Study group: the geometric Langlands correspondence

This group is devoted to the study of the geometric Langlands correspondence. The aim is to study the relation with mirror symmetry and gauge theory established in the recent work of, among others, Witten with Kapustin and Gukov.

The meetings are announced by email. If you would like to be on our e-mailing list, please send an email to one of the organizers.

You can click below to obtain information about the meetings, and then click on the lecture titles to download the corresponding slides.

Meetings

• 17-26 March 2010: School on Geometric Langlands and Gauge Theory at the CRM Barcelona

  Second International School on Geometry and Physics Speakers: David Ben-Zvi (University of Texas at Austin, USA): Gauge theory and representation theory Olivier Biquard (Institute de Mathématique Jussieu, Paris): An introduction to Higgs bundles Tamás Hausel (University of Oxford, UK): Mirror symmetry, Langlands duality and the Hitchin system Anton Kapustin (California Institute of Technology, USA): Langlands duality and topological field theory Tony Pantev (University of Pennsylvania, USA): Lectures on the geometric Langlands conjecture and non-abelian Hodge theory Group picture at the CRM and in Poblet Schedule First week Wednesday 17 March Thursday 18 March Friday 19 March Saturday 20 March Sunday 21 March 10:30–11:30 O. Biquard O. Biquard Outing day (tourism) with calçotada (typical meal) Free day 11:30–12:00 Registration tea/coffee tea/coffee 12:00–13:00 A. Kapustin T. Hausel Lunch Lunch Lunch 15:30–16:30 A. Kapustin O. Biquard A. Kapustin 16:30–17:00 tea/coffee tea/coffee Free Afternoon 17:00–18:00 O. Biquard T. Hausel Second week Monday 22 March Tuesday 23 March Wednesday 24 March Thursday 25 March Friday 26 March 10:30–11:30 O. Biquard D. Ben-Zvi A. Kapustin D. Ben-Zvi D. Ben-Zvi 11:30–12:00 tea/coffee tea/coffee tea/coffee tea/coffee tea/coffee 12:00–13:00 T. Pantev T. Pantev T. Pantev T. Hausel T. Hausel Lunch Lunch Free afternoon Lunch End of School 15:30–16:30 D. Ben-Zvi A. Kapustin T. Pantev 16:30–17:00 tea/coffee tea/coffee tea/coffee 17:00–18:00 T. Pantev T. Hausel D. Ben-Zvi Social Dinner: Tuesday, 23 March, 21:00, at Restaurant Provença in Barcelona. Group picture: Tuesday, March 23, before the morning coffee break. List of participants Poster Sponsors Ingenio MATHEMATICA (ICMAT node), a CONSOLIDER research project of the Spanish Ministerio de Ciencia e Innovación Interactions of Low-Dimensional Topology and Geometry (ITGP), a Research Network of the European Science Foundation Institut de Matemàtica (Universitat de Barcelona) Facultat de Matemàtiques (Universitat de Barcelona) Centre de Recerca Matemàtica Consejo Superior de Investigaciones Científicas Location: Auditori, Centre de Recerca Matemàtica, Campus Universitat Autònoma de Barcelona Website at the CRM

• Thursday 14 January 2010: CRM Barcelona

  11:00 - 12:00 Aritmética de cuerpos locales Artur Travesa (UB) 12:00 - 12:30 COFFEE BREAK 12:30 - 13:30 Aritmética de cuerpos globales (característica cero) Joan Nualart (UB) FREE TIME 15:30 - 16:30 Funciones L aritméticas Pilar Bayer (UB) 16:30 - 17:00 COFFEE BREAK 17:00 - 18:00 Formas automorfas y correspondencia de Langlands para GL(n) Luis Dieulefait (UB) Location: Aula C1/028, Centre de Recerca Matemàtica, Campus Universitat Autònoma de Barcelona Website at the CRM

• Thursday 12 November 2009: ICMAT Madrid

  11:00 - 12:00 Hitchin system and rigid orbits Olivier Biquard (Paris 6) [Abstract] Gukov and Witten observed that there are problems to understand Langlands duality for certain unipotent tame singularities. We shall try to understand this phenomenon from the point of view of Hitchin's system. 12:00 - 12:30 BREAK 12:30 - 13:30 On the geometric Satake isomorphism Ignasi Mundet i Riera (Univ. Barcelona) [Abstract] Satake's isomorphism, which roughly speaking relates the Hecke algebra and the representation algebra of Langlands dual groups, plays a crucial role in the statement of Langland's correspondence for arbitrary reductive Lie groups. We will explain Satake's isomorphism and its geometric version due to Mirkovic and Vilonen. FREE TIME 16:00 - 17:00 Mirror symmetry, Langlands duality and the Hitchin system Luis Álvarez-Cónsul (ICMAT / CSIC) [Abstract] This is an overview of work of T. Hausel and M. Thaddeus (arXiv:math/0205236) on two moduli spaces of twisted local systems on a compact Riemann surface, with structure groups SLr and its Langlands dual PGLr. Using their Hitchin integrable systems, they show that these spaces exhibit mirror symmetry phenomena: first, they satisfy the prescription of Strominger, Yau and Zaslow for mirror Calabi-Yau partners, and second, their (stringy mixed) Hodge numbers (with coefficient system provided by a flat unitary gerbe) are equal. Location: The two morning lectures will take place in the Sala de Juntas (ground floor), in Serrano 121, Madrid. The afternoon lecture will take place in the Sala de Juntas (second floor), in Serrano 113bis, Madrid. These buildings are situated in the main campus of Consejo Superior de Investigaciones Científicas in Madrid. Here are: a map of the main campus of CSIC, a map #1 and a map #2 (bigger) to get to the CSIC main campus. The nearest Metro station is 'República Argentina' (exit 'Juan de la Cierva'), in the Metro line number 6 (grey line). Other Metro stations nearby are `Nuevos Ministerios', `Gregorio Marañón' and `Avenida de América'. Bus routes 16, 19 and 51 each have a stop close to the building. The nearest local train station (Cercanías) is `Nuevos Ministerios'.

