# Student Topology

Organizers: Eric Kilgore & Nikhil Pandit

## Past Events

In this talk, we explore some properties and notable constructions of taut foliations. We give an overview of Palmeira’s theorem, which describes the topology of the foliation induced on the universal cover of a manifold by a taut foliation. We then discuss the notion of branching in the leaf…

The existence of a taut foliation on a three-manifold has significant topological and geometric consequences. In this talk, we will introduce foliations, including several examples, and define the notion of tautness (as well as several equivalent conditions). To illustrate the…

Abstract: One motivation for the theory of 3-manifold foliations is to generalize some constructions in 2 dimensions which greatly simplify the topology of surfaces. In this talk we will discuss this 2-dimensional theory. We will introduce laminations of surfaces and see how they can be used to…

We'll explore some applications of the Aityah-Singer index theorem (Applications to be determined).

We will generalize the Atiyah-Singer index theorem to the case of manifolds with boundary/cylindrical ends, where the operator near the boundary is of the form d/dt + A, with A non-degenerate self-adjoint. The formula for the index is the same as in the classical case, except we must add an…

We will discuss how Atiyah-Singer index theorem implies the Riemann-Roch theorem of complex manifolds, the Hirzebruch signature theorem of 4n-dimensional manifolds and the Rokhlin theorem of spin 4-manifolds.

Up to this point, we’ve covered the construction of the analytical index and compared it to the topological index defined via K-theory. This week, I'll explain how the topological index is computed via characteristic classes and compute some examples depending on available time.

This will be a continuation of last week’s talk about the index theorem. We will finish discussing Atiyah and Singer’s proof that the analytical index coincides with the topological index, and will introduce spaces of Fredholm operators as classifying spaces for K-theory.