Topology seminar - Spring 2011
January 27 and February 3
Ben Braun - A neat way to prove that K_{3,3} is not planar using the Borsuk-Ulam theorem
Simplicial complexes are a family of topological spaces built out of "nice" pieces of Euclidean space. We will consider the following question: If D is a simplicial complex, for which dimensions n can we embed D in n-dimensional Euclidean space? It is well-known that if D is the complete bipartite graph K_{3,3}, one cannot embed D in the plane; we thus say K_{3,3} is not a planar graph. We will outline a beautiful proof of this fact using the Borsuk-Ulam theorem. Along the way, we will introduce without proof the Borsuk-Ulam theorem and the related machinery of Z_2-spaces, the Z_2-index, and deleted joins of simplicial complexes.
February 10
Ben Braun - Glory is fleeting, but a topological invariant is forever
We will discuss the fundamental question "what is an invariant?" In particular, if you have a topological invariant, what is varying? And what is unchanged? Why do we care? We will introduce the Euler characteristic of a space as an example of a topological invariant.
This talk will be accessible to undergraduate math majors. If you have had a topology or analysis course that is a plus, but everyone is welcome!
February 17
Clinton Hines - Unit tangent vector fields on spheres, I
We discuss a lower bound to the number of orthonormal tangent vector fields to the n-sphere. This will be achieved via orthonormal multiplications as they relate to Clifford Algebras. We will talk about some concrete examples of Clifford Algebras and specifically how their structures generate these vector fields on spheres. This talk should be accessible to undergraduate math majors. If you've had a topology or analysis course that would be a plus, but everyone is welcome!
March 7
Kate Ponto - A little bit of stable homotopy theory
As preparation for the topology seminar on March 24, I'll talk a little about generalized cohomology theories and some of the things they lead to (spectra, ring spectra, and highly structured ring spectra).
March 24
Niles Johnson - University of Georgia
Complex Orientations and p-typicality
In joint work with Justin Noel, we give computational results related to the structure of power operations on complex oriented cohomology theories (localized at a prime p), making use of the amazing connection between complex orientations and the theory of formal group laws. After introducing the relevant concepts, we will describe the main results: for primes p less than or equal to 13, orientations factoring non-trivially through the Brown-Peterson spectrum cannot carry power operations, and thus cannot provide MU_(p)-algebra structure. This implies, for example, that if E is a Landweber exact MU_(p)-ring whose associated formal group law is p-typical of positive height, then the canonical map MU_(p) --> E is not a map of H_infty ring spectra. It immediately follows that the standard p-typical orientations on BP, E(n), and E_n do not rigidify to maps of E_infty ring spectra. We conjecture that similar results hold for all primes.
April 7
Angelica Osorno - University of Chicago
2-vector bundles and their classifying space
In recent work of Baas-Dundas-Richter-Rognes, the authors define 2-vector bundles and prove that their classifying spaces, K(Vect) is equivalent to the algebraic K-theory of the connective K-theory spectrum ku. In this talk we will give an introduction to bicategories and 2-vector spaces, explain the construction of the classifying space K(Vect). Finally we will explain how some extra structure in the bicategory of 2-vector spaces translate into an infinite loop space structure on K(Vect).
April 14 and 28
Clinton Hines and Beth Kirby - The Ochanine Genus, Modular Forms, and the Brown-Kervaire Invariant
We consider a refinement of the universal elliptic genus, called the Ochanine or beta Genus. After a brief treatment of modular forms over graded rings, we examine certain modular forms over KO(lower star). We then show that the beta genus applied to a spin manifold is such a modular form. In a follow-up to this discussion, we will use these constructions to look at the Brown-Kervaire Invariant.