My research interests are in arithmetic and algebraic geometry, with a particular focus on tropical and non-Archimedean geometry. This rich and rather new research area allows me (in many cases with my collaborators) to explore many surprising connections to a multitude of areas of modern geometry, such as to the geometry of moduli spaces, to Brill-Noether theory, to Hurwitz theory, and to geometric group theory (in the form of Bass-Serre theory). In order to relate the tropical world to the world of classical algebraic and arithmetic geometry I often employ techniques from analytic geometry (over Archimedean and non-Archimedean fields) in its many disguises, from logarithmic geometry in the sense of Fontaine-Kato-Illusie, from deformation theory, and from the geometry of stacks.
- Tropical double ramification loci (joint with D. Zakharov).
- Divisorial motivic zeta functions for marked stable curves (joint with M. Brandt).
- Abelian tropical covers (joint with Y. Len and D. Zakharov).
- Skeletons of Prym varieties and Brill-Noether theory (joint with Y. Len).
- Symmetric powers of algebraic and tropical curves: a non-Archimedean perspective (joint with M. Brandt).
- A moduli stack of tropical curves (joint with R. Cavalieri, M. Chan, and J. Wise)
- Realizability of tropical canonical divisors (joint with M. Möller and A. Werner). J. Eur. Math. Soc. (JEMS), to appear.
- Logarithmic Picard groups, chip firing, and the combinatorial rank (joint with T. Foster, D. Ranganathan, and M. Talpo). Mathematische Zeitschrift, Volume 291 (2019), Issue 1–2, Pages 313–327.
- Non-Archimedean geometry of Artin fans. Advances in Mathematics, Volume 345 (2019), Pages 346-381.
- Clutching and gluing in tropical and logarithmic geometry (joint with A. Huszar and S. Marcus). Journal of Pure and Applied Algebra, Volume 223 (2019), Issue 5, Pages 2036-2061.
- Tropicalization is a non-Archimedean analytic stack quotient. Mathematical Research Letters 24 (2017) No. 4, 1205-1237.
- Functorial tropicalization of logarithmic schemes: The case of constant coefficients. Proceedings of the London Mathematical Society (2017) 114 (6), 1081-1113.
- Faithful realizability of tropical curves (joint with M. Cheung, L. Fantini, and J. Park). Int. Math. Res. Notices (2016) 2016 (15): 4706-4727.
- Tropical geometry of moduli spaces of weighted stable curves. Journal of the London Mathematical Society (2015) 92 (2): 427-450.
- Tropical compactification in log-regular varieties. Mathematische Zeitschrift, June 2015, Volume 280, Issue 1-2, pp 195-210.
- Towards a tropical Hodge bundle (joint with B. Lin). Combinatorial Algebraic Geometry, 353-368, Fields Institute Communications (2017), Volume 80, Springer 2017, Editors: Greg Smith and Bernd Sturmfels.
- Skeletons and fans of logarithmic structures (joint with D. Abramovich, Q. Chen, S. Marcus, and J. Wise). Non-Archimedean and Tropical Geometry, 287-336, Simons Symposia (2016), Editors: Matt Baker and Sam Payne.
Dan Abramovich, Madeline Brandt, Renzo Cavalieri, Melody Chan, Qile Chen, Man-Wai Cheung, Lorenzo Fantini, Tyler Foster, Alana Huszar, Yoav Len, Bo Lin, Steffen Marcus, Martin Möller, Jennifer Park, Dhruv Ranganathan, Mattia Talpo, Annette Werner, Jonathan Wise, Dmitry Zakharov.
Non-refereed publications, theses, and reports
- Around tropical curves. Oberwolfach Reports, to appear.
- Newton-Okounkov bodies and reified valuations of higher rank (joint with A. Camara, I. Giné, R. Gualdi, N. Kalinin, J. Roé, S. Urbinati, and X. Xarles). Extended Abstracts February 2016, Positivity and Valuations, Trends in Mathematics, Springer.
- Logarithmic structures, Artin fans, and tropical compactifications, Oberwolfach Reports Volume 12, Issue 4, 2015, pp. 3271–3331.
- Tropical geometry of logarithmic schemes, my PhD-Thesis