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Lie Groups and Representation Theory Seminar

[ Seminar 2015 | Past Seminars | Related conferences etc. ]

Upcoming talks

Place: Graduate School of Mathematical Sciences, the University of Tokyo [ Access ]
Intensive lectures (集中講義)
Date: July 7 & 14, 2015, 15:00-16:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Michael Pevzner (Reims Univeristy)
Title: Different aspects of Rankin-Cohen brackets
Abstract: Rankin-Cohen brackets form an infinite family of bi-differential operators having a surprisingly rich internal structure. We shall illustrate some of its manifestations in the first lecture and give some explanations in the second one.

Lecture 1. First examples: differential operations on modular forms and quantization of the one-sheeted hyperboloid.

Lecture 2. Rankin-Cohen brackets as symmetry breaking operators: efficiency of the F-method.

Intensive lectures (集中講義)
Date:July 15-17, 23 & 24, 2015, 15:00-16:30
Place: Room 122 (July 15, 16, 23); Room 126 (July 24); Room 128 (July 17), Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Paul Baum (Penn State Univeristy)
Title: INDEX THEORY AND K-HOMOLOGY
Abstract: This series of five lectures will prove the Atiyah-Singer index theorem as a corollary of Bott periodicity, and then give an exposition of some further developments.

Lecture 1. Dirac Operator.
The Dirac operator of R^n will be constructed. Spin-c manifolds will be introduced.

Lecture 2. Atiyah-Singer Revisited.
First, some classical low-dimensional examples of the Atiyah-Singer theorem will be considered. Next, the Atiyah-Singer theorem for elliptic differential (or pseuo-differential) operators on closed smooth manifolds will be proved as a corollary of Bott periodicity.

Lecture 3. What is K-homology?
K-homology is the dual theory to K-theory. K-homology can be defined in three ways : via homotopy theory, via K-cycles, and (following Atiyah and Kasparov) via functional analysis. The lecture will give the three definitions and will explain why they are equivalent.

Lecture 4. Beyond Ellipticity.
K-homology will be used to prove an index theorem for a naturally arising class of hypoelliptic (but not elliptic) differential operators.

Lecture 5. The Riemann-Roch Theorem.
The Grothendieck-Riemann-Roch (GRR) theorem will be reviewed. K-homology will be used to extend GRR to projective algebraic varieties which may have singularities.

Date: July 14 (Tue), 2015, 17:00-18:30
Place: Room 122, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Paul Baum (Penn State Univeristy)
Title: MORITA EQUIVALENCE REVISITED
Abstract:
[ pdf ]
Let X be a complex affine variety and k its coordinate algebra. A k- algebra is an algebra A over the complex numbers which is a k-module (with an evident compatibility between the algebra structure of A and the k-module structure of A). A is not required to have a unit. A k-algebra A is of finite type if as a k-module A is finitely generated. This talk will review Morita equivalence for k-algebras and will then introduce --- for finite type k-algebras ---a weakening of Morita equivalence called geometric equivalence. The new equivalence relation preserves the primitive ideal space (i.e. the set of isomorphism classes of irreducible A-modules) and the periodic cyclic homology of A. However, the new equivalence relation permits a tearing apart of strata in the primitive ideal space which is not allowed by Morita equivalence.

Let G be a connected split reductive p-adic group, The ABPS (Aubert- Baum-Plymen-Solleveld) conjecture states that the finite type algebra which Bernstein assigns to any given Bernstein component in the smooth dual of G, is geometrically equivalent to the coordinate algebra of the associated extended quotient. The second talk will give an exposition of the ABPS conjecture.

Date: July 21 (Tue), 2015, 15:30-16:30
Place: Room 122, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Toshiaki Hattori (服部俊昭) (Tokyo Institute of Technology)
Title: Shimizu's lemma for SL(3,R)/SO(3) (清水の補題のSL(3,R)/SO(3)の場合への拡張について)
Abstract:
[ pdf ]
We find a generalization of Shimizu's lemma in the case of the symmetric space SL(3, R)/SO(3) of noncompact type of rank 2. We also find a relation between this lemma and displacement of horoballs by isometries.

PSL(2,C)の部分群の離散性に関する必要条件である清水の補題, Jorgensenの不等式を双曲空間から他の階数1の対称空間の場合 に拡張しようという研究が現在進行中であるが, 高階の対称空間 についてそのような結果はまだないようである。階数が2の対称 空間で最も簡単なSL(3,R)/SO(3)の場合に清水の補題を拡張する 試みについてお話しする。

Date: July 21 (Tue), 2015, 17:00-18:30
Place: Room 122, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Paul Baum (Penn State Univeristy)
Title: GEOMETRIC STRUCTURE IN SMOOTH DUAL
Abstract:
[ pdf ]
Let G be a connected split reductive p-adic group. Examples are GL(n, F) , SL(n, F) , SO(n, F) , Sp(2n, F) , PGL(n, F) where n can be any positive integer and F can be any finite extension of the field Q_p of p-adic numbers. The smooth (or admissible) dual of G is the set of equivalence classes of smooth irreducible representations of G. This talk will first review the theory of the Bernstein center. According to this theory, the smooth dual of G is the disjoint union of subsets known as the Bernstein components. The talk will then explain the ABPS (Aubert-Baum-Plymen-Solleveld) conjecture which states that each Bernstein component is a complex affine variety. Each of these complex affine varieties is explicitly identified as the extended quotient associated to the given Bernstein component.

The ABPS conjecture has been proved for GL(n, F), SO(n, F), and Sp(2n, F).

Date: July 28 (Tue), 2015, 17:00-18:30
Place: Room 122, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Fabian Januszewski (Karlsruhe Institute of Technology (KIT))
Title: On (g,K)-modules over arbitrary fields and applications to special values of L-functions
Abstract:
[ pdf ]
I will introduce g,K-modules over arbitrary fields of characteristic 0 and discuss their fundamental properties and constructions, including Zuckerman functors. This may be applied to produce models of certain standard modules over number fields, which has applications to special values of automorphic L-functions, and also furnishes the space of regular algebraic cusp forms of GL(n) with a natural global Q-structure.

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© Toshiyuki Kobayashi