–August 25th:

**Boris Pittel **(Ohio State University)

**Title**: Perfect partitions of a set of random integers.

**Abstract**:

**Andrew Newman**(Carnegie Mellon University)

**Title**: Complexes of nearly maximum diameter

**Abstract**: The combinatorial diameter of a simplicial complex is the diameter of its dual graph. Using a probabilistic approach we determine the right first-order asymptotics for the maximum possible diameter among all d-complexes on n vertices as well as among all d-pseudomanifolds on n vertices. This is joint work with Tom Bohman.

**Parisa Fatheddin**(Ohio State University)

**Title:**Asymptotic behavior of stochastic Navier-Stokes and Schrodinger equations

**Abstract: **We consider the asymptotic limits of two dimensional incompressible stochastic Navier Stokes equation and one dimensional stochastic Schrodinger equation. These limits include large and moderate deviations, Central limit theorem, and the law of the iterated logarithm. For large and moderate deviations, we will discuss both the Azencott method and the weak convergence approach and show how they can be used to derive the Strassen’s compact law of the iterated logarithm. The exit problem will also be given as an application. This is joint work with Zhaoyang Qiu.

At the end of the talk I will also briefly talk about my recently published book: *Teaching and Research in Mathematics: A Guide with Applications to Industry. *

**Wesley Pegden**(CMU)

**Title**: Markov chains and sampling methods for contiguous partitions

**Abstract**: Markov chains have become an essential tool for sampling contiguous partitions of geometric regions, with applications in detection of gerrymandering of political districtings. Nevertheless, there remains a dearth of rigorous results on the mixing times of the chains employed for this purpose. In this talk we’ll discuss a sub-exponential bound on the mixing time of the Glauber dynamics chain for the case of bounded-size contiguous partition classes on certain grid-like classes of graphs.

**Anh Le**(Ohio State University)

**Title**: Bohr sets in sumsets

**Abstract**:

Bohr sets in Z are sets of return times to a neighborhood of 0 in a toral rotation. An important problem in additive combinatorics is to find out when a “large” subset of Z contains a Bohr set. Bergelson and Ruzsa proved that if r + s + t = 0 and a subset A of Z has positive upper density, then rA + sA + tA contains a Bohr set. In this talk, we will present a generalization of this theorem to countable abelian groups and show an analogous result when the set A arises from a partition (instead of having positive upper density).

**Huy T. Pham**(Stanford University)

**Title**: The Kahn—Kalai conjecture and Talagrand’s selector process conjecture

**Abstract**: The threshold of an increasing graph property is the density at which a random graph transitions from unlikely satisfying to likely satisfying the property. Kahn and Kalai conjectured that this threshold is always within a logarithmic factor of the expectation threshold, a natural lower bound to the threshold which is often much easier to compute. In probabilistic combinatorics and random graph theory, the Kahn—Kalai conjecture directly implies a number of difficult results, such as Shamir’s problem on hypergraph matchings, or the threshold for containing a bounded degree spanning tree. I will discuss recent joint work with Jinyoung Park that resolves the Kahn—Kalai conjecture.

Our proof of the Kahn—Kalai conjecture is closely related to the resolution (in joint work with Jinyoung Park) of a conjecture of Talagrand on extreme events of suprema of certain stochastic processes driven by sparse Bernoulli random variables (known as selector processes), and a question of Talagrand on suprema of general positive empirical processes. Given recent advances on chaining and the resolution of the (generalized) Bernoulli conjecture, these results give the first steps towards Talagrand’s last “Unfulfilled dreams’’ in the study of suprema of general empirical processes.

**Sivaguru Sritharan**(Air Force Research Laboratory)

**Title**: The Boltzmann Equation

**Abstract**: This talk will be an introduction to discrete and continuous Boltzmann equations and their mathematical properties. We will also discuss conservation laws that result from various moments and their mathematical characteristics. At least two Fields medalists (P. L. Lions and Cedric Villani) contributing to this subject, making it one of the intense research areas in mathematics. Recent surge of interest in hypersonic aerodynamics has also created major resurgence in research focus among engineers.