Abstract: A key result in the rich theory of rational Cherednik algebras is the deformed Harish-Chandra isomorphism, proposed by Etingof-Ginzburg and proved by Gan-Ginzburg, that identifies the spherical subalgebra of the type A rational Cherednik algebra with a quantized Nakajima quiver variety, the latter of which is defined as a quantum Hamiltonian reduction of a ring of differential operators. I will discuss a multiplicative analogue of this result, wherein the rational Cherednik algebra is replaced with the usual double affine Hecke algebra of GLn and the quantized quiver variety is replaced with a quantized multiplicative quiver variety, as defined by Jordan. This setting is strange because we are no longer working with rings of differential operators, but rather a less familiar ring of quantum differential operators defined by Varagnolo-Vasserot. Nonetheless, I will explain how, via an idea of Varagnolo-Vasserot, Macdonald polynomials can be used to establish the analogous isomorphism in a manner quite similar to that of the rational case.

Abstract: There’s a longstanding connection between gauge theory and quantum groups. In this talk we will show how the semi-classical limits of various quantum groups are encoded by the cohomology of spheres equipped with certain foliated structures. The technical procedure is the homotopy transfer of the product in de Rham models to cohomology, which in the case of a holomorphic structure was carried out by Zeng. I will then explain the relationship of this approach with recent constructions quantum groups via the factorization algebra of observables of (three, four, and five-dimensional) Chern-Simons theory due to Costello and others.

Abstract: Let G be a connected reductive group over an algebraically closed field. Such groups are classified via root data and can be parameterised via Chevalley group schemes over integers. In this talk, we shall first recall the construction of Chevalley group schemes by Lusztig using quantum groups. Then we shall discuss the construction of symmetric subgroup schemes parameterising symmetric subgroups K of G using quantum symmetric pairs. The existence of such group schemes allows us to apply characteristic p methods to study the geometry of K-orbits on the flag variety of G. This leads to a construction of Frobenius splittings via quantum symmetric pairs, generalising the algebraic Frobenius splittings by Kumar-Littelmann. This is based on joint work with Jinfeng Song (NUS). 

Abstract: The a-function of a Coxeter group W is a nonnegative integer valued function on W defined by Lusztig that is constant on Kazhdan--Lusztig cells. It is known that the identity element of W is the only element with a-value 0, and that the elements with a-value 1 are exactly the elements with a unique reduced expression. In this talk, I will report on recent joint work with Richard Green on elements and cells of a-value 2. Our main tool is a type of partially ordered sets called Viennot's heaps.

Abstract: In this talk, we will introduce an element, which we call the Kirillov projector, that connects the topics of the title: on the one hand, it is constructed using the Yangian of gl_N, on the other hand, it defines a canonical projection onto the space of Whittaker vectors for any Whittaker module over the mirabolic subalgebra of gl_N. With the help of the Kirillov projector, we will deduce some categorical properties of Whittaker modules, for instance, a mirabolic analog of Skryabin’s theorem. We will also show that it quantizes a certain non-standard r-matrix as well as the subvariety of companion matrices. Based on arXiv:2310.06669.

Abstract: We will develop techniques for proving holographic entanglement entropy inequalities from Lipshitz contraction mappings. Techniques for improving the proof method will be discussed, and further mathematical extensions will be described.

Abstract: The Brauer category is a diagrammatic monoidal category describing the representation theory of the orthogonal and symplectic groups.  Its endomorphism algebras are Brauer algebras, which replace the group algebra of the symmetric group in the orthogonal and symplectic analogues of Schur-Weyl duality.  However, the Brauer category is missing one important piece of the picture—the spin representation.  We will introduce a larger category, the spin Brauer category, that remedies this deficiency.  This is joint work with Peter McNamara.

Abstract: The classical Torelli problem asks whether a smooth projective variety can be recovered from the Hodge structure on its middle cohomology. A log Calabi-Yau variety U is quasiprojective, so has a Deligne mixed Hodge structure on its middle cohomology, and one can similarly ask to what extent U can be recovered from its mixed Hodge structure. The situation is well-understood in dimension 2 due to work of Gross-Hacking-Keel. I will present partial results in the 3-dimensional case, and applications to the Morrison-Kawamata cone conjecture. This is work in progress with Paul Hacking. 

Abstract: In their paper “Webs and quantum skew-Howe duality", Cautis-Kamnitzer-Morrison gave a generators and relations presentation of the full monoidal subcategory of representations of the quantum special linear group generated by the q-analogues of the exterior powers of the defining representation. This presentation has had applications to geometric representation theory, modular representation theory, and link homology.

I will describe joint work with Haihan Wu, https://arxiv.org/abs/2309.03623, in which we give a generators and relations presentation of the full monoidal subcategory of representations of the quantum orthogonal group generated by q-analogues of the exterior powers.

Abstract: In contrast to prediction, which uses data to infer the future, retrodiction uses data to infer the past. The primary example is given by Bayes' rule P(y|x) P(x) = P(x|y) P(y), where we use prior information, conditional probabilities, and new evidence to update our belief of the state of some system. The question of how to extend this idea to quantum systems has been debated for many years. In this talk, I will lay down precise axioms for (classical and quantum) retrodiction using category theory, thus providing a mathematical definition for this intuitive concept. Among a variety of proposals for quantum retrodiction used in areas such as error-correction, thermodynamics, and the black hole information paradox, only one satisfies these categorical axioms. This talk is based on joint work with Francesco Buscemi (https://doi.org/10.22331/q-2023-05-23-1013).