This page was last updated in April 2014.

My curriculum vitae (updated February 2014) also contains a list of my papers and talks.

### Research Papers

- A Hadwiger Theorem for Simplicial Maps
with P. Christopher Staecker:
We define the notion of
*valuation*on simplicial maps between geometric realizations of simplicial complexes, generalizing both the intrinsic volumes and the Lefschetz number. This allows us to prove a Hadwiger-style classification theorem for all such valuations. (*preprint, 2014*) links - Intrinsic Volumes of Random Cubical Complexes
with Michael Werman:
We give exact polynomial formulae for the expected value and variance of the intrinsic volumes of several models of random cubical complexes. We also prove a central limit theorem for these intrinsic volumes and, for our primary model, an interleaving theorem for the zeros of the expected-value polynomials. (
*preprint, 2014*) links - Hadwiger Integration of Random Fields:
I provide a formula for the expected values of Hadwiger integrals of Gaussian-related random fields, which are both theoretically interesting and potentially useful in applications such as sensor networks and image processing. (
*preprint, 2013*) links - Hadwiger's Theorem for Definable Functions
with Yuliy Baryshnikov and Robert Ghrist:
We generalize the intrinsic volumes to the valuations on real-valued functions and provide a classification theorem for such valuations, analogous to Hadwiger's classic theorem. (published in
*Advances in Mathematics*, 2013) links - Hadwiger Integration of Definable Functions: This is my Ph.D. dissertation, completed in 2011, in which I define Hadwiger integrals and prove a classification theorem for valuations on definable functions. links

### Expository Articles

- Colorful Symmetries
with Brian Bargh and John Chase:
With a focus on the concept of symmetry, this article explains how to count the number of ways that you can color an icosahedron (or another geometric object) with
*n*colors. (published in*Math Horizons*, 2014) links - Cycles of Digits:
Cyclic permutations of digits that appear in repeating fractions can help students understand important concepts in abstract algebra. (
*preprint, 2013*) links

**My Erdös number is 4:**

me → Robert Ghrist → Aaron Abrams → Earl Canfield → Paul Erdös

### Selected Presentations

- Hadwiger and Lefschetz: Valuations on Simplical Maps: given in the Postdoc Seminar at the Institute for Mathematics and its Applications in December 2013
- Hadwiger Integration and Applications: a 50-minute explanation of valuations, intrinsic volumes, Hadwiger's theorem, and Hadwiger integrals given at The Ohio State University in November 2013 links
- Mathematics of Juggling: a 50-minute talk given in the Math Postdoc Seminar at the University of Minnesota in September 2013; slides made in collaboration with John Chase links
- Hadwiger Integration and Applications: a 25-minute talk given at the Applied Topology conference in Będlewo, Poland, in July 2013 links
- Benefits of Collaborative Writing for Learning: a 15-minute talk given at the Joint Mathematics Meetings in San Diego, January 2013
- Hadwiger Integration of Random Fields: a 50-minute talk given in the Geometry Seminar at the University of Illinois at Urbana-Champaign in October 2012
- Math Research: Patterns, Potatoes, and Problem Solving: a 50-minute talk given at Huntington University in September 2012
- Hadwiger Integration and Applications: a 50-minute talk given in the Geometry Seminar at the University of Illinois at Urbana-Champaign in April 2012
- Euler Integration and Applications: a 50-minute talk given for the Purdue University Math Club in April 2012

### Poster

- Hadwiger Integration and Applications: a poster about my research that I made in September 2013 links