Interests

My primary research is in the area of abstract homotopy theory, specifically questions involving (monoidal) model categories, Quillen equivalences, Bousfield localization, cellularization, algebras over (colored) operads, and coalgebras over comonads. My work has applications to (stable) homotopy theory, higher category theory, equivariant homotopy theory, Goodwillie calculus, homological algebra, motivic homotopy theory, and representation theory. My PhD thesis was supervised by Mark Hovey.

As an undergraduate I was lucky to be part of two REUs, so I firmly believe in the value of research for undergraduates. I maintain a list of projects on which I would be happy to collaborate with undergraduates and early graduate students in mathematics or computer science. If you're interested in seeing this list please email me.

To Denison students: I will happily take on research students in applied statistics at any time, ideally during the semester. For pure mathematics or computer science, my research interests tend towards the abstract. Thus, research with me will likely best serve students interested in graduate school. For such students, the best time to do research with me is during the summer after your sophomore year, or as an independent study in your junior or senior year. The summer after your junior year would be best spent at an REU, to best position yourself for applying to graduate school.

A nice collection of open problems about popular games can be gleaned from this MathOverflow question.

My primary research is in the area of abstract homotopy theory, specifically questions involving (monoidal) model categories, Quillen equivalences, Bousfield localization, cellularization, algebras over (colored) operads, and coalgebras over comonads. My work has applications to (stable) homotopy theory, higher category theory, equivariant homotopy theory, Goodwillie calculus, homological algebra, motivic homotopy theory, and representation theory. My PhD thesis was supervised by Mark Hovey.

I
have also done research in computer science and discrete
mathematics.
Under the supervision of Danny
Krizanc, my Master's thesis
settled a conjecture involving autonomous agents
moving on a graph. I have supervised undergraduate research
in computer science on the structure of
social network
graphs, R package development, and streaming algorithms for
statistical
inference.

I have done work in statistics, with two papers so far
focusing on statistical pedagogy, and several applied
statistics
papers. I have supervised undergraduate
research projects related to spatial econometrics,
genomic modeling, data-driven journalism, and epidemiology
(specifically, the opioid epidemic in Ohio).

I have done work in economics, to create a game theoretic model for the collective action problem in revolutions, with a particular focus on the 2014 rebellion in Ukraine.

I have a project in progress focusing on mass shooting data in the USA.

My coauthors have included Mark Hovey, Michael Batanin, Donald Yau, Javier Gutierrez, Boris Chorny, Carles Casacuberta, Don Larson, Kristen Mazur, Carolyn Yarnall, Cary Malkiewich, Mona Merling, Luke Wolcott, Nathan Carter, Olga Nicoara, Tom Bressoud, Matt Kretchmar, Dick DeVeaux, Mahesh Agarwal, Maia Averett, Ben Baumer, Andrew Bray, Lance Bryant, Lei Cheng, Amanda Francis, Robert Gould, Albert Kim, Qin Lu, Ann Moskol, Deb Nolan, Roberto Pelayo, Sean Raleigh, Ricky Sethi, Mutiara Sondjaja, Neelesh Tiruviluamala, Paul Uhlig, Talitha Washington, Curtis Wesley, Ping Ye, Andrew Magyar, J.D. Phillips, Kyle Prifogle, William Young, Lin Ma ('20), Lam Tran ('21), and my brother, Michael White.

I am also an active contributor to MathOverflow.

Books and Chapters

- Introduction to Data
Systems:
Building from Python, with
Thomas
Bressoud,
Springer, 2020.

- Statistics for Mathematicians, Chapter in Data Science for
Mathematicians,
edited by Nathan
Carter,
Taylor and Francis, 2020.

- Monoidal Bousfield Localization and Algebras over Operads,
Wesleyan University
Library, 2014.

- Traversals of Infinite Graphs with Random Local Orientations, Wesleyan University Library, 2012. Available digitally through WesScholar, or as arxiv 1308.1041.

