Motivic Chern classes, Hecke algebras, and stable envelopes
Professor Leonardo Mihalcea, Virginia Tech University, USA
The Chern-Schwartz-MacPherson (CSM) class of a compact (complex) variety X is a homology class which provides an analogue of the total Chern class of the tangent bundle of X, for X singular. Its K-theoretic version, the motivic Chern class, is a class with good functorial properties, and for smooth X it normalizes to the Hirzebruch's lambda-y class of the cotangent bundle of X. One can associate these classes to any constructible subset of X, and in the talk I will discuss how one can use the Demazure-Lusztig operators in the Hecke algebra to calculate the motivic Chern classes for Schubert cells in generalized flag manifolds X=G/B. I will also discuss relations to K theoretic envelopes of Maulik and Okounkov, and a conjectural positivity property. This recovers and extends beyond Lie type A recent results obtained by Fejer, Rimanyi and Weber, using localization techniques. The talk is based on ongoing joint work with Paolo Aluffi, Changjian Su, and Jorg Schurmann.
Ext operators as W-algebra intertwiners: AGT with matter for general surfaces
Professor Andrei Negut, Massachusetts Institute of Technology, USA
Andrei Negut will summarise an algebro-geometric proof, which works for a fairly general smooth projective surface, of the following fact: the Ext operator that Carlsson-Okounkov associate to moduli spaces of sheaves on a surface is a W-algebra intertwiner (as predicted by the work of Nekrasov and Alday-Gaiotto-Tachikawa).
Cherednik operators at infinity
Professor Maxim Nazarov, University of York, UK
Heckman introduced N operators on the space of polynomials in N variables, such that these operators form a covariant set relative to permutations of the operators and variables, and such that Jack symmetric polynomials are eigenfunctions of the power sums of these operators. Maxim Nazarov introduces the analogues of these N operators for Macdonald symmetric polynomials, by using Cherednik operators. The latter operators pairwise commute, and Macdonald polynomials are eigenfunctions of their power sums. The groups compute the limits of their operators when N tends to infinity. These limits yield a Lax operator for Macdonald symmetric functions. This is a joint work with Evgeny Sklyanin.
Helix structures in quantum cohomology of Grassmannians
Dr Giordano Cotti, Max-Planck-Institut für Mathematik, Germany
In occasion of the 1998 ICM in Berlin, B Dubrovin conjectured an intriguing connection between the enumerative geometry of a Fano manifold X with algebro-geometric properties of exceptional collections in the derived category D^b(X). Under the assumption of semisimplicity of the quantum cohomology of X, the conjecture prescribes an explicit form for local invariants of QH*(X), the so-called “monodromy data”, in terms of Gram matrices and characteristic classes of objects of exceptional collections. In this talk, a refinement of this conjecture will be presented, and particular attention will be given to the case of complex Grassmannians. At points of small quantum cohomology, these varieties manifest a coalescence phenomenon, whose occurrence and frequency is surprisingly subordinate to the distribution of prime numbers. A priori, the analytical description of these Frobenius structures cannot be obtained from an immediate application of the theory of isomonodromy deformations. Giordano Cotti will show how, under minimal conditions, the classical theory of M Jimbo, T Miwa, K Ueno (1981) can be extended to describe isomonodromy deformations at a coalescing irregular singularity. Furthermore, a property of quasi-periodicity of Stokes matrices associated to the points of small Quantum Cohomology of complex Grassmannians will be discussed. Based on joint works with B Dubrovin and D Guzzetti.