Organizers: David Lipshutz and Daniel Lacker. Click on a title to display the associated abstract. Non-standard seminar times and locations are emphasized in **bold**.

**Thursday**, September 15, 2016, **4 p.m., Room 118 of 170 Hope Street**:

Nate Eldredge, University of Northern Colorado.

*Sub-Riemannian geometry, Heisenberg groups, and strong hypercontractivity*.

Sub-Riemannian geometry is like doing Riemannian geometry with one hand tied behind your back: you're in a manifold, but there are certain directions you can't go. This talk will start with an introduction to life in a sub-Riemannian world (everyday examples include parallel parking and Asteroids) and why non-commutativity makes it interesting. To bring in some probability, we'll see how to move randomly in such a space. We'll look at Heisenberg groups as a fundamental example. I'll then outline some recent work (joint with Leonard Gross and Laurent Saloff-Coste) on the relationship between logarithmic Sobolev inequalities and so-called strong hypercontractivity in complexified versions of these groups.

Tuesday, September 20, 2016, 11 a.m., Room 110:

Sanjay Ramassamy, Brown University.

*Barak-Erdös graphs and the infinite-bin model*.

Barak-Erdös graphs are the directed acyclic version of Erdös-Rényi random graphs : the vertex set is $\{1,\dots,n\}$ and for each $i<j$ with probability $p$ we add an edge directed from $i$ to $j$, independently for each pair $i<j$. The length of the longest path of Barak-Erdös graphs grows linearly with the
number of vertices, where the growth rate $C(p)$ is a function of the edge probability $p$.

Foss and Konstantopoulos introduced a coupling between Barak-Erdös graphs and a special case of an interacting particle system called the infinite-bin model. Using this coupling, we derive some properties of $C(p)$: analyticity for large $p$, differentiability and absence of second derivative at $p=0$.

If time permits, we will discuss some properties of general infinite-bin models such as freezing and the existence of phase transitions.

This is joint work with Ksenia Chernysh and Bastien Mallein.

**Friday, September 30, 2016, 9 a.m. – 5 p.m., MIT, Cambridge, MA**:

Charles River Lectures on Probability Theory and Related Topics.

Tuesday, October 4, 2016, 11 a.m., Room 110:

Fabrice Baudoin, University of Connecticut.

*Stochastic Lévy areas on the complex projective spaces*.

By its simplicity, its number of far reaching applications, and its connections to many areas of mathematics, the Lévy's area formula is undoubtedly among the most important and beautiful formulas in stochastic calculus. In this talk we describe analogues and consequences of the formula on the complex symmetric spaces $CP^n$ and $CH^n$. This is joint work with Jing Wang (UIUC).

Tuesday, October 18, 2016, 11 a.m., Room 110:

Rami Atar, Technion.

*A Skorohod map on measure-valued paths and priority queues*.

A Skorohod-type transformation that acts on paths with values in the space of measures over the real line is argued to provide a generic model for priority. We will describe how it is useful for scaling limits of earliest-deadline-first, shortest-job-first and related disciplines. This is joint work with Anup Biswas, Haya Kaspi and Kavita Ramanan.

**Thursday**, October 27, 2016, 11 a.m., Room 110:

Li-Cheng Tsai, Columbia University.

*Interacting particle systems with moving boundaries*.

In this talk we will survey a few one-dimensional particle systems with a moving boundary. This includes the infinite Atlas model, Aldous' up-the-river problem, and a modified one-dimensional Diffusion Limited Growth. In these systems, particles perform independent Brownian motions or random walks, and interact only through the boundary via rank-based drift and/or absorption. We will explain how connecting these systems to PDE and Stochastic PDE helps to solve problems regarding large time asymptotics. For systems with absorption, we will demonstrate a new method of utilizing the flux condition to bypass the loss of control on local equilibrium.

This talk is based on joint work with Amir Dembo and Wenpin Tang.

Tuesday, November 1, 2016, 11 a.m., Room 110:

François Delarue, Université Nice-Sophia Antipolis.

*Master equations for mean field games. Classical solutions and convergence problem*.

Mean field games is a theory for describing asymptotic Nash equilibria in games involving a large number of players interacting with one another in a mean field way.

First, I will explain how equilibria may be described by an infinite dimensional PDE set on the space of probability measures, which is called the master equation. This description includes the case when players are submitted to a systemic noise. Second, I will explain how to solve this equation in the classical sense whenever the so-called Lasry Lions monotonicity condition is in force. Third, I will show how classical solutions may be used to establish the convergence of games with finitely many players to mean field games.

Tuesday, November 8, 2016, 11 a.m., Room 110:

Patrick Rebeschini, Yale University.

*Locality and message passing in network optimization *.

The complexity of network optimization depends on the network topology, the nature of the objective function, and what information (local or global) is available to the decision makers. In this talk we introduce a notion of network locality and explore some of its properties and applications to algorithms. In particular, we investigate message-passing, a simple, iterative, distributed, general-purpose paradigm for problems with a graph structure. We establish a general framework to analyze the convergence of message-passing in convex problems with constraints. As an application, we analyze the behavior of message-passing to solve systems of linear equations in the Laplacian matrices of graphs, and to compute electric flows, which are two fundamental primitives that arise in several domains such as computer science, electrical engineering, operations research, and machine learning. We will show that the behavior of message-passing in these instances is connected to certain diffusion properties of random walks on graphs. This is joint work with Sekhar Tatikonda.

**Thursday, November 17 – Friday, November 18, 2016, CUNY, New York, NY**:

Northeast Probability Seminar.

Tuesday, November 22, 2016, 11 a.m., Room 110:

Luc Rey-Bellet, University of Massachusetts, Amherst.

*Scalable uncertainty quantification inequalities for stochastic systems*.

We discuss recent development in information inequalities which allow to estimate systematic bias due to uncertainties in coefficients, model miss-specfication, or error due to approximations/numerical schemes. Of particular interest are spatially extended systems and/or systems in long time regime (steady state).

We illustrate these inequalities to phase diagram of statistical mechanics as well as to numerical splitting schemes for kinetic Monte-Carlo algorithms.

The talk is based on joint works with M. Katsoulakis, K. Gourgoulias and J. Wang and relies on previous work by Paul Dupuis and his collaborators.

Tuesday, December 6, 2016, 11 a.m., Room 110:

Solesne Bourguin, Boston University.

*Recent developments on Wigner integrals*.

This talk aims at presenting the main ideas behind non-commutative probability theory without assuming prior knowledge. In particular, we will go over what a non-commutative probability space is and what independence means in such a framework. We will then move on to the free probability setting (an instance of non-commutative probability theory) and discuss free Brownian motion and Wigner integrals, as well as why these objects are of importance is the field. We will conclude by presenting recent results and some open problems dealing with Wigner integrals.