Gauge theory and representationscategorical representationscharacter theoriessurfaces motto theme. One such theme is the langlands philosophy, a vast nonabelian generalization of the fourier transform of classical harmonic analysis, which serves as a visionary roadmap for the subject and places it at the heart of number theory. The goal of this program is to bring international experts in these areas together to facilitate exchange and development of ideas. New connections of representation theory to algebraic. The kirillov paradigm for the heisenberg group 15 5. Suppose fab 0 where fx p n j0 a jx j with a n 1 and where a and b are relatively prime integers with b0. Prior to this there was some use of the ideas which we can now identify as representation theory characters of cyclic groups as used by. The conference will overview the role of representation theory in various disciplines such as automorphic forms, harmonic analysis, and invariant theory developed over the past half a century, present recent progress in this field, and finally aim to extrapolate it into future. We proceed by induction on the width of the support of f. Brief description of algebraic number theory, algebraic. Actually, as i was recently reminded, we also sometimes know how to do. This second volume contains expanded versions of lecture notes for o.
The spinor representation as we have seen, the groups spinn have a representation on rn given by identifying v. It uses matlab to teach the field s most important. Research seminar in geometric representation theory, symplectic geometry, mathematical physics, gromovwitten theory, integrable systems. Basic representation theory of reductive padic groups. One reason for their ubiquity is that they provide a useful way to organize data. This lecture is a brief introduction to the relationship between the algebraic geometry of flag varieties and the representation theory of reductive lie algebras and real reductive lie groups. The author would like to thank the organizers of this conference for the opportunity to make a presentation. 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 seminar who we are the fields institute is a centre for mathematical research activity a place where mathematicians from canada and abroad, from academia, business, industry and financial institutions, can come together to carry out. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. A conference on geometric representation theory, and related topics, 3rd7th july 2017. Algebraic number theory involves using techniques from mostly commutative algebra and. Algebraic geometry and representation theory seminar room 290c speaker. 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.
A representation of g is said to have finite length n if it admits a composition series of length n. Massachusetts, usa knots invariants and representation theory. Introduction to representation theory by pavel etingof. I am wondering whether someone can tell me how algebraic geometry enters the picture. The tools in this specialty include techniques from analysis for example, theta functions and computational number theory.
Part i the rst two lectures will be largely elementary and expository. Soergels modular category o is a kind of toy model for the representation theory of g. Geometric representation theory lecture 20 posted by john baez. The first conference, gauge theory and representation theory, will be held during the week of november 2630, 2007.
A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. I heard that today algebraic geometry plays some significant role in representation theory, which is a little surprising because when i learnt representation theory it is basically algebra, topology, differential geometry and a little functional analysis. They will deal with the upperhalfplane h and riemann sphere p1 from the points of view of hodge theory, representation theory and complex geometry. Geometric and categorical representation theory matrix. Geometric methods in representation theory springerlink.
In abstract algebra, a representation of an associative algebra is a module for that algebra. The seminar is jointly run by john baez and james dolan. Moduli spaces of sheaves on surfaces hecke correspondences and representation theory. In 1984, physicists knizhnik and zamolodchikov discovered a fundamental differential equation describing correlation functions in conformal field theory. This means that the problem of finding, say, ndimensional representations of a reduces to solving a bunch of nonlinear algebraic equations with. If the algebra is not unital, it may be made so in a standard way see the adjoint functors page. As already noted, in our framework, the existence of a good amodel for yis supposed to ensure that deformation quantization of yproduces an actual deformation of the algebra of holomorphic functions, depending on a complex parameter or 1n, not just a formal. These operatorvalued bessel functions arose grouptheoretically. Arithmetic algebraic geometry, the study of algebraic varieties over number fields, is also represented at lsu. Kostants theorem on polynomial rings 346 chapter 7. The main objects that we study in algebraic number theory are number. Cime activity is carried out with the collaboration and financial support of. They are based on the first authors notes from a lecture series delivered by the second author at the simons centre for geometry and physics in august 2019. A conformal map is a function which preserves angles locally.
