Georgia Institute of Technology College of Sciences

This is a continuation of the previous talk.

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.

We will discuss Etnyre and Honda's proof of the classification of Legendrian positive torus knots in the tight contact 3-sphere up to Legendrian isotopy by using the tools from convex surface theory.

In this talk, we will discuss a result due to Gabai which states that a minimal genus Seifert surface for a knot in 3-sphere can be realized as a leaf of a taut foliation of the knot complement. We will give a fairly detailed outline of the proof. In the process, we will learn how to construct taut foliations on knot complements.