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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, 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/Preprints- Left Bousfield Localization and Eilenberg-Moore Categories, with Donald Yau. Available as arXiv:1609.03635
- Homotopical Adjoint Lifting Theorem, with Donald Yau. 2016. Available as arxiv 1606.01803
- Left Bousfield Localization and Eilenberg-Moore Categories, with Michael Batanin. 2016. Available as arXiv 1606.01537
- Right Bousfield Localization and Operadic Algebras, with Donald Yau. 2015. Available as arXiv 1512.07570
- Bousfield Localization and Algebras over Colored Operads, with Donald Yau. 2015. Submitted. Available as arXiv 1503.06720
- 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.
- Monoidal Bousfield Localization and Algebras Over Operads, 2014. Also available as arXiv 1404.5197.
- Model Structures on Commutative Monoids in General Model Categories, 2014. Also available as arXiv 1403.6759.
- An Alternative Approach to Equivariant Stable Homotopy Theory (with Mark Hovey), 2013. Also available as arxiv 1312.3846.
- 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 Overview of Schema Theory, Expository Paper, 2009. Also available as arxiv 1401.2651.
- An Investigation into the Structure of Digroups (with A. Magyar, K. Prifogle, and W. Young), Proceedings of the Wabash Summer Institute in Algebra, 2007.
- Model Structures on non-Reduced Operads and the Commutative Monoid Axiom, with Michael Batanin, final revisions stage.
- Model Structures on Grothendieck Categories and Left Proper Model Categories of Algebras over Operads, with Michael Batanin, being written up.
- Bousfield Localization and Eilenberg-Moore Categories, with Michael Batanin, being written up.
- Bousfield Localization without Left Properness, final revisions stage.
- Encoding Equivariant
Commutativity via Operads (with Javier Gutiérrez), being written up.
- 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.
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.
Grants
- Pedagogical Practice Projects Grant, SAGE Labs in Calculus, Denison University, spring 2017.
- Pedagogical Practice Projects Grant, Statistical Modeling, Denison University, spring 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.