Recent progress in the field of algebraic K-theory has made it possible to compute the K-theory of certain rings from the cyclic homology and certain versions of their cdh-cohomology. The cdh-cohomology groups are constructed specifically so that they behave well with the methods of resolution of singularities. The idea then will be to write out our various computations on singular schemes, such as K-theory computations, in terms of computations on smooth schemes. Along with some historical context, we will present an overview of the cdh-topology and the construction of the cdh-cohomology groups along with their important properties. In particular, we will use the various versions of cdh-cohomology to develop a framework that makes it possible to compute some K-theory and cyclic homology groups of singular schemes. Afterwards, we will present some recent applications of this computational framework as well as some new results.
Supercharacters and superclasses are coarser versions of the irreducible characters and conjugacy classes of finite groups. I will discuss a supercharacter theory of the group of unipotent upper triangular matrices over a finite field. Classifying the conjugacy classes of this group is a provably 'wild' problem. By considering the orbits of a group action, conjugacy classes and characters can be clumped in a compatible manner such that the resulting superclasses and supercharacters are easily indexed. I will present this construction, as well as an analogous one for the unipotent orthogonal, symplectic and unitary groups.
Supercharacter Theories of Unipotent Groups Constructed Via Group Actions, Part 1
Oct. 08, 2013 2pm (MATH 220)
Agnes Szendrei (CU)
X
Let A be a finite algebra in a residually small variety with a cube term. I will discuss sufficient conditions which ensure that A is dualizable.
We view the group algebra of a finite group as a semisimple Hopf algebra and look at three different types of subalgebra, each of which has interesting duality properties. We focus on Schur rings, and their relationship to the representation theory of the larger algebra.
Hopf algebras, Frobenius algebras, and Schur rings
