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 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 have a project in progress focusing on mass shooting data
in the
USA.
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), and my brother, Michael White.
I am also an active contributor to MathOverflow.
Books and Chapters
- Introduction to Data
Systems:
Building from Python, with
Thomas
Bressoud,
Springer, 2020.
- Statistics for Mathematicians, Chapter in Data Science for
Mathematicians,
edited by Nathan
Carter,
Taylor and Francis, 2020.
- 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.
Research Publications
- Smith Ideals of Operadic Algebras in Monoidal Model
Categories,
with Donald Yau, accepted to Algebraic and Geometric
Topology, arXiv:1703.05377.
- 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.
- Monoidal
Bousfield Localization and Algebras Over Operads. Equivariant Topology and Derived Algebra,
Cambridge University Press, pp. 179-239, 2021. Available
as arXiv:1404.5197.
- 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.
- Substitudes, Bousfield
localization, higher braided operads, and Baez-Dolan
stabilization, Mathematisches
Forschungsinstitut Oberwolfach, Number 46, 2021:
Homotopical Algebra and Higher Structures, pp. 56-60.
- 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.
- 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).
- Right Bousfield Localization and Operadic Algebras, with
Donald Yau. Tbilisi
Mathematics Journal, Special Issue
(HomotopyTheorySpectra), pp. 71-118, 2020. Available as arXiv:1512.07570.
- 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.
- 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.
- 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, pp. 997-1038. Available as arXiv:1802.08858.
- Arrow Categories of Monoidal Model Categories, with Donald
Yau, Math. Scandinavica,
125(2), pp. 185-198, 2019, 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.
- 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.
- A user's guide:
Monoidal Bousfield localizations and algebras over
operads, Enchiridion:
Mathematical Users Guides, Vol. 1, 2015.
Available as arXiv:1801.03191.
- White Paper Research
Report
(title is classified), Internal
NSA Journal, Division R6,
2010.
- 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.
Preprints
- Left Bousfield Localization without Left Properness, with
Michael
Batanin, arXiv:2001.03764,
submitted.
- Comonadic Coalgebras and left Bousfield Localization, with
Donald
Yau, available as arXiv:1805.11536.
- A variant of a Dwyer-Kan theorem for model categories,
with Boris Chorny,
submitted, submitted, available as arXiv:1805.05378.
- Right Bousfield Localization and Eilenberg-Moore
Categories, with Donald
Yau. Submitted. Available as arXiv:1609.03635.
- On
Colimits
in Various Categories of Manifolds, December 2012.
Ongoing
Projects
- 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.
- Localization and Cellularization for Motivic Symmetric Spectra, with Carles Casacuberta.
- A short note on
smallness and
topological monoids.
- The Random
Basic Walk on Infinite Graphs.
- Parallel Search on
Intersection Graphs, with Jessica Tang.
- 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.
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
- RC Good Fellowship, Convenient
Combinatorial Categories of Topological Spaces,
Principal Investigator, spring 2022. Funding for a
one-semester research leave.
- COVID Re-engagement Grant, The McClure-Smith Conjecture, Principal
Investigator, fall 2021. Funding for a summer visit to
Michael Batanin in Prague, Czech Republic.
- 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.
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.