Smooth infinitesimal analysis

Geometry Topology Student Seminar
Friday, May 2, 2014 - 14:00
1 hour (actually 50 minutes)
Skiles 006
Georgia Tech

This is a final project for Dr. Etnyre's Differential Geometry class.

After briefly considering embeddings of the category of smooth manifolds into so called smooth toposes and arguing that we may ignore the details of the embedding and work from axioms if we agree to use intuitionistic logic, we consider axiomatic synthetic differential geometry. Key players are a space R playing the role of the "real line" and a space D consisting of null-square infinitesimals such that every function from D to R is "microlinear". We then define microlinear spaces and translate many definitions from differential geometry to this setting. As an illustration of the ideas, we prove Stokes' theorem. Time permitting, we show how synthetic differential geometry may be considered as an extension of differential geometry in that theorems proven in the synthetic setting may be "pulled back" to theorems about smooth manifolds.