Title: My preferred proof of the Lefschetz fixed point theorem 

Abstract: There are many different proofs of the Lefschetz fixed point theorem.  The most familiar approach uses simplicial approximation and is often a first example of the power of simplicial homology.  I'll talk about a very different proof that I find much more useful.  This proof requires more input, but it generalizes easily. 

Title:  The Freudenthal Suspension Theorem

Abstract:  The Freudenthal suspension theorem asserts that for an (n-1)-connected CW complex X the suspension map from \pi_i(X) to \pi_{i+1}(SX) is an isomorphism for i < 2n - 1 and a surjection for i = 2n - 1. We will introduce relative homotopy groups and the long exact sequence in homotopy groups for a space X and a subspace A. With these tools we will show how the Freudenthal suspension theorem follows from the homotopy excision theorem. Time permitting, we will examine some consequences for homotopy groups of spheres.

Title:  An introduction to operads

Abstract:  Operads first arose in the 60's and 70's for the study of loop spaces, but there was a large resurgence of interest in the 90's once connections with Koszul duality, moduli spaces, and representation theory were realized. I will discuss the definition and familiar examples in both topology and algebra. We will see Stasheff polyhedra in the context of loop spaces as well as examples related to moduli spaces.

Title:  Introduction to vector bundles and their classifications

Abstract:  We will introduce the definition of a vector bundle and look at a few examples. Next we will look at how to make new vector bundles from old bundles using familiar algebraic operations like direct sum, tensor product, and the pullback. Finally we will discuss classifying isomorphism classes of bundles over a topological space X, and time permitting, we will show these isomorphism classes are in bijection with homotopy classes of maps from X to Grassmanians on R infinity.

Title:  Limits, Colimits, and Homotopy . . . Oh, my!

Abstract:  Given maps f: X --> Y and g: X --> Z of topological spaces, we obtain a unique map h: X --> Y x Z that respects the appropriate projections.  This property corresponds more generally to the limit of a diagram of spaces.  In this talk, we will define the limit, colimit, and their homotopy analogs and discuss their universal properties and relative merits/uses.  No prior topological knowledge is assumed.

Title:  Eilenberg-MacLane Spaces

Abstract:  A space X is a K(G,n)  if \pi_n(X)=G and \pi_i(X)=0 if i\neq n. An interesting aspect is that the homotopy type of a CW comples K(G,n) is uniquely determined by G and n. We will investigate the construction of K(G,1), otherwise known as BG, for an arbitrary (discrete) group G, the homology of K(G,1) spaces, and the infinite symmetric product SP(X).