Conference on Representations of Algebras and Related Topics, July 15 - August 10, 2002, Toronto, Canada

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

International Conference on Representations of Algebras and Related Topics, July 15 -- August 10, 2002, Toronto, Canada The PI and the organizing committee plan to provide support for the participation of as many junior US mathematicians as possible in the biennial international meeting in Toronto this summer. Next to the presentation of cutting-edge research, several series of overview lectures by leading experts and an extensive workshop program aim at introducing start-up researchers to the field. Central subjects are the geometry of module varieties, invariants and deformations, homological developments -- specifically, derived categories, representation type, including the in-depth exploration of pivotal classes of tame or wild algebras. Among the applications -- their thorough documentation is one of the highest priorities -- are the theories of complex semisimple Lie algebras and algebraic groups, singularities and categories of perverse sheaves, new insights into the structure of the Bernstein-Gelfand-Gelfand category, as well as the homological treatment of finite group schemes. The representation theory of algebras has its origin in the study of 'groups', originally certain collections of motions of space which allow for concatenation and inversion. The next idea was to represent more abstract algebraic objects, arising in theoretical physics and chemistry for instance, in terms of linear motions of suitable spaces, to render their structure more transparent. In the meantime, the scope of objects studied along this line has been dramatically enlarged, and a big arsenal of methods to analyze such representations has been assembled. Tools which have proved particularly effective include geometrical and combinatorial setups, and variations of a general theme which can be outlined as follows: One starts by establishing optimal situations, which can be completely analyzed, and then measures the deviation of a more challenging situation from the optimal ones to make headway in steps dealing with progressive increases in complexity. Applications extend to almost all parts of mathematics and provide the theoretical frames for models in theoretical physics. In fact, some of the recent initiatives were prompted by physicists and feed back into quantum physics.