Seminários de Lógica Matemática

19 de Março 2009, 18h00

Henson's Conjecture and stable Banach spaces
Complexo Interdisciplinar, Sala B3-01

ALEXANDER USVYATSOV (CMAF - Universidade de Lisboa)

Abstract:

I will speak about the proof of (possibly the first approximation to) Henson's Conjecture, stating that an uncountably categorical elementary class of Banach spaces is "very close" to the class of Hilbert spaces. The proof requires analysis of minimal types over stable Banach spaces. Although this is a talk in the logic seminar, it will not equire virtually any background in logic, and all the basic notions will be defined and explained.

A class of structures is called categorical in a certain cardinality if it has a unique structure (up to isomoprhism) of that cardinality. Typical examples are the class of algebraically closed fields of a fixed characteristic, or vector spaces over a fixed field. These classes are categorical in all uncountable cardinalities. The classical Morley's Theorem states that if a countable first order theory (or rather the class of its models) is categorical in some uncountable cardinality, then it is categorical in all such cardinalities. C. Ward Henson conjectured in 1970's that a similar result should hold for elementary classes of Banach spaces (cardinality should be replaced by density character) and that any such uncountably categorical class is "essentially" the class of Hilbert spaces. I will state an exact formulation of this conjecture and explain the proof strategy.

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