This page was last updated in April 2017.

My curriculum vitae (updated March 2017) also contains a list of my papers.

### Research Papers

- Efficient Computation of the Multigraded Betti Numbers
with Mike Lesnick (
*in preparation*) - Approval Voting in Product Societies
with Kristen Mazur, Mutiara Sondjaja, and Carolyn Yarnall:
In approval voting, individuals vote for all platforms that they find acceptable. We examine scenarios in which voters must make two decisions simultaneously, present a general lower bound on agreement in a two-dimensional voting society, and examine specific results for societies whose spectra are cylinders and tori.
(to appear in the
*American Mathematical Monthly*) links - Interactive Visualization of 2-D Persistence Modules with Mike Lesnick: We introduce RIVET, a software tool for the interactive visualization of 2-D persistence modules, and present the mathematical foundations for this tool. RIVET efficiently computes barcodes along 1-D affine slices of a 2-D persistence module, using a novel data structure based on planar line arrangements. links
- Towards Domain-Specific Semantic Relatedness: A Case Study from Geography
with Shilad Sen, Isaac Johnson, Rebecca Harper, Huy Mai, Samuel Horlbeck Olsen, Benjamin Mathers, Laura Souza Vonessen, and Brent Hecht:
This paper describes an undergraduate research project on domain-specific semantic relatedness at the 2014 MAXIMA REU.
Focusing on the domain of geography, we showed that algorithms that use geographic signals can outperform standard sematic relatedness algorithms for geographic concepts.
(presented at
*IJCAI 2015*) links - 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. (published in
*Discrete and Computational Geometry*, 2016) 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. (published in
*Topological Methods in Nonlinear Analysis*, 2015) 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

**My Erdös number is 3:**

me → Michael Werman → Nathan Linial → Paul Erdös

### 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