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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.

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've carried out statistical research related to policing and protests, wound healing in diabetic individuals, and mass shootings in the USA.

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 maintain a repository of data sets, R code, relevant research papers, and research questions regarding the overdose epidemic in Ohio. There are dozens of research questions (often suitable for undergraduate students) that could be answered using these data sets, that would result in good publications. Many researchers in Ohio have gotten involved, and I welcome others who are interested to join. For more details, please visit my page Statistical Consulting for Harm Reduction.

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), Nancy Rodriguez, Yassin Bahid, Olga Kutsenko, Chad Topaz, Timur Kiguradze, and my brother, Michael White.

I have also contributed to MathOverflow.

Books and Chapters

  1. Introduction to Data Systems: Building from Python, with Thomas Bressoud, Springer, 2020.
  2. Statistics for Mathematicians, Chapter in Data Science for Mathematicians, edited by Nathan Carter, Taylor and Francis, 2020.
  3. Monoidal Bousfield Localization and Algebras over Operads, Wesleyan University Library, 2014.
  4. Traversals of Infinite Graphs with Random Local Orientations, Wesleyan University Library, 2012. Available digitally through WesScholar, or as arXiv 1308.1041.
Research Publications
  1. The statistical and dynamic modeling of protests in Ukraine: the Revolution of Dignity and preceding times, with Nancy Rodriguez, Yassin Bahid, and Olga Kutsenko, PLOS ONE, 19(5): e0301639, 2024.
  2. On Colimits in Various Categories of Manifolds, Graduate Journal of Mathematics, 9(1), 1-16, 2024.
  3. A variant of a Dwyer-Kan theorem for model categories, with Boris Chorny, Algebraic and Geometric Topology, 24(4), 2185-2208, 2024. Available as arXiv:1805.05378.
  4. Left Bousfield Localization without Left Properness, with Michael Batanin. Journal of Pure and Applied Algebra, 228(6), 2024. Available as arXiv:2001.03764.
  5. An analysis of protesting activity and trauma through mathematical and statistical models, with Nancy Rodriguez, Crime Science volume 12, Article number: 17, 2023.
  6. Smith Ideals of Operadic Algebras in Monoidal Model Categories, with Donald Yau. Algebraic and Geometric Topology, 24(1), 341-392, 2024. Available as arXiv:1703.05377.
  7. Right Bousfield Localization and Eilenberg-Moore Categories, with Donald Yau. Higher Structures, 7(1):22-39, 2023. Available as arXiv:1609.03635.
  8. 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.
  9. Monoidal Bousfield Localization and Algebras Over Operads. Equivariant Topology and Derived Algebra, Cambridge University Press, pp. 179-239, 2021. Available as arXiv:1404.5197.
  10. 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.
  11. Substitudes, Bousfield localization, higher braided operads, and Baez-Dolan stabilization, Mathematisches Forschungsinstitut Oberwolfach, Number 46, 2021: Homotopical Algebra and Higher Structures, pp. 56-60.
  12. 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.
  13. 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.
  14. Right Bousfield Localization and Operadic Algebras, with Donald Yau. Tbilisi Mathematics Journal, Special Issue (HomotopyTheorySpectra), pp. 71-118, 2020. Available as arXiv:1512.07570.
  15. 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.
  16. 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.
  17. Homotopical Adjoint Lifting Theorem, with Donald Yau, Applied Categorical Structures, Volume 27, Issue 4, 2019, pages 385-426. Available as arXiv:1606.01803.
  18. A Project Based Approach to Statistics and Data Science, PRIMUS, Volume 29, Issue 9, 2019, pp. 997-1038. Available as arXiv:1802.08858.
  19. Arrow Categories of Monoidal Model Categories, with Donald Yau, Math. Scandinavica, 125(2), pp. 185-198, 2019, available as arXiv:1703.05359.
  20. An Overview of Schema Theory, The Graduate Journal of Mathematics, Volume 3, Issue 2, 2018, 37-59. Available as arXiv:1401.2651.
  21. Encoding Equivariant Commutativity via Operads, with Javier Gutierrez, Algebraic and Geometric Topology, Volume 18, Number 5, 2018, 2919-2962. Available as arXiv:1707.02130.
  22. 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.
  23. 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.
  24. 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.
  25. 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.
  26. 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.
  27. 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.
  28. A user's guide:  Monoidal Bousfield localizations and algebras over operads, Enchiridion: Mathematical Users Guides, Vol. 1, 2015. Available as arXiv:1801.03191.
  29. White Paper Research Report (title is classified), Internal NSA Journal, Division R6, 2010.
  30. 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.

Popular Press (editor reviewed):

1. Foster negotiation between police and protesters for a peaceful DNC, with Nancy Rodriguez and Chad M.  Topaz, op-ed for the Chicago Tribune, August 16, 2024.
2. Will Milwaukee police fuel violence at Republican National Convention protests? with Chad M. Topaz, op-ed for the Milwaukee Journal Sentinel, July 11, 2024.
3. Police must refrain from escalating violence, with Nancy Rodriguez, Chad M.  Topaz, and Timur Kiguradze, op-ed for JAM News, May 27, 2024.
4. Cooperation can end violent protests, with Nancy Rodriguez and Chad M.  Topaz, op-ed for the New York Daily News, May 16, 2024.

Preprints

  1. Comonadic Coalgebras and left Bousfield Localization, with Donald Yau, submitted.
  2. Model structures on operads and algebras from a global perspective, with Michael Batanin and Florian De Leger, submitted, arXiv:2311.07320.
  3. Quasi-tame substitudes and the Grothendieck construction, with Michael Batanin and Florian De Leger, submitted, arXiv:2311.07322.
  4. Homotopical recognition of diagram categories, with Boris Chorny, submitted, available as arXiv:2411.10120.
Ongoing Projects

  1. Model Structures on non-Reduced Operads and the Commutative Monoid Axiom, with Michael Batanin.
  2. Derived localisation and the Grothendieck construction, with Michael Batanin.
  3. Localization and Cellularization for Motivic Symmetric Spectra, with Carles Casacuberta.
  4. A short note on smallness and topological monoids.
  5. The Random Basic Walk on Infinite Graphs.
  6. Parallel Search on Intersection Graphs, with Jessica Tang.
  7. Model structures for the relative stable module category, with Daniel Bravo and Gabriel Valenzuela.
  8. Abelian Left Bousfield localization, with Daniel Bravo.
  9. On multiplicative norms, with Hiroyuki Nakaoka.
  10. State Unintentional Drug Overdose Reporting Surveillance: Opioid Overdose Deaths and Characteristics in Ohio, with Lin Ma and Lam Tran.

Research I have supervised

Grants

Ideas for Undergraduate Research

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.