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Interests

My primary research is in the area of algebraic topology, specifically questions involving localization and model categories. My work has applications to the theory of algebras over (colored) operads, equivariant homotopy theory, and motivic homotopy theory. In the near future I plan to do more work developing the theory of Bousfield localization, to better understand the various model structures which arise in equivariant homotopy theory, and to make further investigations into motivic homotopy 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. At Denison, I supervised an undergraduate research project resulting in a paper (with Jessica Tang) about the structure of social network graphs.

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 related to spatial econometrics, genomic modeling, and R package development.

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. Encoding Equivariant Commutativity via Operads, with Javier Gutierrez, accepted to Algebraic and Geometric Topology. Available as arXiv:1707.02130.
  2. A Project Based Approach to Statistics and Data Science, accepted to PRIMUS. Available as arXiv:1802.08858.
  3. Bousfield Localization and Algebras over Colored Operads, with Donald Yau. Applied Categorical Structures, Volume 26, pages 153-203, 2018. Available as arXiv 1503.06720
  4. 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.
  5. 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, pages 15-30, 2017. Available as arXiv:1801.06814.
  6. 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, vol. 5. Birkhauser, DOI 10.1007/978-3-319-45441-2, 2016.
  7. 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, vol. 5, no. 2, 186-188, 2015. Available as arXiv:1801.06056.
  8. A user's guide:  Monoidal Bousfield localizations and algebras over operads, Enchiridion: Mathematical User’s Guides, Vol. 1, 2015. Available as arXiv:1801.03191.
  9. A Rational Choice Model of the Rise of Self-Proclaimed States Encompassed in Weak Post-Soviet Economies, with Olga Nicoara. Accepted to 2015 Annual Meetings of the Public Choice Society.
  10. Monoidal Bousfield Localization and Algebras over Operads, Wesleyan University Library, 2014.
  11. Traversals of Infinite Graphs with Random Local Orientations, Wesleyan University Library, 2012. Available digitally through WesScholar, or as arxiv 1308.1041.
  12. White Paper Research Report (title is classified), Internal NSA Journal, Division R6, 2010.
  13. 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, submitted, available as arXiv:1703.05377
  3. Arrow Categories of Monoidal Model Categories, with Donald Yau, submitted, available as arXiv:1703.05359
  4. Homotopy Theory of Homotopy Functors, with Boris Chorny, available as arXiv:1805.05378.
  5. Right Bousfield Localization and Eilenberg-Moore Categories, with Donald Yau. Available as arXiv:1609.03635
  6. Homotopical Adjoint Lifting Theorem, with Donald Yau, submitted. Available as arxiv 1606.01803
  7. Left Bousfield Localization and Eilenberg-Moore Categories, with Michael Batanin. Submitted. Available as arXiv 1606.01537.
  8. Right Bousfield Localization and Operadic Algebras, with Donald Yau. Submitted. Available as arXiv 1512.07570
  9. An Alternative Approach to Equivariant Stable Homotopy Theory, with Mark Hovey, submitted. Available as arxiv 1312.3846.
  10. Monoidal Bousfield Localization and Algebras Over Operads, submitted. Available as arXiv 1404.5197.
  11. An Overview of Schema Theory, Expository Paper, submitted. Available as arxiv 1401.2651.
Ongoing Projects

  1. Model Structures on non-Reduced Operads and the Commutative Monoid Axiom, with Michael Batanin, final revisions stage.
  2. Model structures on operads and algebras from a global perspective, with Michael Batanin, being written up.
  3. Bousfield Localization without Left Properness, being written up.
  4. Localization and Cellularization for Motivic Symmetric Spectra, with Carles Casacuberta, being written up.
  5. A short note on smallness and topological monoids, being written up.
  6. The Random Basic Walk on Infinite Graphs, in final revisions.
  7. Parallel Search on Intersection Graphs, with Jessica Tang.
  8. Sequences of Model Structures and the Stable Module Category, with Daniel Bravo.
  9. Model structures for the relative stable module category, with Daniel Bravo and Gabriel Valenzuela.
  10. Abelian Left Bousfield localization, with Daniel Bravo.
  11. On multiplicative norms, with Hiroyuki Nakaoka.

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.