The main result presented in this talk is a plethystic formula for the
specialization at $t = 1/q$ of the $Q_{u,v}$ operators studied in [math
arKiv:1405.0316]. This discovery yields elementary and direct
derivations of several identities relating these operators at $t =1/q$
to the Rational Compositional Shuffle Conjecture of [math arKiv:
1404.4616]. In particular we are able to give a direct derivation of a
simple formula for the symmetric polynomial
$$Q_{km,kn}1|_{t=1/q} \ \mbox{(for all $m,n$ co-prime and $k \geq 1
$.)}$$

We also give an elementary proof that this polynomial is Schur positive.
Moreover, by combining our main result with the Rational Compositional
Shuffle Conjecture, we obtain a completely elementary derivation of the
identity expressing this polynomial in terms of Parking functions in the
$km \times km$ rectangle.

Proof of Thurston's geometrization conjecture has been a major achievement in the study of 3-manifolds and proves the existence of natural geometric structures on compact 3-manifolds. However this proof and the mere knowledge of the existence of such structures does not give a description of the geometry and therefore fails to answer many of the existing questions. We discuss a project that attempts to find an effective solution to the geometrization and therefore produces models of the promised hyperbolic structures. We explain how this can be used to answer a number of unanswered questions and relate topological and geometrical properties of the manifold.

The study of the outer automorphism group Out(F) of a free group has been a very active area of geometric group theory in the past few years, driven on many fronts by natural parallels that exist between Out(F) and the mapping class group Mod(S) of a surface. I will provide an overview of some of the recent developments in the theory of Out(F), while emphasizing distinguishing features of Out(F) that make it an often more challenging group to understand than its Mod(S) counterpart.

The study of risk measures began in a static environment with the papers of Artzner et al. (1999) and Follmer and Schied (2002). To incorporate information structure over time, static risk measures were extended to a dynamic setting in Barrieu and El Karoui (2009), Jobert and Rogers (2008), Yong (2007) and many others.

We are interested in a specific class of dynamic risk measures, namely dynamic risk measures that arise as solutions of certain types of backward stochastic differential equations (BSDEs) or backward stochastic Volterra integral equations (BSVIEs). We will discuss this connection between risk measures, capital allocations and BSDEs/BSVIEs and provide representation results for dynamic risk measures and dynamic capital allocations. These results are based on classical differentiability results for BSDEs/BSVIEs and Girsanov-type change of measure arguments.

Joint work with Ludger Overbeck.

This talk will be about random lozenge tilings of a class of planar domains which I like to call "sawtooth domains." The basic question is: what does a uniformly random tiling of a large sawtooth domain look like? At the first order of randomness, a remarkable form of the law of large numbers emerges: the height function of the tiling converges to a deterministic "limit shape." My talk is about the next order of randomness, where one wants to analyze the fluctuations of tiles around their eventual positions in the limit shape. Quite remarkably, this ostensibly analytic problem can be solved in an essentially combinatorial way, using a desymmetrized version of the double Hurwitz numbers from enumerative algebraic geometry.

I will provide a broad overview of the kinds of problems that I've worked on with particular emphasis on three subjects: understanding the distribution of statistics of random set partitions; the distribution of ranks of selmer groups of elliptic curves; and the study of polynomials in large numbers of variables with random inputs.

Understanding the coupling of physical models at different scales is important and challenging. In this talk, we focus on the issue of kinetic-fluid coupling, in particular, the half-space problems for kinetic equations coming from the boundary layer. We will present some recent progress in algorithm development and analysis for the linear half-space kinetic equations, and its application in coupling of neutron transport equations with diffusion equations. (joint work with Jianfeng Lu and
Weiran Sun).