Abstract:
In the recent years the study of geometric and analytic properties of the loop spaces on manifolds and the related theory of two-dimensional conformal fields greatly influenced many areas of mathematics.
In both theories we encounter "linear phenomena" that are hard to describe by the "standard" approach of topological vector spaces. The characteristic feature is the appearance of "vertex" phenomena, when for two "smooth families of operators" A(s) and B(t), the 2-parametric product family A(s)B(t) may have poles, e.g. on the hypersurface s=t.
We discuss an approach that is "dual" to the approach of topological linear algebra. Instead of (the covariant notion of) topology, the vector space is equipped with the (contravariant) notion of a smooth family of vectors, and a lattice of distinguished vector subspaces. It turns out that the geometry of loop spaces, the calculus of conformal fields, the language of vertex operator algebras, and the Beilinson-Drinfeld factorization condition may all be easily expressed in the language of linear algebra of such vector spaces - as long as the distinguished subspaces satisfy a certain "fatness" property.
This encapsulates all the counter-intuitive properties of the theories above into one notion of fatness. It turns out that while this notion has no finite-dimensional analogues, the classical analysis is full of examples of "fat" lattices, making it easier to study new phenomena via this approach.