Andrew Krenz

After graduating from Purdue University in 2018, I began
studying math as a graduate student at the University of Wisconsin - Madison.
I'm working with Tullia Dymarz and Chenxi Wu in the areas of
Geometric Group Theory and Dynamics. Here is a copy of my CV.
Feel free to contact me; my wisc dot edu e-mail address is "krenz3 at."

Research

There is a generalization of the Birkhoff ergodic theorem for certain Fuchsian groups.
Right now I am working on expanding the list of groups known to enjoy this ergodic
property beyond the Fuchsian examples. The essential property required of the groups,
in order for the proof to go through, is that their action on hyperbolic space be
controlled by some combinatorial data. This combinatorial data manifests in one of (at
least) two distinct forms. It may present as a special kind of partial order, or perhaps
dually as a generalized notion of `median' i.e., algebraic structure. Presently, I am
taking a closer look at the categorical aspects of this duality. In particular, there
is a notion of ultrafilter on these partial orders which, in a lot of ways, mimics the
behavior of ultrafilters on boolean algebras. On the other side, the ideals (in the sense
of universal algebra) of the median operation behave like convex sets. In either case
there are traces of codensity. For instance, the codensity monad of the inclusion of
finite sets into sets is the ultrafilter monad - the proof I know invokes the language
of measure and integration. Similarly, the inclusion of convex subsets into measure
spaces has, as its codensity monad, the Giry monad. Integration is a common thread tying
all of these situations together. In the process of proving that the half-spaces of
ultrafilters functor or the ultrafilters of half-spaces functor admit monad structures and
moreover, arise as the codensity monad of some functor, one expects to learn about integration
operations on the pertinent combinatorial objects. Another example from the literature uses
the concept of codensity to understand the possible (group) actions on a set. This approach
should allow us to push the current techniques (close to) as far as they go in this direction.