Georgia Institute of Technology College of Sciences

Understanding the stable homotopy groups of spheres is one of the great challenges of algebraic topology. They form a ring which, despite its simple definition, carries an amazing amount of structure. A famous theorem of Hopkins and Ravenel states that it is filtered by simpler rings called the chromatic layers. This point of view organizes the homotopy groups into periodic families and reveals patterns. There are many structural conjectures about the chromatic filtration.

We review the basics of hyperbolic geometry in preparation for studying mapping class groups.

In this work, a novel approach is used to study geometric properties of the indicatrix bundle and the natural foliations on the tangent bundle of a Finsler manifold. By using this approach, one can find the necessary and sufficient conditions on the Finsler manifold (M; F) in order that its indicatrix bundle has the Sasakian structure.

The Steenrod algebra consists of all natural transformations of cohomology over a prime field. I will present work of Milnor showing that the Steenrod algebra also has a natural coalgebra structure and giving an explicit description of the dual algebra.

Harer's homology stability theorem states that the homology of the mapping class group for oriented surfaces of genus g with n boundary components is independent of g for low degrees, increasing with g. Therefore the (co)homology of the mapping class group stabilizes. In this talk, we present Tillmann's result that the classifying space of the stable mapping class group is homotopic to an infinite loop space.

Much effort in the past several decades has gone into lifting various algebraic structures into a topological context. I will describe one such lifting: that of the arithmetic theory of elliptic curves. The result is a rich and highly structured family of cohomology theories collectively known as elliptic cohomology. By forming "global sections" one is led to a topological enrichment of the ring of modular forms. Geometric interpretations of these theories are enticing but still conjectural at best.