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

Tuesday, February 2, 2016, 11 a.m., Room 110:

Nayantara Bhatnagar, University of Delaware.

*Limit theorems for monotone subsequences in Mallows permutations*.

The longest increasing subsequence (LIS) of a uniformly random permutation is a well studied problem. Vershik-Kerov and Logan-Shepp first showed that asymptotically the typical length of the LIS is 2sqrt(n). This line of research culminated in the work of Baik-Deift-Johansson who related this length to the GUE Tracy-Widom distribution.

We study the length of the LIS and LDS of random permutations drawn from the Mallows measure, introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation p in S_n is proportional to q^Inv(p) where q is a real parameter and Inv(p) is the number of inversions in p. We determine the typical order of magnitude of the LIS and LDS, large deviation bounds for these lengths and a law of large numbers for the LIS for various regimes of the parameter q. In the regime that q is constant, we make use of the regenerative structure of the permutation to prove a Gaussian CLT for the LIS.

This is based on joint work with Ron Peled and with Riddhi Basu.

Tuesday, February 16, 2016, 11 a.m., Room 110:

Wendell Fleming, Brown University.

*Differential games*.

The first part of this lecture gives a concise historical overview of two-player, zero-sum differential games, beginning with the pioneering work of Isaacs. The Crandall-Lions theory of viscosity solutions for first order nonlinear PDEs has a crucial role in the treatment of upper and lower value functions. for differential games. Differential games arise naturally in risk sensitive stochastic control theory. This theory brings together deterministic and stochastic approaches to disturbances in control systems.

A second part of the lecture concerns recent work with D. Hernandez-Hernandez on mixed strategy differential games. Such games arise naturally when upper and lower game values are different.

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

Marcel Nutz, Columbia University.

*Martingale optimal transport and beyond*.

We study the Monge-Kantorovich transport between two probability measures, where the transport plans are subject to a probabilistic constraint. For instance, in the martingale optimal transport problem, the transports are laws of martingales. Interesting new couplings emerge as optimizers in such problems.

Constrained transport arises in the context of robust hedging in mathematical finance via linear programming duality. We formulate a complete duality theory for general performance functions, including the existence of optimal hedges. This duality leads to an analytic monotonicity principle which describes the geometry of optimal transports. Joint work with Mathias Beiglböck, Florian Stebegg and Nizar Touzi.

Tuesday, March 15, 2016, 11 a.m., Room 110:

Yao Li, University of Massachusetts Amherst.

*Nonequilibrium steady-states for some interacting particle systems*.

In this talk I will present our recent results on non-equilibrium steady states (NESS) for a class of stochastic microscopic heat conduction models, in which energy exchange among particles is mediated by a lattice of "energy tanks". Those stochastic models are derived from mechanical chain models (Eckmann & Young 2006) by randomizing certain chaotic quantities. We proved various rigorous results including the existence and uniqueness of NESS, the exponential convergence towards NESS, the existence of local thermodynamic equilibrium (LTE), and the slow (polynomial) mixing phenomenon under some relaxed conditions.

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

Vadim Gorin, MIT.

*Largest eigenvalues in random matrix beta-ensembles: structures of the limit*.

Despite numerous articles devoted to its study, the universal scaling limit for the largest eigenvalues in general beta log--gases remains a mysterious object. I will present two new approaches to such edge scaling limits. The outcomes include a novel scaling limit for the differences between largest eigenvalues in submatrices of a random matrix and a Feynman-Kac type formula for the semigroup spanned by the Stochastic Airy Operator. (based on joint work with M.Shkolnikov)

**Friday, April 8, 2016, Columbia University, New York, NY**:

Columbia-Princeton Probability Day.

Tuesday, April 19, 2016, 11 a.m., Room 110:

Philippe Robert, INRIA.

*Stochastic models of gene expression*.

Protein production is a key process of prokaryotic and eukaryotic cells consuming more that 80% of their resources. The cytoplasm of the cell being a disorganized medium subject to thermal noise, the protein production process has an important stochastic component. In this context the random fluctuations of the number of copies of a given protein in a cell are of primary importance. We discuss the impact of modeling issues of this production process through the use of Markovian and non-Markovian models. Mathematical models of the production of {\em one} fixed type of proteins is presented. When several classes of proteins are considered, an important additional aspect has to be taken into account, the limited common resources of the cell (polymerases and ribosomes) used by the production process. We focus on the allocation of ribosomes in the case of the production of multiple proteins. Asymptotic results of the equilibrium are obtained under a scaling procedure and a realistic biological assumption of saturation of the ribosomes available in the cell. It is shown in particular that, in the limit, the number of non-allocated ribosomes at equilibrium converges in distribution to a Poisson distribution whose parameter satisfies a fixed point equation. It is in particular shown that the production process of different types of proteins can be seen as independent production processes but with modified parameters. We will conclude by an analysis of the impact of some feedback mechanisms to control the fluctuations of the number of proteins. Joint Work with Renaud Dessalles, Vincent Fromion and Emanuele Leoncini.

Tuesday, April 26, 2016, 11 a.m., Room 110:

Fraydoun Rezakhanlou, University of California, Berkeley.

*Periodic orbits for stationary Hamiltonian systems*.

According to Poincare-Birkhoff Theorem, a periodic twist map of a cylinder has at least two fixed points. V.I. Arnold realized that the correct generalization to higher dimensions concerned the Hamiltonian flows and symplectic maps. Arnold's conjecture in the case of a torus gives a lower bound on the number of periodic orbits of a Hamiltonian system associated with a periodic Hamiltonian function. This conjecture was established by Conley and Zehnder in 1984. A parallel generalization of the classical Poincare-Birkhoff Theorem is to investigate whether it holds in the stochastic setting. In this talk, I discuss a variant of the Poincare-Birkhoff and Conley-Zehnder Theorems for Hamiltonian systems associated with stationary
and ergodic Hamiltonian functions. (Joint work with Alvaro Pelayo.)

**Thursday, April 28, 2015**, 11 a.m., Room 110:

Ruth Williams, University of California, San Diego.

*Criticality and adaptivity in enzymatic networks*.

The contrast between stochasticity of biochemical networks and regularity of cellular behavior suggests that biological networks generate robust behavior from noisy constituents. Identifying the mechanisms that confer this ability on biological networks is essential to understanding cells. Here we use stochastic queueing models to investigate one potential mechanism.

In living cells, enzymes perform the critical function of acting as catalysts to ensure that important reactions occur at rates fast enough to sustain life. We show how competition among different molecular species for the attention of a limited pool of shared enzymes in enzymatic networks can produce strong correlations between the different species when these systems are poised near a critical state where the substrate input flux is equal to the maximum processing capacity. We then consider the enzymatic networks with adaptation, where the limiting resource is produced in proportion to the demand for it. In this setting, we show that strong correlations are robustly produced across a broad range of system parameters. This adaptive queueing motif suggests a natural control mechanism for producing strong correlations in biological systems. Based on joint work with Paul Steiner, Jeff Hasty and Lev Tsimring.