- Smith Ideals of Operadic Algebras in Monoidal Model
Categories,
with Donald Yau, accepted to
*Algebraic and Geometric Topology*, arXiv:1703.05377. - Homotopy theory of algebras of substitudes and their localisation, with Michael Batanin. Transactions of the American Mathematical Society, Volume 375, Number 5, May 2022. Available as arXiv:2001.05432.
- Monoidal Bousfield Localization and Algebras Over Operads. Equivariant Topology and Derived Algebra, Cambridge University Press, pp. 179-239, 2021. Available as arXiv:1404.5197.
- VEGF-A, PDGF-BB and HB-EGF engineered for promiscuous super affinity to the extracellular matrix improve wound healing in a model of type 1 diabetes, joint with Jeffrey Hubbell, Michael White, and Priscilla Briquez, Nature: Regenerative Medicine, 2021.
- Substitudes, Bousfield localization, higher braided operads, and Baez-Dolan stabilization, Mathematisches Forschungsinstitut Oberwolfach, Number 46, 2021: Homotopical Algebra and Higher Structures, pp. 56-60.
- Left Bousfield Localization and Eilenberg-Moore Categories, with Michael Batanin. Homology, Homotopy, and Applications, Volume 23, Issue 2, pp. 299-323, 2021. Available as arXiv:1606.01537.
- A Statistical Analysis of Drug Seizures and Opioid Overdose Deaths in Ohio from 2014 to 2018, with Lin Ma and Lam Tran, Journal of Student Research, Volume 10, Issue 1 (2021).
- Right Bousfield Localization and Operadic Algebras, with Donald Yau. Tbilisi Mathematics Journal, Special Issue (HomotopyTheorySpectra), pp. 71-118, 2020. Available as arXiv:1512.07570.
- An Alternative Approach to Equivariant Stable Homotopy Theory, with Mark Hovey. Tbilisi Mathematics Journal, Special Issue on Homotopy Theory, Spectra, and Structured Ring Spectra, 51-69, 2020. Available as arXiv:1312.3846.
- The
User's Guide Project: Looking Back and Looking Forward,
with Don Larson, Kristen Mazur, and Carolyn Yarnall. Journal of Humanistic
Mathematics, Volume 10, Issue 1, pages 411 -
430, 2020.

- Homotopical Adjoint Lifting Theorem, with Donald Yau, Applied Categorical Structures,
Volume 27, Issue 4, 2019, pages 385-426. Available as arXiv:1606.01803.

- A Project Based Approach to Statistics and Data Science, PRIMUS, Volume 29,
Issue 9, 2019, pp. 997-1038. Available as arXiv:1802.08858.

- Arrow Categories of Monoidal Model Categories, with Donald
Yau, Math. Scandinavica,
125(2), pp. 185-198, 2019, available as arXiv:1703.05359.

- An Overview of Schema Theory, The Graduate Journal of Mathematics, Volume 3, Issue 2, 2018, 37-59. Available as arXiv:1401.2651.
- Encoding Equivariant
Commutativity via Operads, with Javier Gutierrez, Algebraic and Geometric
Topology, Volume 18, Number 5, 2018, 2919-2962.
Available as arXiv:1707.02130.

- Bousfield Localization and Algebras over Colored Operads, with Donald Yau. Applied Categorical Structures, Volume 26, Issue 1, 2018, pages 153-203. Available as arXiv:1503.06720.
- Model Structures on Commutative Monoids in General Model Categories. Journal of Pure and Applied Algebra, Volume 221, Issue 12, 2017, pages 3124-3168. Available as arXiv:1403.6759.
- Curriculum Guidelines
for
Undergraduate Programs in Data Science, with Richard De
Veaux, et al.
Report from Undergraduate Faculty Group at Park City
Mathematics
Institute, Annual
Review of Statistics, Vol. 4, 2017, pages 15-30. Available as arXiv:1801.06814.

- Baez-Dolan Stabilization via (Semi-)Model Categories of Operads, with Michael Batanin. In Interactions between Representation Theory, Algebraic Topology, and Commutative Algebra, ed. Dolors Herbera, Wolfgang Pitsch, and Santiago Zarzuela, Research Perspectives CRM Barcelona, Volume 5, 2015, pages 175-179. Birkhauser, DOI 10.1007/978-3-319-45441-2.
- The User's Guide Project: Giving Experiential Context to Research Papers, with Cary Malkiewich, Mona Merling, Frank Lucas Wolcott, and Carolyn Yarnall. Journal of Humanistic Mathematics, volume 5, number 2, 2015.
- A Rational Choice Model of the Rise of Self-Proclaimed States Encompassed in Weak Post-Soviet Economies, with Olga Nicoara. 2015 Annual Meetings of the Public Choice Society.
- A user's guide:
Monoidal Bousfield localizations and algebras over
operads, Enchiridion:
Mathematical Users Guides, Vol. 1, 2015.
Available as arXiv:1801.03191.

- White Paper Research Report (title is classified), Internal NSA Journal, Division R6, 2010.
- An Investigation into the Structure of Digroups (with A. Magyar, J.D. Phillips, K. Prifogle, and W. Young), Proceedings of the Wabash Summer Institute in Algebra, 2007.

