Algebra and Number Theory Seminar
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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.

Topology Seminar
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Posted February 3, 2017

3:30 pm - 4:20 pm Lockett 233
Jose Ceniceros, Louisiana State University

TBD

Harmonic Analysis Seminar
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Posted April 24, 2017

3:30 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.

Colloquium
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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)).

Algebra and Number Theory Seminar
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Posted April 11, 2017

3:00 pm - 3:50 pm Lockett 277
Ha Tran, University of Calgary

On reduced ideals of a number field

Let F be a number field. The reduced ideals of F can be used for computing its class group and regulator. In this talk, we will introduce reduced ideals first for quadratic fields then for an arbitrary number field. Next, we will discuss a generalization of reduced ideals using the LLL-algorithm. Finally, some open problems relating to this topic will be presented.

Algebra and Number Theory Seminar
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Posted April 22, 2017

3:00 pm - 3:50 pm Lockett 277
David Lax, Virginia Tech

Order Filter Model for Minuscule Plucker Relations

Abstract: The Plucker relations which define the Grassmann manifolds as projective varieties interact nicely with a natural order on the projective coordinates; the resulting homogeneous coordinate ring is an algebra with straightening law. This is a property shared by all minuscule flag manifolds. The order structures on their projective coordinates share common properties and are called minuscule lattices. We study their generalized Plucker relations independent of Lie type through the minuscule lattices. To do this we combinatorially model the Plucker coordinates based on Wildberger's construction of minuscule Lie algebra representations; it uses the colored partially ordered sets known as minuscule posets. We obtain, uniformly across Lie type, descriptions of the Plucker relations of ``extreme weight''. We show that these are supported by ``double-tailed diamond'' sublattices of minuscule lattices. From this, we obtain a complete set of Plucker relations for the exceptional minuscule flag manifolds. These Plucker relations are straightening laws for their respective coordinate rings.