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See also the

GRASP Lecture Notes Page,

containing a large and growing collection of lecture notes, with a strong emphasis on geometric Langlands topics. In particular this collection includes (close to) complete notes from Geometric Langlands conferences in Vienna (2007), Luminy (2006), and Berkeley (2003).

My Selected Lectures:

Lecture notes and videos from the
2007 London Math Society Invited Lecture Series are available here.

Topological Field Theory and Geometric Langlands: notes and audio and video from a lecture series at the KITP Santa Barbara workshop on Geometric Langlands and Gauge Theory, July 2009.

Introduction to Geometric Langlands, notes from lecture at the Vienna workshop on Geometric Langlands and Physics, January 2007.

Lectures from the Gottingen Winterschule on Geometric Langlands: Geometric function theory, Geometric Langlands and Topological Field Theory and Geometric Langlands and Real Groups.

A three-part introduction to the geometric Langlands program I gave in 2002 is available on streaming video courtesy of MSRI: Geometric Class Field Theory, Quantization of Hitchin's Hamiltonians, and Factorization.

The February 2005 Talbot workshop (for which I was the plenary speaker) provided an intense "Geometric Langlands retreat" for graduate students. There are notes available as one huge file or as individual pages courtesy of Megumi Harada.

What's the deal with Geometric Langlands?: lecture notes taken by Parker Lowrey for a talk I gave at the Algebraic Geometry Boot Camp, a warmup workshop for graduate students attending the 2005 AMS Summer Institute in Algebraic Geometry, Seattle.

The Geometric Langlands Program, audio for a lecture I gave at the Workshop on Forms of Homotopy Theory: Elliptic Cohomology and Loop Spaces at the Fields Institute in Toronto, 2004.

Other Geometric Langlands Resources:

The Northwestern Geometric Langlands Page and Chicago Geometric Langlands Page contain many useful links and references.

Here are links to the ICM addresses of Bezrukavnikov and Braverman, closely related to geometric Langlands. Gerard Laumon's Bourbaki talk Travaux de Frenkel, Gaitsgory et Vilonen sur la correspondance de Drinfeld-Langlands surveys the proof of the geometric Langlands conjecture for GL_n.

Edward Frenkel (arxiv search) has written several excellent survey articles on aspects of the geometric Langlands program:

Here are other (also related) surveys by Frenkel:
Here are files of two works by A. Beilinson and V. Drinfeld (posted with the authors' permission). (Most of these are g-zipped postscript - to read these, apply "gunzip (filename)" to obtain a postscript file.) Chiral Algebras (version: 9/00 --- far from the published final version), and Quantization of Hitchin's Integrable System and Hecke Eigensheaves (uncompressed pdf version), Quantization of Hitchin's Integrable System and Hecke Eigensheaves (gzipped ps version) (version: 2/00) These files are quite large - to get 100 pages at a time, download the following: Hitchin Pages 1-100, Hitchin Pages 101-200, Hitchin Pages 201-300, and Hitchin Pages 301-384. Chiral Pages 1-100, Chiral Pages 101-200, and Chiral Pages 201-273.

Geometric Langlands on TV:

In Episode 11 of Season 4 of Fox's "24", we find Drinfeld modules, the central construction in the solution of the Langlands program over function fields (by Drinfeld for GL2 and Lafforgue for GLn), in use by the Counter Terrorist Unit, spearheaded by Kiefer Sutherland's character Jack Bauer. One of the terrorists explains to his boss about CTU foiling their efforts: "Using a Drinfeld module they've already shut down over 90 reactors" (see a blog review). Interestingly, an online transcript of the show differs from the script and subtitles, and mentions instead a "dreen-filled mode tool" (see an unofficial transcript). Disclaimer: To the extent of my knowledge, the relevance of Drinfeld modules to national security is an original contribution of "24".

David Goss kindly provided the backstory for this mysterious appearance of Drinfeld modules: "I grew up in the Detroit area with a number of great people who have gone on to great things; some of them in showbiz. My dear friend Michael Loceff is one such. He and I studied math together at Michigan. After a few years Michael lost interest in his studies and went on to other things. He teaches computer science over the web at Foothill college using software he wrote. Michael's cousin is Joel Surnow and Michael started to write with Joel on "La femme Nikita" and now Michael is an executive producer and write for 24. So you can imagine where the mention of Drinfeld modules arose! ... Finally, Michael likes to use names of people he grew up with in the show. In one episode at the beginning of season 2 or 3 (I never saw it) there is a dead drug dealer named "David Goss" and Jack Bauer goes around asking people if they knew this drug dealer.... One has to have a sense of humor."

