We define a version of algebraic K-theory for bornological algebras, using the recently developed continuous K-theory by Efimov. In the commutative setting, we prove that this invariant satisfies descent for various topologies that arise in analytic geometry, generalising the results of Thomason-Trobaugh for schemes. Finally, we prove a version of the Grothendieck-Riemann-Roch Theorem for analytic spaces. Joint work with Jack Kelly and Federico Bambozzi. 

Let G be a reductive group over a finite field F of characteristic p. I will present work with Tzu-Jan Li in which we determine the endomorphism algebra of the Gelfand-Graev representation of the finite group G(F) where the coefficients are taken to be l-adic integers, for l a good prime of G distinct from p. Our result can be viewed as a finite-field analogue of the local Langlands correspondence in families. 

The "curse of dimensionality" refers to the host of difficulties that occur when we attempt to extend our intuition about what happens in low dimensions (i.e. when there are only a few features or variables)  to very high dimensions (when there are hundreds or thousands of features, such as in genomics or imaging).  With very high-dimensional data, there is often an intuition that although the data is nominally very high dimensional, it is typically concentrated around a much lower dimensional, although non-linear set. There are many approaches to identifying and representing these subsets.  We will discuss topological approaches, which represent non-linear sets with graphs and simplicial complexes, and permit the "measuring of the shape of the data" as a tool for identifying useful lower dimensional representations.