09:30 to 11:00 
Shrawan Kumar (University of North Carolina, USA) 
Eigenvalue problem for reductive groups For any n xn Hermitian matrix A, let e(A) = (e_A(1), ..., e_A(n)) be its set of eigenvalues written in descending order. We recall the following classical problem.
Problem 1. (The Hermitian eigenvalue problem) Given two ntuples of non increasing real numbers: e' = (e'(1), ..., e'(n)) and e" = (e"(1), ..., e"(n)), determine all possible f = (f(1), ..., f(n)) such that there exist Hermitian matrices A, B, C with e(A) = e'; e(B) = e"; e(C) = f and A + B + C = 0. Even though this problem goes back to the nineteenth century, the first significant result was obtained by H. Weyl in 1912. With contributions from several mathematicians over the century, this problem was finally solved by combining the works of Klyachko (1998), KnutsonTao (1999), Belkale (2001) and KnutsonTaoWoodward (2004).
This problem can be generalized for an arbitrary reductive algebraic group as follows: Let G be a connected algebraic group with a maximal compact subgroup K and let g and k be their Lie algebras. Consider the Cartan decomposition g = k+p. Choose a maximal subalgebra (which is necessarily abelian) a of p and let a+ be a dominant chamber in a. Then, any Korbit in p intersects a+ in a unique point. Then the analogue of the above Hermitian eigenvalue problem is the determination of the following subset C_m (for any m > 2) of (a+)^m: C_m := {(a_1, ..., a_m) in (a+)^m such that there exists (x_1, ..., x_m) in (a+)^m with x_1+ ...+ x_m = 0} and x_i in AdK. a_i}. By works of several mathematicians including BerensteinSjamaar (2000), KapovichLeebMillson (2005), BelkaleKumar (2006) and Ressayre (2008), C_m has been determined in terms of an irredundant set of inequalities. This series of talks will give a complete solution of the problem. The main tools used are: Geometric Invariant Theory and Topology. We will assume familiarity with basic algebraic geometry and topology. Otherwise, the lectures will be selfcontained.
We will also discuss the parallel `saturated' tensor product decomposition problem.



11:30 to 12:30 
Anne Moreau (Université ParisSaclay, Paris,France) 
Vertex algebras and associated varieties The aim of the lectures is to give a gentle introduction to the theory of vertex algebras, and present the concept of associated variety. The associated varieties are certain Poisson varieties attached to vertex algebras whose geometry capture important information. They are related to other interesting objects of the theory: the arc spaces and Zhu’s algebras.



14:00 to 15:00 
Arun Ram (University of Melbourne, Australia) 
Representations of quantum affine algebras and Yangians The lectures will focus on the classification, combinatorial construction, characters, and crystals of integrable representations of quantum affine algebras. Following the paper arXiv1907.11796 , the classification fall into three cases: positive level, negative level and level zero, of which the level 0 case is the most “interesting”.



15:30 to 15:50 
G V Krishna Teja (ISIBC, India) 
Short talk  Weights of all highest weight modules over BorcherdsKacMoody algebras Let g = g(A) be a general Borcherds–Kac–Moody (BKM) CLie algebra, for Borcherds–Cartan matrix A, and V a highest weight gmodule with weightset wt V . Over Kac–Moody g, we know wt V for:
1) integrable V classically;
2) simple (highest weight) V by Dhillon and Khare, via parabolic Vermas;
3) all highest weight modules V recently by the author and Khare, via holes & higher order Vermas. Over BKM g, even for integrable V , to our knowledge wt V is unknown.
In this talk, we will determine the weightsets wt V for all highest weight modules V over general BKM g, extending and subsuming at once the formulas in settings 1)–3). Our formula is:
i) explicit, nonrecursive and cancellationfree; and
ii) uniform across all types of g (semisimple to Kac– Moody to general BKM) and all V (and all highest weights λ).
We begin with (the combinatorial problem of) describing the comple ments of the uppersets in Zn(≥0), thereby determining wt V for all V over g = H ⊕ · · · ⊕ H for H the 3dimensional Heisenberg Lie algebra and over g = g(A) with A(ii) = 0 ∀i (in connection to finitedimensional Heisenberg Lie algebras). This reveals a refined notion of holes to study wt V . We construct a novel class (to our knowledge) of “higher order parabolic Vermas” M(λ, H) over BKM g (only singleton holes in H in real directions), subsuming and generalizing i) all integrable V and ii) parabolic Vermas over Kac–Moody g.
We determine wt M(λ, H) over all λ and H, for every BKM g. In particular, we determine wt V for all integrable V – perhaps interestingly in the flavour of Weyl orbit weightformulas for integrable V over Kac–Moody g (see e.g. Kac’s book). For all V over all BKM g, we recover wt V as the union of wt M(λ, H) with H ⊇ HV . (This talk is based on a recent joint work with Souvik Pal.)



15:50 to 16:10 
Divya Setia (IISER Mohali, India) 
Short talk  Demazure filtration of Tensor Product Modules of Current Lie Algebra of type $A_1$. Let g be a finitedimensional simple Lie algebra over the complex field C and g[t] be the Lie algebra of polynomial mappings from C to g, which is its associated current algebra. We study the structure of the finitedimensional representations of
the current Lie algebra of type A1, sl2[t], which are obtained by taking tensor products of local Weyl modules with Demazure modules. We show that these representations admit a Demazure flag and obtain a closed formula for the graded multiplicities of the level 2 Demazure modules in the filtration of the tensor product of two local Weyl modules for sl2[t]. Using Pieri formulas, we have also expressed the product of two specialized Macdonald polynomials in terms of specialized Macdonald polyomials.
Furthermore, we show that the tensor product of a local Weyl module with an irre ducible sl2[t] module admits a Demazure filtration and derive graded character of such tensor product modules. This helps us express the product of a specialized Macdon ald polynomial with a Schur polynomial in terms of Schur polynomial. Our findings provide evidence for the conjecture that the tensor product of Demazure modules of levels m and n respectively has a filtration by Demazure modules of level m + n.



16:10 to 16:30 
Duncan Laurie (University of Oxford, UK) 
Short talk  Quantum toroidal algebras: braid group actions and automorphisms Quantum toroidal algebras Uq(g_tor) occur as the Drinfeld quantum affinizations of quantum affine algebras. In particular, they contain (and are generated by) a horizontal and vertical copy of the affine quantum group. In type A, Miki obtained an automorphism of Uq(g_tor) exchanging these subalgebras, which has since played a crucial role in the investigation of its structure and representation theory.
In this talk, we shall construct an action of the extended double affine braid group B on the quantum toroidal algebra in all untwisted types. In the simply laced cases, using this action and certain involutions of B we obtain automorphisms and antiautomorphisms of Uq(g_tor) which exchange the horizontal and vertical subalgebras, thus generalising the results of Miki.


