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**. Click on the following links for seminar schedules from past semesters: Fall 2016, Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014.

Tuesday, February 14, 2017, 11 a.m., Room 110:

Andrea Agazzi, Stanford University.

*Large deviations theory for chemical reactions networks*.

At the microscopic level, the dynamics of networks of chemical reactions can be modeled as jump Markov processes. The rates of these processes are in general neither uniformly Lipschitz continuous nor bounded away from zero, obstructing the straightforward application of large deviation theory to this framework. We bypass these issues by respectively applying tools of Lyapunov stability theory and recent results on interacting particle systems. This way, we characterize a class of processes obeying a LDP in path space, and extend the latter to infinite time intervals through Wentzell-Freidlin (W-F) theory. Finally, we provide natural sufficient topological conditions on the network of reactions for the applicability of our LDP and W-F results. These conditions can be easily checked algorithmically.

This is joint work with Amir Dembo and Jean-Pierre Eckmann.

Tuesday, February 14, 2017, **4 p.m., 170 Hope Street, Room 108**:

William Massey, Princeton University.

*Dynamic queueing transience*.

Inspired by communication systems and healthcare services, this talk summarizes the methods developed with many collaborators over many decades to understand the transient behavior of dynamic rate queues. This analysis is needed when confronted with the time-varying parameters found in time-inhomogeneous Markovian queueing models. The static, steady-state, equilibrium analysis for constant rate queues no longer applies.

Constant parameters that summarize the transient behavior for steady state systems are now replaced by deterministic dynamical systems. We can then approximate the optimal behavior of these queues by controlling the corresponding family of ordinary differential equations.

Tuesday, February 28, 2017, 11 a.m., Room 110:

Mark Rudelson, University of Michigan.

*Invertibility and condition number of sparse random matrices*.

Consider an n by n linear system Ax=b. If the right-hand side of the system is known up to a certain error, then in process of the solution, this error gets amplified by the condition number of the matrix A, i.e. by the ratio of its largest and smallest singular values. This observation led von Neumann and his collaborators to consider the condition number of a random matrix and conjecture that it should be of order n. This conjecture of von Neumann was proved in full generality a few years ago. In this talk, we will discus whether von Neumann's conjecture can be extended to sparse random matrices. We will also discus invertibility of the adjacency matrix of a directed Erdos-Renyi graph.

Joint work with Anirban Basak.

**Thursday**, March 16, 2017, 11 a.m., Room 110:

Mustazee Rahman, MIT.

*Local statistics of random sorting networks*.

A random sorting network is a uniformly random path of shortest length from the identity to the reverse permutation in the Cayley graph of the symmetric group generated by adjacent transpositions. They are in bijection with staircase shaped Young tableaux via the Edelman-Greene algorithm, a cousin of the RSK algorithm.

I will explain how the local statistics of large random sorting networks can be derived from the eigenvalues of anti-symmetric Gaussian matrices. This involves the local statistics of large staircase shaped tableaux and a local version of the Edelman-Greene algorithm.

Joint work with Vadim Gorin.

Tuesday, April 4, 2017, 11 a.m., Room 110:

Ruoyu Wu, Brown University.

*Some asymptotic results for weakly interacting particle systems*.

Weakly interacting particle systems have been widely used as models in many areas, including communication systems, mathematical finance, chemical and biological systems, and social sciences. A typical such system is given as a collection of Markov processes, representing trajectories of N interacting particles, given as the solution of stochastic differential equations (SDE) driven by mutually independent Poisson random measures or Brownian motions. The interaction between particles occurs through the coefficients of the SDE in that these coefficients depend, in addition to the particle's current state, on the empirical distribution of all particles in the collection. In this talk, I will present law of large numbers, central limit theorems, large deviation principles and moderate deviation principles for several types of such systems.
Joint work with Shankar Bhamidi, Amarjit Budhiraja and Wai-Tong (Louis) Fan.

Tuesday, April 11, 2017, 11 a.m., Room 110:

Mykola Portenko, Institute of Mathematics of National Academy of Sciences of Ukraine.

*On some Markov processes related to a symmetric stable process*.

**Thursday**, April 20, 2017, **12 p.m., 170 Hope Street, Room 108**:

Thaleia Zariphopoulou, University of Texas at Austin.

**Thursday**, April 27, 2017, 11 a.m., Room 110:

Louigi Addario-Berry, McGill University.

Tuesday, May 2, 2017, 11 a.m., Room 110:

David Kelly, NYU.

Tuesday, May 9, 2017, 11 a.m., Room 110:

Mickey Salins, Boston University.