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13-17 Jun 2022 Paris (France)
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home The geometric representation theory has developed in many directions recently. The aim of this school is to give an introduction to several new technics in complex geometry with possible applications to representation theory. In particular, we’ll focus on Quantum cohomology and K-theory, Stable envelopes, Quiver varieties, Cluster algebras…The school will consists of 4 courses given by L. Mihalcea, R. Rimanyi, A. Smirnov, C. Robles, T. Lam, A. Shapiro with titles - Quantum K theory of flag manifolds (L.Mihalcea) - Hodge structures - their representation theory and applications (C.Robles and T.Lam) - Vertex functions, stable envelopes, 3d mirror symmetry (R.Rimanyi and A.Smirnov) - K-theoretic Coulomb branches and cluster algebras (A.Shapiro) After the school there will be a 3-days workshop (in June 20, 21 and 22) with research talks on the same topics. The speakers will be P. Bousseau, J. Eberhardt, M. Gorsky, A. Mellit, L. Mihalcea, R. Rimanyi, C. Robles, A. Shapiro, A. Smirnov, S. Spenko, A. Weber. Accommodation can be provided by the organisation subject to availability. The support of your accomodation will be confirmed after your registration has been examined. Meal and transport costs are at your own expense. The school will take place in the University of Paris (building Sophie Germain, Amphi Turing, level -1). It is located in the center of Paris (see https://www.math.univ-paris-diderot.fr/ufr/localisation for details). The school will be held in a hybrid format form, with courses and talks given on-site and remotely and all streamed simultaneously through Zoom and recorded. The Zoom link is https://u-paris.zoom.us/j/89210819260?pwd=VGIrYmlUTUszdVVDZmtFSW4xNlUwZz09 Organizers : N. Perrin (Univ. Paris-Saclay) O. Schiffmann (Univ. Paris-Saclay) M. Varagnolo (Univ. Cergy-Paris) E. Vasserot (Univ. Paris-Cite) PROGRAM OF THE SCHOOL : 1) K-theoretic Coulomb branches and cluster algebras (A.Shapiro, 4 hours) Braverman, Finkelberg, and Nakajima have proposed a mathematical definition of the Coulomb branch of a 3d N=4 SUSY gauge theory of cotangent type. In their work, the (quantized K-theoretic) Coulomb branch is defined as the equivariant K-theory of a certain moduli space specified by a complex reductive group G and its complex representation N. It was subsequently conjectured by Gaiotto, that such a Coulomb branch carries the structure of a quantum cluster variety. I will outline a proof of this conjecture in the case of quiver gauge theories, i.e. when N is a representation of a quiver, and G is its gauge group. My talks will be based on joint work with Gus Schrader. 2) Quantum K theory of flag manifolds (L.Mihalcea, 4 hours) Since its definition in the early 2000's by Givental and Lee, the study of quantum K (QK) theory rings found applications in many parts of mathematics and physics. In this series of lectures aimed to graduate students and early career researchers, we will focus on the QK ring of flag manifolds, with an emphasis on Grassmannians. The aim is twofold. First, I wish to establish the computational foundations of the subject. I will explain how geometric properties of the Kontsevich moduli space of stable maps lead to combinatorial formulae for QK multiplications of Schubert classes. This is based on the `quantum = classical' statement, and on algorithms based on multiplication by divisor Schubert classes (Chevalley formulae). The second aim is to showcase some of the structural properties of the QK ring, such as an unexpected functoriality property, and a presentation of the QK ring as a Jacobi ring, inspired by mirror symmetry. If time permits, I will also mention the `quantum=affine' statement, relating the QK ring to the K-homology of the affine Grassmannian. I will not assume prior knowledge on quantum K theory. 3) Vertex functions, stable envelopes, 3d mirror symmetry (R.Rimanyi and A.Smirnov, 6 hours) Lecture 1: Quiver varieties, bow varieties Lecture 2: Equivariant cohomology, K-theory, elliptic cohomology Lecture 3: Stable envelopes: definition, examples Lecture 4: Vertex functions, differential equations Lecture 5: 3d mirror symmetry Lecture 6: Summary, conclusions 4) Hodge structures - their representation theory and applications (C.Robles and T.Lam, 6 hours) Lecture 1 (Robles): Introduction to Hodge structures. I will introduce (pure, polarized) Hodge structures and variations of Hodge structure. We will cover period domains, compact duals, the infinitesimal period relation (IPR). If time allows, we will also cover Mumford—Tate domains and Hodge representations (else this will be postponed to Lecture 3). Lecture 2 (Lam) : Mixed Hodge structures and the curious Lefschetz theorem. I will introduce mixed Hodge structures on the cohomology of complex algebraic varieties, with a focus on smooth, affine varieties of Hodge-Tate type. I will discuss an analogue of the Lefschetz theorem, called "curious Lefschetz", for such spaces. I will also discuss the Grothendieck-Lefschetz trace formula and the relation to point counting over finite fields. Lecture 3 (Robles): B. Gross’s representation theoretic construction of Calabi-Yau-type VHS. I’ll review the construction and introduce the “geometric realization problem”. The question is almost entirely open. I’ll review what is know, and the tools that have been developed to tackle the problem. Lecture 4 (Lam): Mixed Hodge structures of cluster varieties. I will introduce cluster algebras and cluster varieties, and discuss their geometry, cohomology, and point counts. I will talk about the curious Lefschetz theorem for the class of Louise cluster varieties. Lecture 5 (Robles): Characteristic cohomology of the infinitesimal period relation. The IPR is a geometric PDE constraining a VHS. To such PDE there is associated a characteristic cohomology. I will discuss it’s relationship to Berstein-Gelfand-Gelfand resolutions and an open question. Lecture 6 (Lam): Mixed Hodge structures of Richardson and positroid varieties. Open Richardson varieties are intersections of opposing Schubert cells in the flag variety and are conjecturally (sometimes known) to be cluster varieties. Their cohomology and mixed Hodge structures are appear as Ext-groups in Category O, and also as knot homology groups of certain links.
