GARTS Spring 2024

Feb 2, 2024

Speaker: Ching Hsien Lee

Title: Serre's Theorem

Feb 9, 2024

Speaker: Yuqiao Huang

Title: Lie group--Lie algebra correspondence

Abstract: We will talk about the correspondence between Lie groups and Lie algebras. This answers the question of how the representation theory of Lie groups reduces the representation theory of Lie algebras.

Feb 16, 2024

Speaker: Sumit Singh

Title: Macdonald Polynomials

Feb 23, 2024

Speaker: Hong Chen

Title: Interpolation Jack and Macdonald Polynomials

Abstract: Jack and Macdonald polynomials are important generalizations of Schur polynomials: they are orthogonal w.r.t. some generalized scalar product on the ring of symmetric functions, and they are common eigenfunctions to certain operators. Interpolation Jack and Macdonald polynomials are further generalizations of the ordinary ones: they are inhomogeneous, with top degree terms being the ordinary ones, they interpolate the Kronecker delta function, and they satisfy an unexpected extra vanishing property. In this talk, I will introduce these polynomials, in particular, a surprising binomial formula due to Okounkov--Olshanski that relates the ordinary polynomials and the interpolation ones.

Mar 8, 2024

Speaker: Muhammad Haris Rao (U Melbourne)

Title: Affine Paving for Hessenberg Varieties

Abstract: Flag varieties lie at the intersection of geometry, representation theory and combinatorics. In this talk, I will introduce the family of regular semisimple Hessenberg varieties as a generalisation of flag varieties, and present an affine paving for them. Using this, I will walk through explicit computations of the cohomology of specific Hessenberg varieties before presenting more general formulas in terms of the combinatorics of the symmetric group.

Mar 29, 2024

Speaker: Jinfeng Song (NUS)

Title: Poisson homogeneous spaces and quantum symmetric pairs

Abstract: It is known that the Drinfeldâ€“Jimbo quantum groups can be viewed as quantized coordinate algebras of the dual Poisson-Lie groups. A quantum symmetric pair consists of a quantum group and a coideal subalgebra, called an i-quantum group. In this talk, I will explain that i-quantum groups can be viewed as quantized coordinate algebras of Poisson homogeneous spaces of dual Poisson groups. The construction fits into the general picture, called the quantum duality principle.

slidesApril 19, 2024

Speaker: Nick Backes

Title: Supercuspidal Representations and the Weil Representation

Abstract: Every irreducible admissible representation of GL(2,R), and in fact every real reductive group, can be constructed by parabolic induction. The representation theory of algebraic groups over nonarchimedean fields is distinguished by the existence of supercuspidal representations: precisely those which are not obtained through parabolic induction. Supercuspidals have Kirillov models, in which the vector space is explicit, but the action of the group is not fully described. However, there exists another construction of supercuspidal representations on GL(2,Q_p) through the Weil representation, in which the action of the group is concretely realized. This construction is what I will show in this talk.

Spring 2021 Talks (Jason Saied)

Spring 2019 Talks (Alejandro Ginory)