Mike Shulman, University of San Diego

Abstract: Topology is ubiquitous in mathematics: many naturally-occurring sets are actually non-discrete spaces, and it is more usual for functions to be continuous than not.  Why, then, do we base mathematics on set theory, whose basic objects have no topology?  (This question becomes more serious when formalizing mathematics in a computer, where proof obligations of continuity cannot be ignored.)  An alternative is to use a "synthetic" way of doing mathematics, where the basic objects are spaces and all functions are continuous by default.  Discontinuity is controlled by a form of modal logic, and becomes surprisingly powerful when combined with a synthetic form of homotopy theory.  I will sketch this approach to topology and some of the recent progress in the area, including relative consistency theorems showing that the synthetic theory can be interpreted into classical topology.