Held in Salón Magallanes, Hotel Diego de Almagro, Punta Arenas
Welcome and coffee
True motives and graded representation theory
Recent advances in the theory of motivic sheaves due to Cisinsky-Deglise allow very direct geometric constructions of graded categories of representations.
Local hard Lefschetz for Soergel bimodules
I will give a introduction to the local hard Lefschetz property for Soergel bimodules and outline joint work with Ben Elias giving a proof. As in our proof of Soergel's conjecture, signs (in the form of the Hodge-Riemann bilinear relations) are crucial. I will also describe results of Soergel and Kuebel which link local hard Lefschetz to the Jantzen and Andersen filtrations.
Light leaves and Lusztig's conjecture
I will define the light leaves basis, a basis of the Hom spaces in the category of Soergel bimodules. I will explain their relation to Lusztig's conjecture in modular representation theory and in what way can one use them to give counterexamples to this conjecture (they were found by Geordie Williamson). The relevant groups in this context are the Weyl and affine Weyl groups. Finally, I will explain joint work with Ben Elias giving deeper understanding of the baby case of Universal Coxeter groups.
Afternoon coffee and registration
Nichols algebras of diagonal type
Nichols algebras are Hopf algebras in braided tensor categories with very particular properties; for instance, the positive parts of quantized enveloping algebras at a generic parameter, and their finite-dimensional counterparts when the parameter is a root of one, are Nichols algebras. The input datum to define a Nichols algebra is a braided vector space, that is a solution of the braid equation or equivalently of the quantum Yang-Baxter equation. Nichols algebras of diagonal type are by definition those corresponding to solutions of the braid equation given by a perturbation of the usual transposition given by a matrix of non-zero scalars. These Nichols algebras appear in the classification of pointed Hopf algebras with abelian group, within the method proposed by Hans-Jürgen Schneider and myself. The main questions concerning them are: (*) Classify the Nichols algebras of diagonal type with finite dimension; (*) give for them an optimal presentation by generators and relations. The first question was solved by István Heckenberger using notably the Weyl groupoid introduced by himself; the second was answered by Iván Angiono in his thesis at the University of Córdoba. Currently Angiono and myself are working in clarifications of some aspects of these results, including relations with contragredient Lie (super) algebras, either in zero or in positive characteristic. In this talk I will survey from scratch the notion of Nichols algebras of diagonal type and the main results evoked above and will report on the recent work in progress with Angiono.
Representation theory of quantized quiver varieties
I will describe known results on the representation theory of quantizations of Nakajima quiver varieties. The talk is partly based on my joint work with Bezrukavnikov, and partly on my work in progress.
Cherednik algebras and torus knots
The Cherednik algebra B(c,n), generated by symmetric polynomials and the quantum Calogero-Moser Hamiltonian, appears in many areas of mathematics. It depends on two parameters - the coupling constant c and number of variables n. I will talk about representations of this algebra, and in particular about a mysterious isomorphism between the representations of B(m/n,n) and B(n/m,m) of minimal functional dimension. This symmetry between m and n is made manifest by the fact that the characters of these representations can be expressed in terms of the colored HOMFLY polynomial of the torus knot T(m/d,n/d), where d=GCD(m,n). The talk is based on my joint work with E. Gorsky and I. Losev.
Geometric (q_1, q_2)-characters and gauge theory
Stable envelopes in K-theory and beyond
Irreducible modules for Witt algebras
A classification of all irreducible weight modules with finite weight multiplicities over the Lie algebra of vector fields on n-dimensional torus will be discussed. This is a joint result with Y.Billig (Carleton University, Canada). Our classificaton is obtained for an arbitrary n, generalizing a well known result of O.Mathieu for the first Witt algebra (or for the Virasoro algebra) when n=1. V.Mazorchuk and K.Zhao showed that any such irreducible module is either uniformly bounded (cuspidal) or a highest weight type module. Moreover, highest weight type modules are induced from cuspidal modules (for n-1-dimensional torus). We provide a description of all irreducible cuspidal modules completing the classification.
Cherednik algebras and elliptic pairs
I will describe a conjecture on relating the category of representations of a type A rational DAHA to perverse sheaves. The conjecture is partly motivated by geometric Langlands duality for an elliptic curve, though it involves aspects of Langlands duality which have not yet been worked out.
Rigid DAHA-modules in rank one and their applications
To put this topic into perspective, we will begin with a brief review of relatively recent applications of DAHA of type A_1, including the monodromy of the Lame equation, refined Jones polynomials, relations to the WZWH model and a link to the nonsymmetric q-Whittaker functions and the PBW-filtration, as well as the connections with Lusztig's quantum groups. The key example will be the generalized nonsymmetric Verlinde algebra; when t=q and upon the symmetrization it describes the reduced category of Lusztig's quantum group. The purpose of this talk is to define the action of the absolute Galois group in the latter and similar rigid DAHA modules of type A_1, which for instance results in certain arithmetic refined Jones polynomials. Considering here rigid modules of dimension 3 reproduces the Livne triangular groups, which will be explained from scratch.
Bus leaves Hotel Diego de Almagro for Puerto Natales