
February 10
Arthur Jaffe (Harvard)
On reflection positivity
We review how reflection positivity plays a fundamental role in the relation between probability and quantum theory. Also we describe the two new ways that it occurs: in the study of complex functionals and the study of Majoranas.

February 17
Presidents' day

February 24
Alan Edelman (MIT)
Hermite, Laguerre and Jacobi
Listen to Random Matrix Theory:It's Trying to Tell Us Something
In orthogonal polynomial theory one quickly learns about the special place occupied by Hermite, Laguerre, and Jacobi polynomials. I have often wondered why they are so special. I have heard dozens of correct answers. I am still not satisfied that I really know. Somehow in probability, Hermite, Laguerre, and Jacobi takes the form of Gaussian, Chi, and Beta Random variables. In computation, the triad takes the form of the symmetric eigenvalue problem, the singular value decomposition, and the lesser known, but very useful, generalized singular value decomposition. In Random Matrix Theory we have the Gaussian ensembles, the Wishart Matrices, and the Manova Matrices. Their limits are the Semicircle Law, the MarcenkoPastur Law, and the Wachter Law. In combinatorics, we have the Catalan Numbers, the Narayana Trinagle, and a pyramid. This talk, intended to be accessible to a wide audience, will illustrate these ideas from a random matrix theory viewpoint. Discuss multivariate orthogonal polynomials, and the random matrix method of "Ghosts and Shadows."

March 3
Charles Newman (NYU)
Statistical Mechanics and the Riemann Hypothesis
In this talk we review a number of old results concerning
certain statistical mechanics models and their possible connections to the
Riemann Hypothesis.
A standard reformulation of the Riemann Hypothesis
(RH) is: The (twosided) Laplace transform of a certain specific function
Psi on the real line is automatically an entire function on the complex
plane; the RH is equivalent to this transform having only pure imaginary
zeros. Also Psi is a positive integrable function, so
(modulo a multiplicative constant C) is a probability density function.
A (finite) Ising model is a specific type of probability measure P on
the points S=(S_1,...,S_N) with each S_j = +1 or 1. The LeeYang
theorem implies that that for nonnegative a_1, ..., a_N, the Laplace
transform of the induced probability distribution of a_1 S_1 + ... +
a_N S_N has only pure imaginary zeros.
The big question here is whether it's possible to find a sequence of Ising
models so that the limit as N tends to infinity of such distributions has
density exactly C Psi. We'll discuss some hints as to how one might
try to do this.

March 10
Brian Rider (Temple University)
Universality for the Stochastic Airy Operator
The Stochastic Airy Operator first arose as the continuum limit of certain (generalizations of) ensembles of symmetric Gaussian random matrices in the vicinity of their spectral edge. We show that this picture persists for the general logarithmic gas on the line with uniformly convex polynomial potential. Based on joint work with Manjunath Krishnapur and Balint Virag.

March 17
Yuri Bakhtin (NYU)
Burgers equation with random forcing in noncompact setting
The Burgers equation is one of the basic nonlinear
evolutionary PDEs. The study of ergodic properties of the Burgers
equation with random forcing began in 1990's. The natural approach is
based on the analysis of optimal paths in the random landscape
generated by the random force potential. For a long time only compact
cases of the Burgers dynamics on a circle or bounded interval were
understood well. In this talk I will discuss the Burgers dynamics on
the entire real line with no compactness or periodicity assumption on
the random forcing. The main result is the description of the ergodic
components and existence of a global attracting random solution in
each component. The proof is based on ideas from the theory of first
or last passage percolation. This is a joint work with Eric Cator and
Kostya Khanin.

March 24
Spring Vacation

March 31
Oleksandr Kutovyi (MIT)
Markov evolutions for interacting particle systems in continuum
We analyze an interacting particle system with a Markov
evolution in continuum. The corresponding Vlasovtype
scaling, which is based on a proper scaling of corresponding
Markov generators and has an algorithmic realization in
terms of related hierarchical chains of correlation
functions equations is studied. The existence of rescaled
and limiting evolutions of correlation functions as well as
convergence to the limiting evolution are shown.

