Geometric representation theory of nite and padic groups. Representation theory resources and references representation theory of finite groups c. Recent progress in geometric langlands theory dennis gaitsgory abstract. Geometric representation theory, math 267y, fall 2005. Newest geometricrepresentationtheory questions feed subscribe to rss to subscribe to this rss feed, copy and paste this url into your rss reader. This is a glossary of representation theory in mathematics. Enumerative geometry and geometric representation theory. Moreover, we study the central degeneration the degeneration that shows up in local models of shimura varieties and gaitsgorys central sheaves of semiinfinite orbits, mirkovicvilonen mv cycles, and iwahori orbits in the. Gaitsgory, on ginzburgs lagrangian construction of representations of gln. Geometric representation theory, math 224, spring 2017.
Equivariant cohomology and divided differences 5 3. This book is an introduction to geometric representation theory. The latter takes as an input an algebraic curve xover a xed base eld kassumed algebraically closed and of characteristic 0 and a reductive group g. To this day, manifestations of langlands duality in geometric representation theory serve as a source of motivation to most problems that i study.
Verdier duality and geometric langlands correspondence geometric langlands equivalence is a still conjectural equivalence between the derived category of dmodules on the moduli stack of g. Such cocycles are crucial for applications in both geometry and, provocatively, for the elliptic frontier in representation theory. A unifying theme is the geometry of homogeneous complex manifolds. Introduction to geometric representation theory lectures presented at the workshop on geometry and integrability, university of melbourne, 615 feb, 2008 peter bouwknegt abstract. A geometric model for complex analytic equivariant elliptic cohomology a longstanding question in the study of elliptic cohomology or topological modular forms has been the search for geometric cocycles. In rozanskywitten theory on m, speczs1 is the thing we want. Preface the goal of this book is to present several central topics in geometric group theory,primarilyrelatedtothelargescalegeometryofin. Gaitsgory, geometric representation theory, math 267y, fall 2005. Rozenblyum, a study in derived algebraic geometry, web it develops some aspects of the theory of derived algebraic geometry with the categorical geometric langlands conjecture in mind see also. Harvard university department of mathematics one oxford street cambridge, ma 028, usa office. Geometric representation theory and beyond september 26 30. Written for the 2015 algebraic geometry summer institute. The workshop will be held in room g10 of the cmsa, located at 20 garden street, cambridge, ma. Localization and represention theory of reductive lie groups.
Representation theory, automorphic forms, and complex geometry. Geometric representation theory, math 224, spring 2017 dennis gaitsgory. Dennis gaitsgory and joseph bernstein during the school on automorphic froms in jerusalem, april 2001. Geometric representation theory and beyond september 26. Ams representation theory of the american mathematical society. In this unified field theory, called e8 theory, all forces and matter are described as the twisting of a single geometric object.
Conference on algebraic geometry, representation theory. A conference in honor of the 70th birthday of wilfried schmid. Recall that speczs2 only detects the a nization of m. Introduction to lie algebras and representation theory. One of the reasons for interest in geometric representation theory, of which the above is our first example, is that it produces representations with canonical bases. Geometric representation theory notes by dennis gaitsgory. Quick reminder on reductive groups and lie algebras, h1, ov, sp or j, section ii. It is according to professor hermann a readable book, so it would be appropriate for this plannedtobe reading course. These are the class s theories, and some of them have speci c names.
It develops some aspects of the theory of derived algebraic geometry with the categorical geometric langlands conjecture in mind. This winter quarter at northwestern im organising a reading seminar in geometric representation theory. An introduction to geometric group theory pristina matthieu dussaule mars 2017 this is a eight hours course that i gave at the university of pristina. These lectures are centered around the subjects of hodge theory and representation theory and their relationship.
The is the english version of the text of the talk at s eminaire bourbaki on february 16, 2016. Algebraic geometry and representation theory seminar the. Sengupta, notes on representations of algebras and finite groups. Verdier duality and geometric langlands correspondence geometric langlands equivalence is a still conjectural equivalence between the derived category of dmodules on the moduli stack of gbundles on a curve x the geometric side and a certain modification of the derived category of quasicoherent sheaves on the moduli. Representation theory is the study of the basic symmetries of. Geometric construction of the enveloping algebra uslnc 193 4. Newest geometric representation theory questions feed subscribe to rss to subscribe to this rss feed, copy and paste this url into your rss reader.
Dennis gaitsgory, geometric representation theory, 61 pp. In other words, id be curious to see something using geometry that is meatier than, say, just using geometric techniques to construct the exceptional isomorphisms between lowdimensional lie groups but something that can still be stated in the framework of classical representation theory unlike, say, the borelweil theorem and friends. Dennis gaitsgory, notes on geometric langlands, web d. One objective is to present, in a general context, some of the recent work of carayol c1, c2, c3. Before getting into the true geometric part of geometric group theory, lets give a look at the combinatorial. Applications of toric geometry to geometric representation theory.
Local geometric langlands correspondence and affine kac. Geometric representation theory studies representations of various. Reading seminar in geometric representation theory. The seminar is jointly run by john baez and james dolan. Gauge theory, geometric langlands and vertex operator algebras. Gaitsgory, geometric representation theory notes for a course on highest weight categories. Moreover, we study the central degeneration the degeneration that shows up in local models of shimura varieties and gaitsgorys central sheaves of semiin nite orbits, mirkovi c. We propose a conjectural description of these categories as categories of representations of the corresponding affine kacmoody. Ivan mirkovi\c, dmitriy rumynin, geometric representation theory of restricted lie algebras of classical type.
