Title: Quantum Integrable Systems and Enumerative Geometry

Date: 02/05/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Anton M. Zeitlin, Louisiana State University

The correspondence between integrable systems and enumerative geometry
started roughly 25 years ago in the works of Givental and his collaborators,
studying quantum cohomology and quantum K-theory. Around 10 years ago,
physicists Nekrasov and Shatashvili proposed an unexpected relation between
quantum K-theory and quantum integrable systems based on quantum groups
within their studies of 3-dimensional gauge theories. Their bold proposal
led to a lot of interesting developments in mathematics, bringing a new life
to older ideas of Givental, and enriching it with flavors of geometric
representation theory via the results of Braverman, Maulik, Nakajima, Okounkov
and many others. In this talk I will focus on recent breakthroughs in the
subject, leading to the proper mathematical understanding of Nekrasov-Shatashvili
original papers as well as some other subsequent conjectures made by physicists.
Our main illustration of such a relation is an interplay between equivariant quantum K-theory of the cotangent bundles to Grassmanians and the Heisenberg XXZ spin chain. We will also
discuss relation of equivariant quantum K-theory of flag varieties and
many-body integrable systems of Ruijsenaars-Schneider and Toda.

Title: Rectifiability and Minkowski bounds for the singular sets of multiple-valued harmonic spinors

Date: 02/06/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

Speaker: Boyu Zhang, Harvard University

We prove that the singular set of a multiple-valued harmonic spinor on a 4-manifold is 2-rectifiable and has finite Minkowski content. This result improves a regularity result of Taubes in 2014. It implies more precise descriptions for the limit behavior of non-convergent sequences of solutions to many important gauge-theoretic equations, such as the Kapustin-Witten equations, the Vafa-Witten equations, and the Seiberg-Witten equations with multiple spinors.

In this talk I will give a constructive proof to " Let k be a knot in S1 ×S2 freely homotopic to S1 ×pt then S1 × pt bounds an invertible concordance and k splits (S1 × pt) × [0, 1]."

Title: Dynamical System Seminar Almost sure invariance principle for hyperbolic systems with singularities.

Date: 02/08/2018

Time: 3:00 PM - 4:00 PM

Place: C517 Wells Hall

Speaker:

Speaker: Jianyu Chen, University of Massachusetts Amherst
Title: Almost sure invariance principle for hyperbolic systems with singularities.
Abstract: We investigate a wide class of two-dimensional hyperbolic systems with singularities, and prove the almost sure invariance principle (ASIP) for the random process generated by sequences of dynamically H\"older observables. The observables could be unbounded, and the process may be non-stationary
and need not have linearly growing variances.
Our results apply to Sinai dispersing billiards and their conservative perturbations, as well as the induced systems of Bunimovich billiards. The random processes are not restricted to the ergodic sum, but applicable to entropy fluctuation, shrinking target problems, etc.

Title: Optimal mixing and irregular transport by incompressible flows

Date: 02/09/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Anna Mazzucato, Pennsylvania State University

I will discuss transport of passive scalars by incompressible flows (such as a die in a fluid) and measures of optimal mixing and stirring under physical constraint on the flow. In particular, I will present recent results concerning examples of flows that achieve the optimal theoretical rate in the case of flows with a prescribed bound on certain Sobolev norms of the associated velocity, such as under an energy or an enstrophy budget. These examples are related to examples of (instantaneous) loss of Sobolev regularity for solutions to linear transport equation with non-Lipschitz velocity.