- Left Bousfield Localization without Left Properness, with
Michael
Batanin, arXiv:2001.03764,
submitted.

- Comonadic Coalgebras and left Bousfield Localization, with
Donald
Yau, available as arXiv:1805.11536.

- A variant of a Dwyer-Kan theorem for model categories, with Boris Chorny, submitted, submitted, available as arXiv:1805.05378.
- Right Bousfield Localization and Eilenberg-Moore Categories, with Donald Yau. Submitted. Available as arXiv:1609.03635.
- On Colimits in Various Categories of Manifolds, December 2012.

- Model Structures on non-Reduced Operads and the Commutative Monoid Axiom, with Michael Batanin.
- Model structures on operads and algebras from a global perspective, with Michael Batanin.
- Derived localisation
and the
Grothendieck construction, with Michael Batanin.

- Localization and Cellularization for Motivic Symmetric Spectra, with Carles Casacuberta.
- A short note on smallness and topological monoids.
- The Random Basic Walk on Infinite Graphs.
- Parallel Search on Intersection Graphs, with Jessica Tang.
- Model structures for the relative stable module category, with Daniel Bravo and Gabriel Valenzuela.
- Abelian Left Bousfield localization, with Daniel Bravo.
- On multiplicative norms, with Hiroyuki Nakaoka.

Research I have
supervised

- An Overview of Spatial
Econometrics, Alex Tybl, arxiv
1605.03486 and Social
Science
Research Network (SSRN) number 2778679.

- Using Genomics to Predict Learning Disabilities, Trevor Masters.
- Parallel Search on Intersection Graphs, with Jessica Tang.
- Data Driven Journalism
and the
Opioid Epidemic (2019), with Lin Ma and Lam Tran.

- Streaming Statistical
Tests
(2019) with Colin Smith.

Grants

- RC Good Fellowship, Convenient Combinatorial Categories of Topological Spaces, Principal Investigator, spring 2022. Funding for a one-semester research leave.
- COVID Re-engagement Grant, The McClure-Smith Conjecture, Principal
Investigator, fall 2021. Funding for a summer visit to
Michael Batanin in Prague, Czech Republic.

- Pedagogical Practice Projects Grant, Inviting Data Analytics Majors into Introductory Computer Science, Denison University, fall 2019.
- Center for Teaching and Learning, funding for Pedagogy and Resilience reading group, Denison University, 2019-2020.
- Pedagogical Practice Projects Grant, SAGE Labs in Calculus, Denison University, spring 2017.
- Center for Teaching and Learning, funding for Teaching Statistical Concepts
reading group, Denison University, 2016-2017.

- Pedagogical Practice Projects Grant, Statistical Modeling, Denison University, spring 2016.
- Center for Mathematics and Scientific Computation, Equivariant cellularization
and
nullification, co-Principal Investigator with Boris
Chorny,
funding for research visit in July 2015.

- Denison University Research Table Grant, Ethics in Cyber Space, co-Principal Investigator with Joan Krone and John McHugh. Funding for interdisciplinary research, undergraduate research, developing pedagogy, and to bring several external speakers to campus in 2015-2016.
- Project NExT Fellow, 2015-2016.

- National Science Foundation, East Asia and Pacific Summer
Institutes for U.S. Graduate Students (EAPSI): "Studying the
Interplay
between Localization and Categorical Algebra via Algebraic
Topology,'' Principal
Investigator,
award number IIA-1414942.
Funded to be a visiting scholar at Macquarie University
working with
Michael Batanin. Additional funding provided by Australian
Academy of
Sciences. 2014.

- AMS Graduate Student Travel Grant for travel to Joint Mathematics Meetings, January 2014.
- National Science Foundation Travel Grant for Type Theory, Homotopy Theory, and Univalent Foundations conference held in Barcelona, Spain, September, 2013.
- AMS Funding to start Graduate
Student
Chapter at Wesleyan, 2013.

As an undergraduate I was lucky to be part of two REUs, so I firmly believe in the value of research for undergraduates. I maintain a list of projects on which I would be happy to collaborate with undergraduates and early graduate students in mathematics or computer science. If you're interested in seeing this list please email me.

To Denison students: I will happily take on research students in applied statistics at any time, ideally during the semester. For pure mathematics or computer science, my research interests tend towards the abstract. Thus, research with me will likely best serve students interested in graduate school. For such students, the best time to do research with me is during the summer after your sophomore year, or as an independent study in your junior or senior year. The summer after your junior year would be best spent at an REU, to best position yourself for applying to graduate school.

A nice collection of open problems about popular games can be gleaned from this MathOverflow question.