##### Department of Mathematics,

University of California San Diego

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### Department of Mathematics Colloquium

## Andrew Blumberg

#### University of Texas, Austin

## Algebraic K-theory and the geometry of module categories

##### Abstract:

Algebraic K-theory is a deep and subtle invariant of rings and schemes, carrying information about arithmetic and geometry. When applied to the group ring of the loop space of a manifold, it captures information about the diffeomorphism group. Over the past 25 years, the study of algebraic K-theory has been revolutionized by the introduction of trace methods, which use trace (or Chern character) maps to the simpler but related theories of (topological) cyclic and Hochschild homology. A unifying perspective on the properties of algebraic K-theory and these related theories is afforded by viewing the input as a category of compact modules (i.e., a piece of an enhanced triangulated category). This talk will survey recent work describing the structural properties of these theories using various models of the homotopical category of module categories.

### January 9, 2014

### 3:00 PM

### AP&M 6402

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