• Friday 16 October 2009: ICMAT Madrid

  11:00 - 12:00 The Hitchin system and Langlands duality Óscar García-Prada (ICMAT / CSIC) [Abstract] We will explain the Hitchin integrable system for a complex semisimple Lie group G and its relation to the Hitchin integrable system of the Langlands dual of G. 12:00 - 12:30 BREAK 12:30 - 13:30 Geometric Langlands conjecture as an equivalence of derived categories Tomás Gómez (ICMAT / CSIC) [Abstract] The Geometric Langlands conjecture of Beilinson and Drinfeld attaches, to each LG-local system on X, a D-module on the moduli of G-bundles on a curve X (the Hecke eigensheaf). This is expected to induce an equivalence of categories between the derived category of O-modules on the space of LG-local systems to the derived category of D-modules on the moduli of G-bundles (sending the skyscreaper over a local system to the corresponding Hecke eighensheaf). This equivalence has a "classical limit", from which we deduce that the fibers of the Hitchin integrable system corresponding to the groups G and LG should be dual. FREE TIME 16:00 - 17:00 Hecke correspondence José Ignacio Burgos (ICMAT / CSIC) [Abstract] In this talk we will give an introduction to classical Hecke correspondence from the arithmetic and geometric view point and study some of its properties. Location: Salón de Actos (ground floor), Serrano 119 (Edificio Rocasolano), Madrid. This building is situated in the main campus of Consejo Superior de Investigaciones Científicas in Madrid. Here are: a map of the main campus of CSIC, a map #1 and a map #2 (bigger) to get to the CSIC main campus. The nearest Metro station is 'República Argentina' (exit 'Juan de la Cierva'), in the Metro line number 6 (grey line). Other Metro stations nearby are `Nuevos Ministerios', `Gregorio Marañón' and `Avenida de América'. Bus routes 16, 19 and 51 each have a stop close to the building. The nearest local train station (Cercanías) is `Nuevos Ministerios'.

• Thursday 03 September 2009: CRM Barcelona

  11:00 - 12:00 The Hitchin integrable system Óscar García-Prada (ICMAT / CSIC) 12:00 - 12:30 COFFEE BREAK 12:30 - 13:30 An introduction to the Langlands correspondence for GLn over function fields Jochen Heinloth (Univ. Amsterdam) FREE TIME 15:30 - 16:30 S-duality and Geometric Langlands Luis Álvarez-Cónsul (ICMAT / CSIC) 16:30 - 17:00 COFFEE BREAK 17:00 - 18:00 Some ideas of the proofs of Drinfeld and Lafforgue Jochen Heinloth (Univ. Amsterdam) Location: Aula C1/028, Centre de Recerca Matemàtica, Campus Universitat Autònoma de Barcelona Website at the CRM

• Friday 29 May 2009: CRM Barcelona

  10:30 - 11:30 Introduction to algebraic stacks (article) Tomás Gómez (ICMAT / CSIC) 11:30 - 12:00 COFFEE BREAK 12:00 - 13:00 Introduction to D-modules José Ignacio Burgos (Univ. Barcelona) FREE TIME 15:30 - 16:30 Basic notions on the geometric theory of representations Ignasi Mundet i Riera (Univ. Barcelona) 16:30 - 17:00 COFFEE BREAK 17:00 - 18:00 The work of Beilinson and Drinfeld in the Langlands programme (article) Tomás Gómez (ICMAT / CSIC) Location: Aula C1/028, Centre de Recerca Matemàtica, Campus Universitat Autònoma de Barcelona Website at the CRM

Organizers

• Luis Álvarez-Cónsul (ICMAT / CSIC Madrid)
• José Ignacio Burgos (ICMAT / CSIC Madrid)
• Óscar García-Prada (ICMAT / CSIC Madrid)
• Ignasi Mundet i Riera (Universitat de Barcelona)

Links

• Autumn School "Towards a p-adic Langlands Correspondence". Seville, 26-30 October 2009.
• Research Programme in Arithmetic Geometry. CRM, Barcelona, Sept 2009 - July 2010.

[Last updated 01 Jul 2010]