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.

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, 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- 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, pages 997-1038. Available as arXiv:1802.08858.

- Arrow Categories of Monoidal Model Categories, with Donald Yau, accepted to Math. Scandinavica, 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, pages 186-188. Available as arXiv:1801.06056.
- A user's guide: Monoidal Bousfield localizations and algebras over operads, Enchiridion: Mathematical Users Guides, Vol. 1, 2015. Available as arXiv:1801.03191.

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

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

- Smith Ideals of Operadic Algebras in Monoidal Model Categories, with Donald Yau, available as arXiv:1703.05377.

- Homotopy Theory of Homotopy Functors, with Boris Chorny, submitted, available as arXiv:1805.05378.
- Right Bousfield Localization and Eilenberg-Moore Categories, with Donald Yau. Available as arXiv:1609.03635.
- Left Bousfield Localization and Eilenberg-Moore Categories, with Michael Batanin. Submitted. Available as arXiv:1606.01537.
- Right Bousfield Localization and Operadic Algebras, with Donald Yau. Submitted. Available as arXiv:1512.07570.
- An Alternative Approach to Equivariant Stable Homotopy Theory, with Mark Hovey, submitted. Available as arXiv:1312.3846.
- Monoidal Bousfield Localization and Algebras Over Operads, submitted. Available as arXiv:1404.5197.
- On Colimits in Various Categories of Manifolds, December 2012.

- Statistics for Mathematicians, chapter in Data Science for Mathematicians.

- 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.

- Bousfield Localization without Left Properness, with Michael Batanin.
- Homotopy theory of higher braided operads, with Michael Batanin.

- Localization and Cellularization for Motivic Symmetric Spectra, with Carles Casacuberta, being written up.
- A short note on smallness and topological monoids, being written up.
- The Random Basic Walk on Infinite Graphs, in final revisions.
- Parallel Search on Intersection Graphs, with Jessica Tang.
- Sequences of Model Structures and the Stable Module Category, with Daniel Bravo.
- 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.
- Learning from Partner Disciplines when Teaching Statistical Concepts: how to organize a reading and discussion group.
- A course on Randomized Algorithms fit for a liberal arts college.

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

- 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.