This page was last updated in February 2015.

My curriculum vitae (updated March 2015) also contains a list of my papers and talks.

### Research Papers

- Efficient Representation and Visualization of Rank Invariants for 2-D Persistent Homology
with Mike Lesnick (
*in preparation*) - Efficient Computation of the Multi-Graded Betti Numbers
with Mike Lesnick (
*in preparation*) - 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, February 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, updated January 2015*) links - Hadwiger Integration of Random Fields:
I provide a formula for the expected values of Hadwiger integrals (and, using Hadwiger's Theorem, more general valuations) of Gaussian-related random fields, which are both theoretically interesting and potentially useful in applications such as sensor networks and image processing. (to appear in
*Topological Methods in Nonlinear Analysis*) 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

- How Many Ways are there to Juggle?: An introduction to the mathematics of juggling, given in the mathematics seminar at Macalaster College in February 2015; slides made in collaboration with John Chase links
- Computing Persistent Homology: an introduction to the computation of persistent homology, given at CIMAT in January 2015 links
- Euler Integration and Data Analysis: computer science and mathematics lecture at Bryn Mawr College in November 2014
- Visualizing Multidimensional Persistent Homology: given in the Industrial and Applied Mathematics Seminar at the University of Oxford in November 2014
- Euler Integration and Applications: given in the Mathematics Colloquium at the University of Mary Washington in October 2014
- Intrinsic Volumes of Random Cubical Complexes: an introduction to random cubical complexes and their intrinsic volumes, given at the Institute for Mathematics and its Applications in April 2014 and at the Technion (in Haifa, Israel) in May 2014 links
- 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
- 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
- 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