the entropy
of simple random walk on G grows linearly in time; Varopoulos (1985)
showed that this is equivalent to the random walk escaping with a
positive asymptotic speed. Kaimanovich and Vershik also presented the
lamplighter groups (groups of exponential growth consisting of finite
lattice configurations) where (in dimension at least 3) the simple
random walk has positive speed, yet the probability of returning to
the starting point does not decay exponentially. They conjectured a
complete description of the bounded harmonic functions on these
groups; in dimensions 5 and above, their conjecture was proved by
Erschler (2011). I will discuss the background and present a simple
proof of the KaimanovichVershik conjecture for all dimensions,
obtained in joint work with Yuval Peres.

Thursday November 20, Room E17133 at 3:00
Adrian Banner (INTECH)
Rankbased portfolios, the "size effect", and an identity for the exponential distribution
In an equity market with stable capital distribution, a capitalizationweighted index of small stocks tends to outperform a capitalizationweighted index of large stocks. This is a somewhat careful statement of the socalled “size effect”, which has been documented empirically and for which several explanations have been advanced over the years. We review the analysis of this phenomenon by Fernholz (2001) who showed that, in the presence of (a suitably defined) stability for the capital structure, this phenomenon can be attributed entirely to portfolio rebalancing effects, and will occur regardless of whether or not small stocks are riskier than their larger brethren. Collision local times play a critical role in this analysis, as they capture the turnover at the various ranks on the capitalization ladder.
We shall provide a rather complete study of this phenomenon in the context of a simple model with stable capital distribution, the socalled “Atlas model”. As a corollary we shall obtain a hithertounknown identity for the exponential distribution. (Joint work with R. Fernholz, I. Karatzas, V. Papathanakos and P. Whitman.)

November 24 at 3:40
Natesh
Pillai (Harvard)
Probabilistic Challenges in MCMC algorithms
(Markov Chain Monte Carlo) algorithms are an extremely powerful set of tools for sampling from complex probability distributions. In this two talk we will discuss two research themes for studying the efficiency of commonly used algorithms. We will discuss optimal scaling of MCMC algorithms in high dimensions where the key idea is to study the properties of the proposal distribution as a function of the dimension. This point of view gives us new insights on the behavior of the algorithm, such as precise estimates of the number of steps required to explore the target measure, in stationarity, as a function of the dimension of the state space. In the second part, we will introduce `adaptive Markov chains', and discuss their finite sample properties.

December 1
Leonid Koralov
(UMD)
Transition between averaging and homogenization
regimes for periodic flows and averaging for flows with ergodic
components
In this talk we'll discuss two asymptotic problems that are related by common techniques. First, we'll
consider elliptic PDEs with with a small diffusion term in a large domain. The coefficients are assumed to be
periodic. Depending on the relation between the parameters, either averaging or homogenization need to be
applied in order to describe the behavior of solutions. We'll discuss the transition regime.
The second problem concerns equations with a small diffusion term, where the firstorder term corresponds to
an incompressible flow, possibly with a complicated structure of flow lines. Here we prove an extension of the
classical averaging principle of Freidlin and Wentzell.
Different parts of the talk are based on joint results
with M. Hairer, Z. PajorGuylai, D. Dolgopyat, and M. Freidlin.

December 8
Kavita Ramanan (Brown University)
Obliquely reflected diffusions in rough planar domains
Obliquely reflected diffusions in smooth domains are
classical objects that have been well understood for half a century.
Motivated by applications in a variety of fields ranging from
mathematical physics to stochastic networks, a theory for obliquely
reflected diffusions in piecewise smooth domains has also been
developed over the last two decades. However, in domains with rough
boundaries, even the definition of obliquely reflected diffusions is a
challenge. We discuss an approach to constructing obliquely reflected
Brownian motions (ORBMs) in a large class of bounded, simply connected
planar domains that, as a byproduct, also provides a new
characterization of ORBMs in bounded smooth planar domains. The class
of processes we construct also includes certain processes with jumps
like excursionreflected Brownian motions, which have arisen in the
study of SLE. This talk is based on works with Chris Burdzy, Zhenqing
Chen and Donald Marshall.