Sept. 24
10:3011:30 in SL 011

Virgil Chan (IUPUI)
Multiplicative Structures in Algebraic Ktheory
Abstract [+]
In his dissertation, Loday gave the first systematic study of products in algebraic Ktheory of rings using Quillen's Plus Construction. The idea is that the direct sum of all algebraic Kgroups should be a graded commutative ring whenever the underlying ring is commutative, and should coincide with the products given by Milnor in lower degrees. I will survey Loday's construction and explain how to promote it to the level of spectra. I will also show this product structure coincides with most products defined by Milnor.

Oct. 1
10:3011:30 in SL 011

Virgil Chan (IUPUI)
An Explicit Formula for the Loday Assembly
Abstract [+]
We give an explicit description of the Loday assembly map on homotopy groups when restricted to a certain subgroup coming from the AtiyahHirzebruch spectral sequence. This proves and generalises a formula about the Loday assembly map on the first homotopy group written down in the survey article by Lueck and Reich. Furthermore, we use this new result to prove the Loday assembly for the integral group ring of cyclic group of order 2 is a bijection on the second homotopy group, and is an injection when one considers nfold direct sum of cyclic group of order 2.

Oct. 29
10:3011:30 in SL 011

Eric Samperton (UIUC)
Coloring invariants of knots and links are often intractable
Abstract [+]
I'll give a quick intro to my recent results with Greg Kuperberg concerning the computational complexity of enumerating homomorphisms to a fixed finite group G. We showed that when G is nonabelian simple, it is NPhard to decide if a knot complement in S^{3} has a map to G with noncyclic image. We also proved a similar theorem for closed 3manifolds. However, I really just want to use our theorem as motivation to discuss in detail some algebraic topological techniques we used in the process. In particular, we needed to determine the orbits of the action of the braid group of a punctured surface S on the set of homomorphisms from the the fundamental group of S to G, in both the manypuncture limit, and the large genus limit. I'll discuss my theorem that does this. It is a mutual generalization of theorems of EllenbergVenkateshWesterland and DunfieldThurston.
