Posted January 23, 2017
Last modified March 17, 2017
Jie Zhou, Perimeter institute
Periods and Gromov-Witten invariants
Abstract: The mirror symmetry conjectures asserts that the generating series of Gromov-Witten invariants (curve counting) of a Calabi-Yau variety are identical to some "universal" differential polynomials of period integrals of its mirror Calabi-Yau variety. I will explain in detail how these "universal" polynomials can be read off from the Picard-Fuchs system of the mirror Calabi-Yau variety, for the genus zero and one cases which are so far the only cases proved rigorously in mathematics. I will also discuss some nice ingredients (e.g., generating series of point counting, polylogarithms, Feynman diagrams and manipulation on Picard-Fuchs equations) which seem to have a motivic nature. A particularly interesting example of Calabi-Yau 3-fold will be emphasized, in which modular forms arise naturally.
Posted February 3, 20173:30 pm - 4:20 pm Lockett 233
Jose Ceniceros, Louisiana State University
Posted April 24, 20173:30 pm - 4:20 pm Lockett 285
Hongyu He, Department of Mathematics, LSU
Interlacing relations in Representation theory
Given an irreducible representation of U(n) with highest weight $\lambda$, its restriction to U(n-1) decomposes into a direct sum of irreducible representations of U(n-1) with highest weights $\mu$. It is well-known that $\lambda$ and $\mu$ must satisfy the Cauchy interlacing relations $$\lambda_1 \geq \mu_1 \geq \lambda_2 \geq \mu_2...$$ and vice versa. In this talk, I shall discuss the noncompact analogue for the discrete series of $U(p,q)$ as conjectured by Gan, Gross and Prasad. I will introduce the Gan-Gross-Prasad interlacing relations and discuss some recent progress.
Posted February 23, 2017
Last modified April 20, 2017
Earl Taft, Rutgers University and UC Berkeley
Left quantum groups
Abstract: Quantum groups are Hopf algebras, and possess an antipode, which is the analogue of inversion in a group. The antipode satisfies a left and a right condition. In the interest of breaking symmetry in physics, S. Rodriguez and E. Taft constructed a version of quantum SL_2 which has a left antipode but not a right antipode. This was not of interest to physicists
(we will explain why not), but turned out to be of interest to combinatorists, as it was used to give a quantum version of the MacMahon master theorem(also known as the boson-fermion correspondence). A. Lauve and E. Taft extended the Rodriguez-Taft construction to a version of quantum SL_n. Using continuous duals, we relate our work to the quantum universal enveloping algebra of the Lie algebra sl_2, and discuss an open question of whether or not there is a left quantum group containing (U_q)(sl(2)).
Posted April 24, 20173:30 pm - 4:30 pm Lockett 10