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source: SNS Channel Mathematical and Natural Sciences 2015年2月6日

Minicourses of the session "Lie Theory and Representation Theory" (2015)

Winter School of the University of Padova

http://www.crm.sns.it/event/323/

http://www.crm.sns.it/event/293/timetable.html?week=0#tim...

Planned Activities

This INDAM Intensive Period will be divided into three sessions on different topics, the timetable is available here

1) "Vertex algebras, W-algebras, and applications" from December 9th to December 23th 2014 and from January 12th to January 18th 2015 (seminars and mini-courses);

2) "Lie Theory and Representation Theory'' from January 19th to February 6th 2015 (seminars and mini-courses);

3) "Algebraic topology, geometric and combinatorial group theory'' from February 8th 2015 to February 28th 2015 (seminars and mini-courses).

The idea of the Intensive Period is to have some interesting mini courses for PhD students and researchers and regular talks, but also time for discussions and collaboration in the beautiful setting of the city of Pisa. A preliminary calendar of the mini-courses is available here.

There will be other special activities organized during the intensive period: A Cluster Algebras Day on 2th February 2015 A Super Quantum Lie Day on 4th February 2015. The workshop Combinatorics and Algebraic Topology of Configurations from 17th February to 20th February 2015. Two satellite workshop will be part of this INDAM Intensive Period: INdAM Meeting "Configuration spaces: Geometry, Topology and Representation Theory", in Cortona from August 31st to September 6th, 2014 (http://www.dm.unipi.it/~gaiffi/configurationscortona/); INdAM Workshop "Aspects of Lie Theory", in Rome from January 7th to January 10th, 2015 (http://www1.mat.uniroma1.it/people/bravi/indam2015.html).

Second session: Lie Theory and Representation Theory

The study of geometry and representations of affine Lie groups and algebras achieved an increasing importance in the last 30 years starting with the connection with modular representation theory (Lusztig conjecture) and to the study of vector bundles on curves (solution of the KP equation and proof of Verlinde's formula).

In certain cases the theory for affine algebras, even if presenting many more technical subtleties, follows closely the finite case. Indeed, it is possible to generalize some of the constructions and results which hold for semisimple Lie algebras: the theory of highest weight representations, topological properties of the singular locus of Schubert varieties in connection with Hecke algebras, the standard monomial theory applied to the description of the algebraic geometric aspects of the affine Grassmannian and Schubert varieties. Further aspects, while well-understood in the finite-dimensional case, turn out to give rise to new interesting phenomena, as for the Springer fibers. Finally, for this class of algebras completely new questions arise, such as the theory of critical level representations.

The theory of affine Lie algebras and affine Lie groups find applications to many subjects in Lie theory and algebraic geometry. We have already mentioned the relations with modular representation theory and the moduli space of vector bundles on curves, but they are also the central object in the geometric Langlands program, and they are intimately connected with the local models of Shimura varieties.

The intent of finding an algebraic proof for the Kazhdan-Lusztig conjecture on characters of irriducible modules for semisimple and affine Lie algebras, motivated Soergel conjecture. This consists in a purely algebraic statement, in terms of ranks of certain bimodules, which Soergel proved to imply the Kazhdan-Lusztig conjecture. Soergel's conjecture has been recently proven by Elias and Williamson, by adapting to this setting ideas from the work of de Cataldo and Migliorini and making use of categorification techniques.

Geordie Williamson, Lecture I - 19 January 2015 47:50 Geordie Williamson (Max-Planck-Institute, Bonn) - Lecture I http://www.crm.sns.it/course/4033/

This mini-course will be an introduction to perverse sheaves, with emphasis on examples from representation theory. It will be a course full of pictures and examples, with the aim of trying to get some feeling for the fundamentals of perverse sheaves: t-structures and gluing, intersection cohomology complexes, the decomposition theorem, vanishing and nearby cycles. As a grand finale I hope to cover de Cataldo and Migliorini's proof of the decomposition theorem. Although the most important concepts will be recalled, I will assume a basic knowledge of sheaves and derived categories. (If you have never worked with sheaves, cohomology or derived categories before the learning curve will be steep.)

Alexander Premet, Lecture I - 19 January 2015 1:45:03

Vera Serganova, Lecture I - 20 January 2015 1:47:32

Jing Song Huang, Research talk - 20 January 2015 47:50

Paul Levy, Research talk - 20 January 2015 51:03

Geordie Williamson, Lecture II - 21 January 2015 1:04:10

Alexander Premet, Lecture II - 21 January 2015 1:45:53

Serganova, Vera, Lecture II - 22 January 2015 1:02:21

Nicolas Libedinsky, Research talk - 22 January 2015 47:36

Geordie Williamson, Lecture III - 23 January 2015 1:48:51

Geordie Williamson, Lecture IV - 26 January 2015 1:50:04

Serganova, Vera, Lecture III - 26 January 2015 1:49:22

Alexander Premet, Lecture III - 27 January 2015 1:51:01

Peng Shan, Research talk - 27 January 2015 53:03

Geordie Williamson, Lecture V - 28 January 2015 1:38:41

Alexander Premet, Lecture IV - 29 January 2015 1:54:44

Maria Chlouveraki, Research talk - 29 January 2015 50:02

Geordie Williamson, Lecture VI - 30 January 2015 1:00:20

Oksana Yakimova, Research talk - 30 January 2015 50:19

Yann Palu, Research talk - 2 February 2015 41:26

Serganova, Vera, Lecture IV - 28 January 2015 1:55:40

Serganova, Vera, Lecture V - 3 February 2015 1:54:26

Stephane Gaussent, Research talk - 3 February 2015 50:04

Alexander Premet, Lecture V - 5 February 2015 1:47:53

Pierre-Guy Plamondon, Research talk - 2 February 2015 46:35

Alexander Premet, Lecture VI - 6 February 2015 1:54:26

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