Department of

Mathematics


Seminar Calendar
for events the day of Thursday, April 29, 2021.

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Thursday, April 29, 2021

11:00 am in zoom,Thursday, April 29, 2021

Elliptic genera and circle actions

Silvia Sabatini (University of Cologne)

Abstract: Consider a compact symplectic manifold with nonzero first Chern class. Its index is defined to be the largest integer dividing the first Chern class in the second cohomology group (modulo torsion). In the algebraic geometry setting there is a question of relating the index to the Betti numbers of the manifold in the case in which the manifold is a Fano variety, which we refer to as a ``positivity condition on the first Chern class''. This question is not fully answered and there are still open conjectures about it, for instance the Mukai conjecture. In the first part of this talk I will introduce the audience to results which are already known in the symplectic setting, relating the index to the Betti numbers for manifolds admitting some special Hamiltonian circle action. In the second part of the talk I will introduce elliptic genera and show how their behaviour can be used to deduce relations between the Betti numbers and the index without assuming the aforementioned positivity condition on the first Chern class.

3:00 pm in Zoom,Thursday, April 29, 2021

The Unitary Dual Problem and Combinatorics

Wai Ling Yee   [email] (University of Windsor)

Abstract: One of the most important open problems in mathematics is the Unitary Dual Problem: given a group, classify its irreducible unitary representations. The most common approach to classifying unitary representations is to study representations admitting non-degenerate invariant Hermitian forms (these are classified), compute the signatures of the forms, and then determine when the forms are definite. Signature character formulas are very complicated due to the recursive nature of the formulas. However, it turns out that the complexity may be encoded by well-known combinatorial objects: signature character formulas involve signed Kazhdan-Lusztig polynomials (which turn out to be classical Kazhdan-Lusztig polynomials evaluated at $-q$ times a sign) and pieces of Hall-Littlewood polynomials (called Hall-Littlewood polynomial summands) evaluated at $q=-1$. Some of the material covered in this talk is joint work with Justin Lariviere. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.