Enter your mobile number or email address below and well send you a link to download the free kindle app. Geometric representation theory studies representations of various symmetry objects like algebraic groups, hecke algebras, quantum groups, quivers etc. The main goal of our conference is to gather leading specialists in representation theory as well as beginning researchers and advanced graduate students, to create a forum where scientists can exchange new ideas, communicate recent advances and assist younger participants in developing successful research strategies. Geometry is a powerful tool for extracting information from data sets, and a beautiful subject in its own right. The principal goal of the introductory workshop will be to provide a gateway for graduate students and new postdocs to the rich and exciting, but potentially daunting, world of geometric representation theory. These notes provide a concise introduction to the representation theory of reductive algebraic groups in positive characteristic, with an emphasis on lusztigs character formula and geometric representation theory. The point of view of these notes on the topic is to bring out the flavor that representation theory is an extension of the first course on group theory. Then you can start reading kindle books on your smartphone, tablet, or computer. There are powerful new tools and ideas at the forefront of geometric representation theory, particularly categorical incarnations of ideas from mathematical physics, algebraic, arithmetic and symplectic geometry, and topology. This book is an introduction to the representation theory of finite groups from an algebraic point of view, regarding representations as modules over the group algebra. This has been one of the fundamental paradigms of geometric representation. 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.
Representation theory authorstitles recent submissions arxiv. This book recounts the connections between multidimensional hypergeometric functions and representation theory. Representation theory of geometric extension algebras 5 proof. Achar louisiana state university modular category o and parity sheaves on flag varieties. Here an associative algebra is a not necessarily unital ring. The approach is to develop the requisite algebra in reasonable generality and then to specialize it to the case of group representations.
A key new development is the work of gelfand, graev, kapranov, and zelevinsky in the late 80s and early 90s which provided a unifying foundation for the theory of multivariate hypergeometric series. The representation theory of nite groups has a long history, going back to the 19th century and earlier. The focus of the year will be on related recent developments in representation theory, algebraic geometry and physics. These notes cover completely the theory over complex numbers which is character theory. The fundamental aims of geometric representation theory are to uncover the deeper geometric and categorical structures underlying the familiar objects of representation theory and harmonic analysis, and to apply the resulting insights to the resolution of classical. How important is representation theory to algebraic. I sometimes joke that the only thing mathematicians really know how to do is linear algebra, and we do everything else by trying to reduce harder questions to linear algebra. Borel subgroups, parabolic subgroups, the bruhat decomposition 10 4. For the hypergeometric function 2f 1 we have an integral representation due to euler. Gross and kunze 8, 9 developed a theory of operatorvalued bessel functions of matrix argument over real division algebras, which included as special cases the bessel functions considered by herz and james. This classic monograph provides an overview of modern advances in representation theory from a geometric standpoint. Let g be a semisimple, simply connected group over an algebraically closed field of characteristic p, where p is larger than the coxeter number for g. An affine version of robinsonschensted correspondence for kazhdanlusztig theory abstract. Schiffmans course, as well as ten research or survey articles, presented at the 2008 summer school, geometric methods in representation theory grenoble, france, june 16july 4, 2008.
In all these facets of algebraic geometry, the main focus is the interplay between the geometry and the algebra. If is a rational number which is also an algebraic integer, then 2 z. Lectures on the geometry and modular representation theory. Geometric and categorical representation theory are advancing rapidly, with a growing number of connections to the wider mathematical universe.
Geometric representation theory is a very active field, at the center of recent advances in number theory and theoretical physics. In the study of infinitedimensional representation theory. For any researcher in representation theory, algebraic or arithmetic geometry, this book will be a valuable resource. In this, the final lecture of the falls geometric representation theory seminar, i tried to wrap up by giving a correct statement of the fundamental theorem of hecke operators. Representation theory and complex geometry springerlink. This support is the set of integers isuch that hif 6 0, where hiis the cohomological functor associated with the usual tstructure.
Algebraic geometry and representation theory seminar the. Poisson structures arising from noncommutative algebras 26 1. In a book that will appeal to beginners and experts alike, oxford university s nick trefethen presents approximation theory using a fresh approach for this established field approximation theory and approximation practice is a textbook on classical polynomial and rational approximation theory for the twentyfirst century. Geometric function theory david benzvi dear cafe patrons, in this guest post i want to brie. Equivariant ktheory of the steinberg variety 311 6. Geometric representation theory in positive characteristic. Representation theory is gauge theory in low dimensions. In the most common case the function has a domain and range in the complex plane more formally, a map. Gwyn bellamy and ben davison some photos from the conference dinner are available here the goal of this conference is bring together experts in geometric representation theory and related topics, to hear about the latest developments in the field. Lie groups and the derivation of his character formula, alongside one of. Equivariant cohomology and divided differences 5 3.
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