Chairs
Professor Eric Vasserot, Institute of Mathematics of Jussieu - Paris Left Bank, France
Professor Eric Vasserot, Institute of Mathematics of Jussieu - Paris Left Bank, France
Professor Eric Vasserot was a student at the Ecole Normale Superieure in Paris, then successively a researcher at the CNRS at the Ecole Normale Superieure in Paris. Between 1995 and 2005, he was a Professor at the University of Cergy-Pontoise. Since 2005, he has been a Professor at the University Paris-Diderot. Eric works on geometric representation theory and categorifications. Eric was an invited speaker at the ICM in Seoul in 2014.
13:30-14:15
Motivic Chern classes, Hecke algebras, and stable envelopes
Professor Leonardo Mihalcea, Virginia Tech University, USA
Abstract
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.
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Professor Leonardo Mihalcea, Virginia Tech University, USA
Professor Leonardo Mihalcea, Virginia Tech University, USA
Dr Leonardo Mihalcea obtained his bachelor degree from Babes-Bolyai University in Cluj-Napoca, Romania, and his PhD from University of Michigan, USA, in 2005, under the direction of Professor William Fulton. After graduation he held positions at Florida State University, Duke University and Baylor University. He joined the Mathematics Department at Virginia Tech in 2011, and since 2015 he is an Associate Professor. His interests lie in Schubert Calculus and related problems. Recent contributions focus on the quantum K theory ring of cominuscule Grassmann manifolds (finiteness, formulas for Schubert multiplication) and on the study of the characteristic classes of Schubert cells (the Chern-Schwartz-MacPherson classes) for generalized flag manifolds, in relation to certain lagrangian cycles on the cotangent bundle, and to stable envelopes.
14:15-15:00
Ext operators as W-algebra intertwiners: AGT with matter for general surfaces
Professor Andrei Negut, Massachusetts Institute of Technology, USA
Abstract
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).
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Professor Andrei Negut, Massachusetts Institute of Technology, USA
Professor Andrei Negut, Massachusetts Institute of Technology, USA
Andrei Negut’s overall program concentrates on problems in geometric representation theory, an area that overlaps studies in algebraic geometry and representation theory. His results connect to areas in mathematical physics, symplectic geometry, combinatorics and probability theory. His current research focuses on moduli of sheaves, quiver varieties, quantum algebras and knot invariants.
15:30-16:15
Cherednik operators at infinity
Professor Maxim Nazarov, University of York, UK
Abstract
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.
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Professor Maxim Nazarov, University of York, UK
Professor Maxim Nazarov, University of York, UK
Maxim Nazarov started his research in Moscow at the seminar of Israel Gelfand. In 1991 he obtained PhD in Mathematics from Moscow State University, supervised by Alexandre Kirillov and Grigori Olshanski. In 1990 he became Assistant Professor at the Moscow State University. In 1992–1993 he spent 12 months in Kyoto at the Research Institute for Mathematical Sciences as JSPS Postdoctoral Fellow. In 1993 he obtained a University of Wales Research Fellowship to work in Swansea for 2 years. In 1995 he was awarded an EPSRC Advanced Research Fellowship. In 1996 he moved to the University of York, where in 2000 he became a Lecturer. He gave an Invited Talk at the International Congress of Mathematicians in 2002. In 2003 he was awarded a Whitehead Prize by the London Mathematical Society. In 2003 he became a Reader in Mathematics at the University of York. In 2006 he became a Professor.
16:15-17:00
Helix structures in quantum cohomology of Grassmannians
Dr Giordano Cotti, Max-Planck-Institut für Mathematik, Germany
Abstract
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.
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Dr Giordano Cotti, Max-Planck-Institut für Mathematik, Germany
Dr Giordano Cotti, Max-Planck-Institut für Mathematik, Germany
Giordano Cotti earned his PhD in September 2017 at the Scuola Internazionale Superiore di Studi Avanzati (SISSA) in Trieste, Italy, where he studied under the supervision of Professors Boris A Dubrovin and Davide Guzzetti. He is currently a postdoctoral fellow at the Max-Planck-Institut für Mathematik in Bonn, Germany, under the supervision of Professor Yuri I Manin. His research interests lie primarily in the interplay of analytic and algebraic geometry and differential equations arising from mathematical physics. More concretely, his work fits into the general research areas of Gromov-Witten and Frobenius manifolds theories, mirror symmetry, and the theory of integrable systems, namely the theories of isomonodromic deformations and asymptotic aspects of complex differential equations, as well their (conjectural) relationships with the derived geometry of smooth projective varieties (eg derived categories of coherent sheaves and their exceptional collections) and related mirror phenomena.