We consider stochastic three-dimensional rotating Navier-Stokes equations and prove averaging theorems for stochastic problems in the case of strong rotation. Regularity results are established by bootstrapping from global regularity of the limit stochastic equations and convergence theorems. The energy injected in the system by the noise is large, the initial condition has large energy, and the regularization time horizon is long. Regularization is the consequence of a precise mechanism of relevant three-dimensional nonlinear interactions. We establish multiscale averaging and convergence theorems for the stochastic dynamics. References [1] Flandoli F. , Mahalov A. , “Stochastic 3D Rotating Navier-Stokes Equations: Averaging, Convergence and Regularity,” Archive for Rational Mechanics and Analysis, 205, No. 1, 195–237 (2012). [2] Cheng B. , Mahalov A. , “Euler Equations on a Fast Rotating Sphere – Time- Averages and Zonal Flows,” European Journal of Mechanics B/Fluids, 37, 48-58 (2013). [3] Mahalov A. Multiscale modeling and nested simulations of three-dimensional ionospheric plasmas: Rayleigh-Taylor turbulence and nonequilibrium layer dynamics at fine scales, Physica Scripta, Phys. Scr. 89 (2014) 098001 (22pp), Royal Swedish Academy of Sciences.

I will discuss certain remarkable q-difference equations with regular singularities that appear in enumerative K-theory and representation theory. This class includes, in particular, the quantum Knizhnik-Zamolodchikov equations of Frenkel and Reshetikhin. Remarkably, the geometric origin of these equations helps with the computations of the monodromy, as shown in our join work with Mina Aganagic.

For the Dirichlet problem on the k Hessian equation, Caffarelli-Nirenberg-Spruck (1986) obtained the existence of the admissible classical solution when the smooth domain is strictly k-1 convex in R^n. In this talk, we prove the existence of a classical admissible solution to a class of Neumann boundary value problems for k Hessian equations in strictly convex domain in R^n, this was asked by Prof.  N. Trudinger in 1987. The method depends upon the establishment of a priori derivative estimates up to second order. This is the joint work with Qiu Guohuan.

This lecture addresses two central problems in classical ergodic theory: the classification problem and the realization problem. An historical focus of ergodic theory has been the structure and properties of single transformations. Perhaps the most prominent is the Furstenberg-Zimmer structure theorem which describes ergodic transformations in terms of limits of compact and weakly mixing extensions.

This lecture discusses a new phenomenon, Global Structure Theorems. We define two categories: the Odometer Based Systems (finite entropy transformations that have an odometer factor) and Circular Systems (those diffeomorphisms built using a version of the Anosov-Katok technique.) The morphisms  in each category are measure-theoretic joinings.

The main result is that these two categories are isomorphic by a composition-preserving bijection that that takes conjugacies to conjugacies, extensions to extensions, weakly mixing extensions to weakly mixing extensions, compact extensions to compact extensions, distal towers to distal towers (and more). In short, all of the structure present in the odometer based systems is exactly reflected in the Circular Systems and vice versa.

The lecture will conclude with a provocative conjecture.

This is joint work with B. Weiss.

My talk will be related to the stable intersections of dynamically defined Cantor sets (as well as to the finding intervals in their sums). In the famous work of Morreira and Yoccoz it is shown that for a generic pair of dynamically defined Cantor sets with sum of their Hausdorff dimensions greater than one, their intersection (provided that they do intersect) is stable under small perturbations.

However, it would be nice to be able to check this for a particular couple of sets, and up to this moment the only explicitly checkable sufficient condition that is known is that the product of the thicknesses is greater than one.

I will present an approach to finding such a sufficient condition (eventually, in a computer-assisted way) that can be hopefully used to attack the conjecture that claims that F(2)+F(4) contains an interval (that is currently open).

My talk will be based on a joint project will A. Gorodetski, A. Gordenko and E. Nesterova.