Lecture (2007) /The 21st Century COE Program /Math of OCU(J)

The 21st Century COE Program

Constitution of wide-angle mathematical basis focused on knots

Department of Mathematics and Physics
Graduate School of Science
Osaka City University

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Lecture (2007)(2006）

COE21 Selected Topics I
Lecturer：	HARADA, Megumi （McMaster University）
Title：	An introduction to symplectic geometry
Date：	July 17(Tue.)〜July 20(Fri.), 2007 July 23(Mon.)〜July 27(Fri.), 2007
Time：	9:30〜11:30
Place：	Dept. of Mathematics, Sci. Bldg., 3040

Abstracts

The intent of this series of lectures is two-fold: in the first week, we will provide a quick overview of equivariant symplectic geometry, starting at the very beginning (i.e. with the definition of a symplectic structure).
In the second week, we will give a series of loosely connected expository overviews of some themes that consistently arise in current research in this field. The purpose is to familiarize the audience with the basic tools and language of the field.

Topics will include (time permitting): In the first week, we will discuss the definition of a symplectic structure, examples of symplectic manifolds, local normal forms, group actions, Hamiltonian actions, moment maps, symplectic quotients, and Delzant's construction of symplectic toric manifolds. In the second week, we will discuss equivariant cohomology and equivariant Morse theory, localization, moment graphs and GKM theory, Duistermaat-Heckman measure, Kirwan surjectivity. I hope to also discuss related quotient theories (e.g. Kahler and hyperKahler quotients), time permitting.

Lecture
Lecturer：	Taras Panov (Moscow State University)
Title：	Toric topology and complex cobordism
Date：	July 17(Tue.)〜July 20(Fri.), 2007
Time：	13：30〜15：30 （July　18（Wed.）　13：00〜14：00）
Place：	Dept. of Mathematics, Sci. Bldg., 3040

Abstracts

We plan to discuss how the ideas and methodology of Toric Topology can be applied to one of the classical subjects of algebraic topology:

finding nice representatives in complex cobordism classes.

Toric and quasitoric manifolds are the key players in the emerging field of Toric Topology, and they constitute a sufficiently wide class of stably complex manifolds to additively generate the whole complex cobordism ring.
In other words, every stably complex manifold is cobordant to a manifold with a nicely behaving torus action.

An informative setting for applications of toric topology to complex cobordism is provided by the combinatorial and convex-geometrical study of analogous polytopes. By way of application, we give an explicit construction of a quasitoric representative for every complex cobordism class as the quotient of a free torus action on a real quadratic complete intersection.
The latter is a yet another disguise of the moment-angle manifold, another familiar object of toric topology.

We suggest a systematic description for

omnioriented quasitoric manifolds in terms of combinatorial data, and explain the relationship with non-singular projective toric varieties (otherwise known as toric manifolds).

Lecture
Lecturer：	Alexander Premet (Manchester University)
Title：	Premet's Mini Course on W-Algebras and Modular Representations of Lie Algebras
Date：	September 10(Mon.), 12(Wed.), 14(Fri.), 2007
Time：	16:00〜17:30, 15:00〜17:30 (Sep. 14)
Place：	Dept. of Mathematics, Sci. Bldg., 3040

Abstracts

Abstract:

The course will be on finite W-algebras and their relationship with modular representations and primitive ideals in the characteristic zero case. This area is rapidly becoming very popular; people involved here (apart from physisists) are Arakawa, D'Andrea, DeConcini, DeSole, Kac, Brundan, Kleshchev, Losev, Premet and some others (especially those involved with the Yangians). There has been some important progress lately in this area and there are links with the modular theory as well.

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