Title: Stochastic Three-Dimensional Rotating Navier-Stokes Equations: Averaging, Convergence and Regularity

Date: 10/24/2016

Time: 4:02 PM - 5:00 PM

Place: C517 Wells Hall

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.

Title: How to define the Torelli group of a surface with boundary?

Date: 10/24/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

The Torelli group of a closed surface S is defined as the group of diffeomorphisms of S acting trivially on homology and considered up to isotopy. These groups naturally arise
in topology of 3-manifolds and in algebraic geometry. At the same time they are quite interesting groups by themselves. In order to study them, it is highly desirable to have also Torelli groups of surfaces with non-empty boundary. It turns out that the naive definition is not a good one.

This talk will review the perspective of strings on quivers and mutations.
Starting from a brief mention of the closed string which was well studied in the 80s and early 90s we will move to a discussion on the open string and the natural way it gives rise to the notion of quivers - the so called quiver gauge theories. This perspective allows the study of various dynamical questions in brane physics, but also brings contact with fields in mathematics in representation theory and algebraic geometry. We will discuss quivers that arise on branes at singularities, brane tilings, and the phenomenon of Seiberg duality - the physics counterpart of a quiver mutation.

Speaker: Maitreyee Kulkarni, Louisiana State University

Title: Categorification of cluster structure on partial flag varieties

Date: 10/25/2016

Time: 1:00 PM - 1:50 PM

Place: C304 Wells Hall

Let G be a Lie group of type ADE and P be a parabolic subgroup. It is known that there exists a cluster structure on the coordinate ring of the partial flag variety G/P (see the work of Geiss, Leclerc, and Schroer). Since then there has been a great deal of activity towards categorifying these cluster algebras. Jensen, King, and Su gave a direct categorification of the cluster structure on the homogeneous coordinate ring for Grassmannians (that is, when G is of type A and P is a maximal parabolic subgroup). In this setting, Baur, King, and Marsh gave an interpretation of this categorification in terms of dimer models. In this talk, I will give an analog of dimer models for groups in other types by introducing a technique called “constructing sheets over Dynkin diagrams”, which can (conjecturally) be used to generalize the result of Baur, King, and Marsh.

We will give an introduction to Poisson algebras and Poisson manifolds, and discuss the relationship with symplectic geometry and the Hamiltonian formalism in classical mechanics. We will also state some well-known results in the subject, and, time-permitting, discuss some related algebraic questions pertaining to Poisson varieties, which are defined analogously in the algebraic category.

Title: Curvature free rigidity for higher rank three manifolds.

Date: 10/27/2016

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

Fixing K=-1,0, or 1, a complete Riemannian manifold is said to have higher rank if each geodesic admits a parallel vector field making curvature K with the geodesic.
Locally symmetric spaces provide examples. Rank rigidity theorems aim to show that these are the only examples of manifolds of higher rank, usually with additional curvature assumptions.
After discussing historical results, I'll discuss how rank rigidity results hold in dimension three without additional curvature assumptions.

Speaker: Tom Chou, University of California at Los Angeles

Title: Path integral-based inference of PDEs and bond energies and mobility in Dynamic Force Spectroscopy

Date: 10/28/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

A Bayesian interpretation is given for regularization terms for
parameter functions in inverse problems. Fluctuations about the
extremal solution depend on the regularization terms - which encode
prior knowledge - provide quantification of uncertainty. After
reviewing a general path-integral framework, we discuss an application
that arises in molecular biophysics: The inference of bond energies
and bond coordinate mobilities from dynamic force spectroscopy
experiments.