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."

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. This approach

should allow us to push the current techniques (close to) as far as they go in this direction.

Term | Course | Position |
---|---|---|

Fall 2018 | Math 221: Calculus and Analytic Geometry 1 | Teaching Assistant |

Spring 2019 | Math 221 | TA coordinator |

Fall 2019 | Math 234: Calculus -- Functions of Several Variables | TA coordinator |

Spring 2020 | Math 222: Calculus and Analytic Geometry 2 | TA coordinator |

Summer 2020 | Math 221 | Teaching Assistant |

Fall 2020 | Math 234 | TA coordinator |

Spring 2021 | Math 340: Elementary Matrix and Linear Algebra | Teaching Assistant |

Summer 2021 | Math 240: Introduction to Discrete Mathematics | Insructor of Record |

Fall 2021 | Math 112: Algebra | Instructor of Record |

Spring 2022 | Math 112 | Instructor of Record |

Summer 2022 | Math 222 | Instructor of Record |

Fall 2022 | Math 221 | TA coordinator |

Spring 2023 | Math 222 | Teaching Assistant |

Summer 2023 | Math 240 | Instructor of Record |

I have noticed certian browsers tend to omit some pen strokes (especially

single points/dots). This does not seem to be a problem for me when browsing

mobile Safari, for example.

FToC | U-Sub | By Parts | Trig |
---|---|---|---|

Partial Fractions | Linear ODEs |

Chapter 1 | Chapter 6 |
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Lastly, to cite the sources used to create the documents linked in the research section.

Everything in the document about hyperbolic space lives in the canonical refrence for

the subject, "Metric Spaces of Nonpositive Curvature" by Bridson and Haefliger.

The introduction to ultrafilters follows the book by J.L. Bell and A.B. Slomson titled

"Models and Ultraproducts an Introduction" just far enought to see the Ultrafilter

lemma. Along the way, I illustrate examples and solve all of the exercises. This

document should be understandable to anyone who has read through lesson 8 say,

of the discrete math notes above. The main ingredient in the famous proof of Arrow's

Impossibility Theorem is the ultrafilter lemma. The remainder of the document treats

this theorem following the presentation in this video.