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Wednesday, February 08, 2006

More material for a philosophy of real mathematics

I hope I didn't give the impression in the last post that I think there's little mathematical exposition to be had. There could always be more, and what there is could be better organised, but there are plenty of shiny pebbles lying about for us to pick up. From today's Derived categories of coherent sheaves by Tom Bridgeland:
In the usual approach to the study of algebraic varieties one focuses directly on geometric properties of the varieties in question. Thus one considers embedded curves, hyperplane sections, branched covers and so on. A more algebraic approach is to study the varieties indirectly via their (derived) categories of coherent sheaves.
So, two approaches to the study of varieties. Why would we prefer the latter?
Firstly, algebraic geometers have been attempting to understand string theory. The conformal field theory associated to a variety in string theory contains a huge amount of non-trivial information. However this information seems to be packaged in a categorical way rather than in directly geometric terms.
For a summary of one group's approach to the category theoretical formulation of conformal field theory, read Categorification and correlation functions in conformal field theory.
A second motivation to study varieties via their sheaves is that this approach is expected to generalise more easily to non-commutative varieties. Although the definition of such objects is not yet clear, there are many interesting examples. In general non-commutative objects have no points in the usual sense, so that direct geometrical methods do not apply.

This points us to the non-commutative geometry programme (or programmes). A philosopher of geometry wanting to understand the relationship between Connes' and Grothendieck's visions of space might begin with Pierre Cartier's A Mad Day's Work.

A third reason is that recent results leave the impression that categorical methods enable one to obtain a truer description of certain varieties than current geometric techniques allow. For example many equivalences relating the derived categories of pairs of varieties are now known to exist. Any such equivalence points to a close relationship between the two varieties in question, and these relationships are often impossible to describe by other methods. Similarly, some varieties have been found to have interesting groups of derived autoequivalences, implying the existence of symmetries associated to the variety that are not visible in the geometry.

This points us to the theme of mathematical reality.

Four very rich paragraphs.


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