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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, and my coauthors in homotopy theory have included Michael Batanin, Donald Yau, Javier Gutierrez, Gabriel Valenzuela, Daniel Bravo, Hiroyuki Nakaoka, Boris Chorny, and Carles Casacuberta.

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 begun to dabble in statistics, with two papers so far focusing on statistical pedagogy. I have also supervised undergraduate research projects in statistics and data science, related to spatial econometrics, genomic modeling, and a data-driven journalism approach to the opioid epidemic.

I have done work in Economics, with my co-author Olga Nicoara, to create a game theoretic model for the collective action problem in revolutions, with a particular focus on the rebellion in Ukraine.

I am also an active contributor to MathOverflow.

Publications
  1. Homotopical Adjoint Lifting Theorem, with Donald Yau, Applied Categorical Structures, Volume 27, Issue 4, 2019, pages 385-426. Available as arxiv 1606.01803.
  2. A Project Based Approach to Statistics and Data Science, PRIMUS, Volume 29, Issue 9, 2019, pages 997-1038. Available as arXiv:1802.08858.
  3. Arrow Categories of Monoidal Model Categories, with Donald Yau, accepted to Math. Scandinavica, available as arXiv:1703.05359.
  4. An Overview of Schema Theory, The Graduate Journal of Mathematics, Volume 3, Issue 2, 2018, 37-59. Available as arxiv 1401.2651.
  5. Encoding Equivariant Commutativity via Operads, with Javier Gutierrez, Algebraic and Geometric Topology, Volume 18, Number 5, 2018, 2919-2962. Available as arXiv:1707.02130.
  6. 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.
  7. 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.
  8. 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.
  9. 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.
  10. 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, pages 186-188. Available as arXiv:1801.06056.
  11. A user's guide:  Monoidal Bousfield localizations and algebras over operads, Enchiridion: Mathematical Users Guides, Vol. 1, 2015. Available as arXiv:1801.03191.
  12. 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.
  13. Monoidal Bousfield Localization and Algebras over Operads, Wesleyan University Library, 2014.
  14. Traversals of Infinite Graphs with Random Local Orientations, Wesleyan University Library, 2012. Available digitally through WesScholar, or as arxiv 1308.1041.
  15. White Paper Research Report (title is classified), Internal NSA Journal, Division R6, 2010.
  16. An Investigation into the Structure of Digroups (with A. Magyar, K. Prifogle, and W. Young), Proceedings of the Wabash Summer Institute in Algebra, 2007.
Preprints

  1. Comonadic Coalgebras and left Bousfield Localization, with Donald Yau, available as arXiv:1805.11536.
  2. Smith Ideals of Operadic Algebras in Monoidal Model Categories, with Donald Yau, available as arXiv:1703.05377.
  3. Homotopy Theory of Homotopy Functors, with Boris Chorny, submitted, available as arXiv:1805.05378.
  4. Right Bousfield Localization and Eilenberg-Moore Categories, with Donald Yau. Available as arXiv:1609.03635.
  5. Left Bousfield Localization and Eilenberg-Moore Categories, with Michael Batanin. Submitted. Available as arXiv:1606.01537.
  6. Right Bousfield Localization and Operadic Algebras, with Donald Yau. Submitted. Available as arXiv:1512.07570.
  7. An Alternative Approach to Equivariant Stable Homotopy Theory, with Mark Hovey, submitted. Available as arXiv:1312.3846.
  8. Monoidal Bousfield Localization and Algebras Over Operads, submitted. Available as arXiv:1404.5197.
  9. On Colimits in Various Categories of Manifolds, December 2012.
Ongoing Projects

  1. Statistics for Mathematicians, chapter in Data Science for Mathematicians.
  2. Model Structures on non-Reduced Operads and the Commutative Monoid Axiom, with Michael Batanin.
  3. Model structures on operads and algebras from a global perspective, with Michael Batanin.
  4. Derived localisation and the Grothendieck construction, with Michael Batanin.
  5. Bousfield Localization without Left Properness, with Michael Batanin.
  6. Homotopy theory of higher braided operads, with Michael Batanin.
  7. Localization and Cellularization for Motivic Symmetric Spectra, with Carles Casacuberta, being written up.
  8. A short note on smallness and topological monoids, being written up.
  9. The Random Basic Walk on Infinite Graphs, in final revisions.
  10. Parallel Search on Intersection Graphs, with Jessica Tang.
  11. Sequences of Model Structures and the Stable Module Category, with Daniel Bravo.
  12. Model structures for the relative stable module category, with Daniel Bravo and Gabriel Valenzuela.
  13. Abelian Left Bousfield localization, with Daniel Bravo.
  14. On multiplicative norms, with Hiroyuki Nakaoka.
  15. Learning from Partner Disciplines when Teaching Statistical Concepts: how to organize a reading and discussion group.
  16. A course on Randomized Algorithms fit for a liberal arts college.

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.