Assorted introductions and surveys on related topics:

(Please send suggestions to add!) See also Dennis Gaitsgory's suggested background readings.


-Angelo Vistoli: Notes on Grothendieck topologies, fibered categories and descent theory
-Herb Clemens, Aaron Bertram et al. Park City Math Institute Notes on Stacks
-Tomas Gomez: Algebraic stacks
-Barabara Fantechi: Stacks for Everybody (Park City 2001 proceedings)
-Bertrand Toen: Course on stacks
-William Fulton: Introduction to Stacks (from a long-term book project)
-Claudia Centazzo and Enrico Vitale: Sheaf Theory (from a topos theoretic point of view).


-Joseph Bernstein: Course on D-modules. (The classic reference.)
-Dragan Milicic: Lectures on Algebraic Theory of D-Modules
-Philippe Maisonobe and Claude Sabbah: Aspects of the theory of D-modules
-Jean-Pierre Schneiders: An introduction to D-modules

Geometric Representation Theory:

-Dennis Gaitsgory: Course on Beilinson-Bernstein theory
-Dragan Milicic: Algebraic D-modules and representation theory of semisimple Lie groups. (Overview article)
-Dragan Milicic: Localization and represention theory of reductive Lie groups. (Book.)
-Wolfgang Soergel: Gradings on Representation Categories (ICM address)
-Kari Vilonen: Geometric Methods in Representation Theory (Park City lectures). See also his ICM address Topological Methods in Representation Theory.
-Matvei Libine and Wilfried Schmid, Geometric Methods in Representation Theory.
-Vernon Bolton and Wilfried Schmid, Discrete Series.
-Hiraku Nakajima: Geometric construction of representations of affine algebas (ICM address).

Perverse Sheaves:

-Konstanze Rietsch: An introduction to perverse sheaves.
-David Massey: Notes on perverse sheaves and vanishing cycles.
-David Massey: Stratified Morse theory: past and present.

Derived Categories, Model Categories etc:

-Bernhard Keller: On Differential Graded Categories (ICM address).
-Alexei Bondal and Dimitry Orlov: Derived categories of coherent sheaves (ICM address).
-Andrei Caldararu: Derived categories of sheaves: a skimming.
-Richard Thomas: Derived Categories for the Working Mathematician
-Paul Goerss and Kristen Schemmerhorn: Model Categories and Simplicial Methods
-Henning Krause: Derived categories, resolutions and Brown representability (see also the exercises)
-John Greenlees: Spectra for commutative algebraists
-Joseph Lipman: Notes on Derived Categories and Derived Functors
-Charles Weibel: History of homological algebra
-Ivan Mirkovic: course on homological algebra (see also his course on algebraic geometry)
-Pierre Schapira: Courses on categories and homological algebra and algebraic topology

Moduli of Bundles:

-Christoph Sorger: ICTP Lectures on moduli of principal G-bundles over algebraic curves.
-Gerd Faltings: Vector Bundles on Curves (course notes).
-Ron Donagi and Eyal Markman: Spectral curves, algebraically completely integrable Hamiltonian systems, and moduli of bundles.

Hecke Algebras:

-Tom Haines, Robert Kottwitz and Amritanshu Prasad: Iwahori-Hecke algebras.
-Victor Ginzburg: Geometric methods in representation theory of Hecke algebras and quantum groups


-Alexandre Stefanov maintains an excellent collection of links to online textbooks in math, see here.
-Igor Dolgachev's lecture notes page has excellent courses on physics and string theory, invariant theory, and algebraic geometry.
-Franz Lemmermeyer's lecture notes page and course notes page host a large collection of links to courses and notes, mainly related to algebraic geometry and number theory.

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Up: David Ben-Zvi

Department of Mathematics
University of Texas, Austin
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Up: University of Texas, Austin Mathematics Department
Created: March 9 2005 --- Last modified: May 28 2006