I am an Assistant Professor in the
Department of Mathematics at the Ohio
State University. I'm primarily interested in tropical geometry, arithmetic
geometry, and combinatorial
Before coming to the Ohio State University I was an Associate Professor in the Departments of Combinatorics and Optimization and Pure Mathematics at the University of Waterloo.
Prior to that, I was as an RTG Postdoc at The University of Texas, a Postdoctoral member at Mathematical Sciences Research Institute, and an Assistant Research Professor at Duke University. I was a graduate student at Stanford University and an undergraduate at the Ohio State University. Needless to say, Go Bucks!
My curriculum vitae is available here.
Here are some
In case you're wondering, "What is Tropical Geometry?"
My research statement (without future research plans).
Hodge Theory of Matroids (with Karim Adiprasito and June Huh), Notices of the AMS, 64 (2017), 26-30.
Newton-Okounkov Bodies over Discrete Valuation Rings (with Stefano Urbinati), submitted.
Diophantine and tropical geometry, and uniformity of rational points on curves (with Joseph Rabinoff and David Zureick-Brown), submitted.
Hodge theory for combinatorial
geometries (with Karim Adiprasito and June Huh), submitted.
Uniform bounds for the number of rational points on
curves of small Mordell-Weil rank (with Joseph Rabinoff and David
Zureick-Brown), Duke Mathematical Journal, 165 (2016), 3189-3240.
Tropical geometry, the motivic nearby fiber and limit
mixed Hodge numbers of hypersurfaces (with Alan Stapledon), Research in the Mathematical Sciences, 3:10 (2016).
Local h-polynomials, invariants of subdivisions, and
mixed Ehrhart theory (with Alan Stapledon), Adv. Math.,
286 (2016), 181-239.
Matroid theory for algebraic geometers, Simons Symposium Proceedings, accepted.
A Non-Abelian Analogue of Whitney's 2-isomorphism theorem, J. Algebraic Combinatorics, 39 (2014), 683-690.
The Chabauty-Coleman bound at a prime of bad reduction and clifford bounds for geometric rank functions (with David Zureick-Brown), Compositio Math. 149 (2013), 1818-1838.
Tropical Realization Spaces for Polyhedral Complexes, Proceedings of the Workshop on Tropical Geometry. Contemp. Math. 589 (2013), 235-251.
Log-concavity of characteristic
polynomials and the Bergman fan of matroids (with June Huh), Math Ann. 354 (2012), 1103-1116.
Obstructions to lifting tropical curves surfaces in 3-space (with Tristram Bogart), SIAM J. Discrete Math. 26 (2012), 1050-1067.
Lifting Tropical Curves in Space and Linear Systems on Graphs, Adv. Math. 230 (2012), 853-875.
Tropical Geometry and the Motivic Nearby Fiber (with Alan Stapledon), Compositio Math. 148 (2012), 269-294.
Tropical Intersection Theory from Toric Varieties, Collect. Math. 63 (2012), 29-44.
Realization Spaces for Tropical Fans (with Sam Payne), Proceedings of the Abel Symposium, vol. 6, 2011
Monodromy Filtrations and the Topology of Tropical Varieties (with David Helm), Canadian J. Math. 64 (2011), 845-868.
The tropical j-invariant (with Hannah Markwig and Thomas Markwig), LMS J. Comput. Math. 12 (2009), 275-294.
A Tropical Toolkit, Expo. Math. 27 (2009) 1-36.
Tropical Invariants from the Secondary Fan, (withdrawn).
The j-invariant of a Plane Tropical Cubic (with Hannah Markwig and Thomas Markwig), J. Algebra 320 (2008) 3832-3848.
Piecewise polynomials, Minkowski weights, and localization on toric varieties (with Sam Payne), Algebra Number Theory 2 (2008) 135-155.
An Algebraic Formulation of Symplectic Field Theory, Journal of Symplectic Geometry 2 (2008) 135-155.
Line-Bundles on Stacks of Relative Maps, unpublished note.
Topological Recursion Relations by Localization, unpublished note.
I will be using carmen for all communication involving teaching.
Office Hours: TBA
- Math 3345: Foundations of Higher Mathematics, probably
It's often hard to keep up with emails from students in my undergraduate
classes. If your email can be better answered in a brief chat with me, I'd
prefer that you talk to me before or after class or in my office hours.
Please note that Adiprasito, Huh, and Katz resolved Rota's log-concavity conjecture which is distinct from Rota's excluded minors for matroids conjecture, a problem considered to be important by the structural combinatorics community.
Kevin Hartnett's A Path Less Taken to the Peak of the Math World (about June Huh and the resolution of Rota's Conjecture), Quanta Magazine
What is Tropical Geometry?, Notices of the AMS, 64 (2017), 380-382.
Matt Baker's Current Events Bulletin talk on Hodge Theory in Combinatorics at 2017 JMM
Matt Baker's blog post on the effective Chabauty method describing my work with Zureick-Brown
University of Waterloo story about the proof of Rota's conjecture by Adiprasito, Huh, and Katz
Gil Kalai announces the proof of Rota's conjecture by Adiprasito, Huh, and Katz
Matt Baker's blog post on the proof of Rota's conjecture
Article on the Chromatic Polynomial and Adiprasito-Huh-Katz from Nieuw Archief voor Wiskunde (in Dutch)