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).

Title:  Homotopy groups of character varieties

Abstract:  Given a discrete group \Gamma and a (complex reductive or compact) Lie group G, the character variety X_r (G) is the quotient for the conjugation action of G on Hom(\Gamma, G). When G is complex reductive, this quotient should be interpreted in the sense of Geometric Invariant Theory. When G = GL(n) or SL(n), the subspace of irreducible representation coincides with the smooth locus of X_r (G). The rational homology of these spaces has been studied in various cases by a number of authors, and when G = U(n) or SU(n), the homotopy type of the stable moduli spaces X_r (U) and X_r (SU) are explicitly known. In this talk I'll discuss recent progress on understanding low-dimensional homotopy (and integral homology) of character varieties and of their subspaces of irreducible representations. This is joint work with Indranil Biswas, Carlos Florentino, and Sean Lawton.

Title:  The Other Signature Theorem

Abstract:  The celebrated Hirzebruch’s Signature Theorem expresses the signature of an oriented 4k-dimensional manifold as a characteristic number in cohomology with rational coefficients. I will discuss a similar result for the Kervaire invariant of a spin manifold that involves characteristic numbers in real K-theory.  The presentation will be non-technical and will require very little knowledge of algebraic topology.

Title:  The Jones Polynomial

Abstract:  Knot Theory is a subject in topology that studies embeddings of $S^1$ in $\mathbb{R}^3$. We call these embeddings 'knots' (hence 'Knot Theory').  In this talk, we will discuss some of the basic ideas in the subject of Knot Theory.  We will then discuss and give a construction for a useful knot invariant called the Jones Polynomial.

Title:  Chromatic Levels in the Homotopy Groups of Spheres

Abstract:  Understanding the homotopy groups of spheres $\pi_nS^k$ is one of the great challenges of algebraic topology. One of the fundamental theorems in this field is the Freudenthal suspension theorem. It states that $\pi_{n+k}S^k$ is isomorphic to $\pi_{n+k+1}S^{k+1} $ when $k$ is large. Homotopy theorists call this phenomena \emph{stabilization}. The stable homotopy groups of spheres are defined to be these families of isomorphic groups. They form a ring, commonly denoted by $\pi_*S$. Despite its simple definition, this ring is extremely complex; there is no hope of computing it completely. However, it carries an amazing amount of structure. A famous theorem of Hopkins and Ravenel states that it is filtered by simpler rings called the \emph{chromatic layers}. There are many structural conjectures about the chromatic filtration. In this talk, I will give an overview of chromatic theory and talk about one of the structural conjectures, the \emph{chromatic splitting conjecture}.

Title:  Algebraic K-theory and crossed objects

Abstract:  After reviewing the classical lower K-groups, Milnor's K_2, and Quillen's plus construction (stopping for examples along the way), we will look at definitions of crossed modules and crossed complexes. After showing that certain K-groups can be regarded as these crossed objects, we will see how this might give insight into explicit descriptions of the plus construction in terms of generators and relations of the Steinberg group.