# The "dual" way of doing geometry

## Tomasz Miller

The talk will be about some weird ways of doing geometry and physics. In a first course on topology and differential geometry, one constructs geometrical objects by endowing a “base” set X with an appropriate geometrical structure. Only then one might start analysing some other things living on X, for example the commutative algebras of smooth functions C∞(X) or of continuous functions C(X). It turns out, however, that one can do geometry the other way around, that is, begin with a certain commutative algebra A and then interpret it as an algebra of functions on a set X which itself is already “encoded” in A. In a sense, this “dual” approach makes geometry a part of algebra and might have far-reaching consequences in physics. Moreover, the latter approach allows an interesting generalization. Namely, by allowing A to be a noncommutative algebra, we enter the field of noncommutative geometry, where spaces no longer consist of points and where the key to better understanding of the quantum world might lie.