April 7
Christophe Garban (ENS Lyon)
Liouville Brownian motion
Let X be a Gaussian Free Field (GFF) in two dimensions. I
will introduce a Feller process (P_{t}^{X})
on the plane which, a.s. in the realization of the GFF X,
preserves the socalled Liouville measure defined formally
by "M(dx)=e^{γX}dx" (with γ <
γ_{c}=2). The Liouville measure was
popularized a few years ago by Duplantier and Sheffield in
the context of Liouville Quantum gravity. I will discuss the
construction and the properties of this Feller process
called the Liouville Brownian motion as well as some recent
progresses on the supercritical Liouville Brownian motion,
i.e. when γ > γ_{c}=2. This is based
on joint works with N. Berestycki, R. Rhodes, and V. Vargas.

April 14
Omer Tamuz (MSR / MIT)
Majority Dynamics and the Period Two Property
A group of people connected by a social network each start
with some opinion in {0,1}. They then proceed to repeatedly
update their opinions by conforming to those of the majority
of their neighbors. This model, which has been studied for a
few decades now in various fields (cf. synchronous zero
temperature Glauber dynamics), has the curious property that
each person eventually either converges to a fixed opinion
or else, from some point on, oscillates between the two
possible opinions with period two. We will study this model
on infinite graphs and random graphs, showing some old
results, some new ones, and some nice open questions.

April 21
Patriots' day

Thursday April 24
Leonid Pastur (Kharkov) room E17133
On the Law of Large Numbers and the Central Limit Theorem for Linear Eigenvalue Statistics of Sample Covariance Matrices with Dependent Entries
We consider sample covariance matrices whose data matrices
have independent columns but dependent column components,
notably columns with the logconcave distribution. We find
conditions on the column distribution for a limiting
Normalized Counting Measure of eigenvalues to exist (this
proves to be the standard MP law) and for the validity of
the Central Limit Theorem for a wide class of linear
eigenvalue statistics and we comment on related topics of
geometric asymptotic analysis.

April 28th
Philippe Biane (CNRS)
Concavification of free entropy
I will give an overview of free entropy, which concerns how
random matrices approximate traces on von Neumann algebras, and its
use in operator algebra theory, as well as some new developments
leading to the solution of the "additivity problem".

May 5
Victor Kleptsyn (UMR)
Towards rigorous construction for random metrics : the cutoff process. (On a joint work with M. Khristoforov and M. Triestino)
One of the open problems in the domain of quantum gravity is the one of constructing a random metric on a manifold as a limit of a multiplicative cascade; if constructed for the case of a disc or of a sphere, it can be thought as the realization of "exp(DGFF) dz".
Though this problem is wellknown, there are very few rigorous known results. One of them is the work of Benjamini and Schramm for the multiplicative cascades on the interval, where the sequence of distances forms a martingale. The (martingalerelated) convergence of measures (going back to the works of Kahane) is a key element in a work of Duplantier and Sheffield on the KPZ formula. Finally, the results of Le Gall and Miermont show that one can consider a random metric on the sphere as a limit of random planar maps.
The main result of our work is the rigorous construction of a random metric via multiplicative cascades on hierarchical graphs (like Benjamini's eight graph, dihedral hierarchical lattice, etc.), this situation being both still accessible due to the graph structure, but already complicated due to the high nonuniqueness of candidates for geodesic lines.
A key argument, that allows to find a stationary law for the glueing process, is the cutoff process: instead of looking for a critical renormalization value, we "stabilize" the process in the supercritical regime by adding a "shortcut", and then pass to the "diagonal" limit (the renormalization parameter tends to the critical value, and at the same time the influence of the shortcut tends to zero).

May 12
Lionel Levine (Cornell)
Sandpiles and systemspanning avalanches
A sandpile on a graph is an integervalued function on the
vertices. It evolves according to local moves called topplings. Some
sandpiles stabilize after a finite number of topplings, while others
topple forever. For any sandpile s_{0} if we repeatedly add a
grain of sand at an independent random vertex, we eventually reach a
sandpile s_{τ} that topples forever. Statistical
physicists Poghosyan, Poghosyan, Priezzhev and Ruelle conjectured a
precise value for the expected amount of sand in this "threshold
state" s_{τ}; in the limit as s_{0} goes to
negative infinity. I will outline the proof of this conjecture in
http://arxiv.org/abs/1402.3283
and explain the bigpicture motivation, which is to give more
predictive power to the theory of "selforganized criticality".