Introduction to geometric representation theory lectures presented at the workshop on geometry and integrability, university of melbourne, 615 feb, 2008 peter bouwknegt abstract. Dennis gaitsgory, notes on geometric langlands, web d. We will cover about half of the book over the course of this semester. Identifying the sum of equivariant homology groups of c2n with the fock space, we interpret geometrically some important elements of the fock space. Course notes by michel brion lecture notes for courses on algebraic geometry and representation theory. 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. Ams representation theory of the american mathematical. This classic monograph provides an overview of modern advances in representation theory from a geometric standpoint. A unifying theme is the geometry of homogeneous complex manifolds. An introduction to geometric group theory pristina matthieu dussaule. Edward frenkel and dennis gaitsgory, local geometric langlands correspondence and affine kacmoody algebras, algebraic geometry and number theory, progr.
To this day, manifestations of langlands duality in geometric representation theory serve as a source of motivation to most problems that i study. Recall that speczs2 only detects the a nization of m. Geometric representation theory lecture 2 the ncategory cafe. Gauge theory, geometric langlands and vertex operator. Download book pdf algebraic geometry and number theory pp 69260 cite as. Gaitsgory, on ginzburgs lagrangian construction of representations of gln. Algebraic dmodules and representation theory of semisimple lie groups. Course on beilinsonbernstein theory dragan milicic. Recent progress in geometric langlands theory dennis gaitsgory abstract. The is the english version of the text of the talk at s eminaire bourbaki on february 16, 2016. Algebraic geometry and representation theory seminarroom 155.
Introduction to lie algebras and 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. Representation theory online resources columbia university. 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. Relation between two geometrically defined bases in. 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. Gaitsgory, poincarebirkhoffwitt theorem for quadratic algebras of koszul type. Representation theory is the study of the basic symmetries of. Representation theory, automorphic forms, and complex geometry. But zs1 dcohm with its tensor product knows all of m where for now were ignoring the subtlety with the braided monoidal structure. 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. Retrieve articles in representation theory of the american. 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.
These lectures are centered around the subjects of hodge theory and representation theory and their relationship. From april 29 to may 1, 2019 the cmsa will be hosting a conference on algebraic geometry, representation theory and mathematical physics. This is a lecture course for beginners on representation theory of semisimple finite dimensional. For more see page 2 of this 5 page paper by lawvere. Reading seminar in geometric representation theory. These are the class s theories, and some of them have speci c names. Geometry and group theory abstract inthiscourse, wedevelopthebasicnotionsofmanifoldsandgeometry, withapplications in physics, and also we develop the basic notions of. Nineteenthcentury representation theory helped to explain the structure of electron orbitals, and 1920s representation theory is at the heart of quantum chromodynamics. Newest geometricrepresentationtheory questions feed subscribe to rss to subscribe to this rss feed, copy and paste this url into your rss reader. Conference on algebraic geometry, representation theory. Conference on algebraic geometry, representation theory and.
Gauge theory, geometric langlands and vertex operator algebras. Localization and represention theory of reductive lie groups. The objects of study become moduli spaces of sheaves on surfaces, and the goal is to. Applications of toric geometry to 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. Pdf semiinfinite combinatorics in representation theory. Geometric representation theory of the hilbert schemes part ii alexander tsymbaliuk abstract. Preface the goal of this book is to present several central topics in geometric group theory,primarilyrelatedtothelargescalegeometryofin. Geometric representation theory, math 267y, fall 2005. Equivariant cohomology and divided differences 5 3. 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. A conference in honor of the 70th birthday of wilfried schmid. Diaconis, group representations in probability and statistics w. Geometric representation theory and beyond september 26.
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. 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. Geometric representation theory notes by dennis gaitsgory. Dennis gaitsgory and joseph bernstein during the school on automorphic froms in jerusalem, april 2001. Ivan mirkovi\c, dmitriy rumynin, geometric representation theory of restricted lie algebras of classical type. Harvard university department of mathematics one oxford street cambridge, ma 028, usa office. Newest geometric representation theory questions feed subscribe to rss to subscribe to this rss feed, copy and paste this url into your rss reader. 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. 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. Dmodules on the affine flag variety and representations of affine kacmoody algebras.
Local geometric langlands correspondence and affine kacmoody. 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. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. In rozanskywitten theory on m, speczs1 is the thing we want. 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. Borel subgroups, parabolic subgroups, the bruhat decomposition 10 4. Geometric representation theory, math 224, spring 2017. 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. It develops some aspects of the theory of derived algebraic geometry with the categorical geometric langlands conjecture in mind. 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. Geometric representation theory, math 224, spring 2017 dennis gaitsgory. Geometric representation theory of nite and padic groups.
These intersections have interesting geometric and topological properties, and are related to 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. 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. This is a glossary of representation theory in mathematics. Ams representation theory of the american mathematical society. Department website dennis gaitsgory harvard 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. One objective is to present, in a general context, some of the recent work of carayol c1, c2, c3. Geometric representation theory of the hilbert schemes part i alexander tsymbaliuk abstract. In this unified field theory, called e8 theory, all forces and matter are described as the twisting of a single geometric object. Gaitsgory, geometric representation theory, math 267y, fall 2005. The goal is to use geometric techniques to tackle questions in the langlands program in number theory. Written for the 2015 algebraic geometry summer institute. Newest geometricrepresentationtheory questions mathoverflow. We propose a conjectural description of these categories as categories of representations of the corresponding affine kacmoody. Local geometric langlands correspondence and affine kac.
The geometric study of representation theory of kacmoody algebras via. The basic idea is to extend grand unified theories and include gravity as part of a consistent geometric framework. Sengupta, notes on representations of algebras and finite groups. Geometric construction of the enveloping algebra uslnc 193 4. The seminar is jointly run by john baez and james dolan. Representation theory resources and references representation theory of finite groups c. This workshop is organized by bong lian brandeis and artan sheshmani cmsa. 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. Representation theory and complex geometry springerlink.
When t2 we get the topological twist of n 4 super yangmills. This study at tempted to broaden the scope of models traditionally. Before getting into the true geometric part of geometric group theory, lets give a look at the combinatorial. Gaitsgory, geometric representation theory notes for a course on highest weight categories. 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. Dennis gaitsgory, geometric representation theory, 61 pp. Geometric representation theory and beyond september 26 30. Gaitsgory, another realization of the category of modules over the small quantum group, adv. Lectures on geometric group theory cornelia drutu and michael kapovich. 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.
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. Representations corresponding to the bilinear form which is equal to minus one half of the. The workshop will be held in room g10 of the cmsa, located at 20 garden street, cambridge, ma. Enumerative geometry and geometric representation theory. Dennis gaitsgory, notes on geometric langlands, web. Our goal is to read and discuss some classic and important papers in geometric representation theory from the last forty years.
Applications of toric geometry to geometric representation. Reading seminar in geometric representation theory ihes. Introduction throughout the talk we x xto be a smooth connected complete curve and ga reductive group over a ground eld k. The cornerstone on which is built the presentation theory is the concept of free groups. Theory x will be conformal in the 2manifold but topological in the 4manifold. 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. Geometric representation theory lecture 1 the ncategory cafe. Geometric representation theory studies representations of various symmetry objects like algebraic groups, hecke algebras, quantum groups, quivers etc. Geometric representation theory, math 267y, fall 2005 dennis gaitsgory. This book is an introduction to geometric representation theory. This winter quarter at northwestern im organising a reading seminar in geometric representation theory.
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