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Presenter: Dr. Finnur Larusson - University of Western Ontario
Supervisor:

Date: Fri, September 21, 2001
Time: 15:00:00 - 16:00:00
Place: CLE A 211

ABSTRACT

Abstract homotopy theory takes place in a category satisfying a list of axioms due to Quillen, giving a reasonable notion of two arrows being homotopic. This encompasses ordinary homotopy theory of topological spaces and simplicial sets, homological algebra, and much more. Recent applications in arithmetic geometry have attracted much attention.

The Oka Principle is an important theme in complex analysis with a long history. Roughly speaking, it states that on a complex submanifold of Euclidean space, analytic problems of a cohomological nature have only topological obstructions. A famous theorem of Gromov is an instance of this, giving a sufficient condition for any continuous map between two complex manifolds to be homotopic to a holomorphic map.

The talk will give a brief, non-technical overview of these two topics and describe how Gromov's Oka Principle can be placed in an abstract homotopy-theoretic context and interpreted in purely holomorphic terms, without reference to continuous maps. To this end, we embed the category of complex manifolds into a Quillen category, where we can then do homotopy theory with them. The conclusion of Gromov's theorem turns out to be equivalent to a property called excision, which is familiar from topology and appears nowadays in algebraic geometry.

Refreshments will be served at 2:30.