A geometricallyoriented treatment of the subject is very timely and has long been desired, especially since the discovery of dmodules in the early 1980s and the quiver approach to quantum groups in the early 1990s. Geometric representation theory, math 267y, fall 2005 dennis gaitsgory. Relation between two geometrically defined bases in. The basic idea is to extend grand unified theories and include gravity as part of a consistent geometric framework.
Our goal is to read and discuss some classic and important papers in geometric representation theory from the last forty years. Geometric representation theory lecture 2 the topos of all functors fop sets is the classifier for boolean algebras in that its points correspond uniquely to such algebras and the topos itself is not boolean. Download book pdf algebraic geometry and number theory pp 69260 cite as. Local geometric langlands correspondence and affine kacmoody.
These notes are a rough writeup of lectures presented at the workshop on geometry and integrability, held at the university of melbourne, 615 feb, 2008. But zs1 dcohm with its tensor product knows all of m where for now were ignoring the subtlety with the braided monoidal structure. Geometric representation theory of the hilbert schemes part i alexander tsymbaliuk abstract. Geometric representation theory seminar fall 2007 john baez and james dolan this fall, our seminar is tackling geometric representation theory the marvelous borderland where geometry, groupoid theory and logic merge into a single subject. These are lecture notes for a lecture series at the summer school on geometric representation theory and extended affine lie algebras held at university of ottawa in june 2009. For more see page 2 of this 5 page paper by lawvere. Newest geometricrepresentationtheory questions mathoverflow. This study at tempted to broaden the scope of models traditionally. One of the most important problems in geometric representation theory is the geometric langlands correspondence 5, 6, 26, 27. Dennis gaitsgory, notes on geometric representation theory. Geometric representation theory of the hilbert schemes part ii alexander tsymbaliuk abstract. Geometry and group theory abstract inthiscourse, wedevelopthebasicnotionsofmanifoldsandgeometry, withapplications in physics, and also we develop the basic notions of.
Diaconis, group representations in probability and statistics w. Conference on algebraic geometry, representation theory and. One of the main driving forces for geometric representation theory has been the representation theory of nite and padic reductive groups the groups obtained by taking the points of an algebraic group, such as the group of invertible matrices, over a nite or padic eld. These intersections have interesting geometric and topological properties, and are related to representation theory. I will survey a program that seeks to translate the aldaygaiottotachikawa correspondence between gauge theory on r4 and conformal field theory into the language of algebraic geometry. Moreover, we study the central degeneration the degeneration that shows up in local models of shimura varieties and gaitsgory s central sheaves of semiinfinite orbits, mirkovicvilonen mv cycles, and iwahori orbits in the.
Retrieve articles in representation theory of the american. Edward frenkel and dennis gaitsgory, local geometric langlands correspondence and affine kacmoody algebras, algebraic geometry and number theory, progr. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. Introduction throughout the talk we x xto be a smooth connected complete curve and ga reductive group over a ground eld k. From april 29 to may 1, 2019 the cmsa will be hosting a conference on algebraic geometry, representation theory and mathematical physics. This classic monograph provides an overview of modern advances in representation theory from a geometric standpoint. The cornerstone on which is built the presentation theory is the concept of free groups. We recall the classical action of the heisenberg algebra h on the usual and equivariant homology of the hilbert scheme of points on c2, due to grojnowskinakajima. Borel subgroups, parabolic subgroups, the bruhat decomposition 10 4. Algebraic dmodules and representation theory of semisimple lie groups. Geometric representation theory lecture 1 the ncategory cafe. This workshop is organized by bong lian brandeis and artan sheshmani cmsa. The geometric study of representation theory of kacmoody algebras via.
Pdf semiinfinite combinatorics in representation theory. William crawleyboeveys homepage notes for quivers, representation theory, etc. Geometric representation theory lecture 2 the ncategory cafe. Course notes by michel brion lecture notes for courses on algebraic geometry and representation theory. The objects of study become moduli spaces of sheaves on surfaces, and the goal is to. Gauge theory, geometric langlands and vertex operator. Nineteenthcentury representation theory helped to explain the structure of electron orbitals, and 1920s representation theory is at the heart of quantum chromodynamics. Representations corresponding to the bilinear form which is equal to minus one half of the. Algebraic geometry and representation theory seminarroom 155. In number theory, p adic representation theory is central the langlands program, a family of conjectures that have guided a large part of number theory for the past forty years. An introduction to geometric group theory pristina matthieu dussaule. The goal is to use geometric techniques to tackle questions in the langlands program in number theory.
Tree versus geometric representation of tests and items michal beller national institute for testing and evaluation, israel factoranalytic techniques and multidimensional scaling models are the traditional ways of representing the interrelations among tests and items. Reading seminar in geometric representation theory ihes. This is a lecture course for beginners on representation theory of semisimple finite dimensional. Representation theory online resources columbia university. Aug 20, 2005 by a local geometric langlands correspondence for a complex reductive group g we understand a construction which assigns to a local system on the punctured disc for the langlands dual group of g, a category equipped with an action of the formal loop group gt. Course on beilinsonbernstein theory dragan milicic. Gaitsgory, another realization of the category of modules over the small quantum group, adv. Download book pdf representation theory, complex analysis, and integral geometry pp 972 cite as. Applications of toric geometry to geometric representation. Ams representation theory of the american mathematical. Department website dennis gaitsgory harvard university. Dmodules on the affine flag variety and representations of affine kacmoody algebras. Representation theory and complex geometry springerlink.
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