VIDEOS OF THE LECTURES : 1) A. Shapiro 2) L.Mihalcea 3) R.Rimanyi and A.Smirnov 4) C. Robles and T. Lam
PHOTOS OF THE SCHOOL : Some photos of the school are here https://photos.app.goo.gl/EvevWvzvTPEgHKKS9
PRACTICAL INFORMATIONS FOR THE SCHOOL : - A diner will be organized at the restaurant La Dame de Canton on Thursday 16th which is here https://www.google.com/maps/dir/Bâtiment+Sophie+Germain+-+Université+Paris+Cité,+Pl.+Aurélie+Nemours,+75013+Paris/La+Dame+de+Canton,+Port+de+la+Gare,+75013+Paris/@48.8307907,2.3760394,16z/data=!3m1!4b1!4m14!4m13!1m5!1m1!1s0x47e6734cceba0e5b:0x967ba563f64e97ca!2m2!1d2.3807335!2d48.8271044!1m5!1m1!1s0x47e6723d5068a1df:0x5704d36dfae86607!2m2!1d2.3774085!2d48.8345595!3e2 - A few places where it is possible to get lunch : Salads or Sandwiches : Pret A Manger, 110 Avenue de France Boulangerie Eric Kayser - Bibliothèque, 77 Quai Panhard et Levassor Boulangerie Honorine, 48 Avenue de France Sandwiches : Pomme de pain, 13 Rue Marie-Andrée Lagroua Weill-Hallé Asian : Ang, 9 Rue Marie-Andrée Lagroua Weill-Hallé Street Bida, 1 Rue Nicole-Reine Lepaute Crepes : Chocco'Là Paris, 48 Avenue de France
PROGRAM AND VIDEOS OF THE WORKSHOP : P. Bousseau : Cluster varieties and mirror symmetry J. Eberhardt : Motivic Springer Theory M. Gorsky : Cluster structures on braid varieties O. Kivinen : Weight polynomials of affine Springer fibers in type A and the elliptic Hall algebra A. Mellit : Affine Springer fibers, open Hessenberg varieties, and nabla positivity L. Mihalcea : Motivic Chern classes of Schubert cells: applications and experiments T. Padurariu : Donaldson-Thomas theory of C^3 via matrix factorizations R. Rimanyi : Shuffle structure for stable envelopes and applications C. Robles : Global asymptotic structure of period mappings A. Shapiro : Positive representation theory A. Smirnov : TBA S. Spenko : Remarks on cohomological and K-theoretical Hall algebras A. Weber : Elliptic characteristic classes of Schubert varieties and beyond
SCHEDULE OF THE WORKSHOP : Monday 20th : 9h30-10h30 : C. Robles 10h30-11h00 : Coffee break 11h-12h00 : L. Mihalcea 14h00-15h00 : A. Shapiro 15h00-16h00 : Coffee break 16h00-17h00 : T. Padurariu
Tuesday 21th : 9h30-10h30 : O. Kivinen 10h30-11h00 : Coffee break 11h-12h00 : A. Mellit 14h00-15h00 : A. Weber 15h00-16h00 : Coffee break 16h00-17h00 : S. Spenko
Wednesday 22th : 9h30-10h30 : R. Rimanyi 10h30-11h00 : Coffee break 11h-12h00 : M. Gorsky 13h30-14h30 : J. Eberhardt 14h30-15h00 : Coffee break 15h00-16h00 : A. Smirnov 16h15-17h15 : P. Bousseau
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