Research

My research interests are in combinatorial algebraic geometry with a particular focus on tropical and non-Archimedean geometry and the geometry of moduli spaces. I enjoy exploring its many surprising interactions with other areas of modern algebra, geometry, and topology, for example to the geometry of flat surfaces, to Brill-Noether theory, to enumerative geometry, to Hurwitz theory, and to geometric group theory (in the form of Bass-Serre theory and the geometry of Outer space). In order to relate the tropical world to the world of classical algebraic and arithmetic geometry I often draw from techniques in toric and logarithmic geometry, analytic geometry over Archimedean and non-Archimedean fields, as well as from deformation theory and the geometry of stacks.

My preprints are all on the arXiv. Feel free to check out my Google Scholar profile.

Preprints

  1. A non-Archimedean analogue of Teichmüller space and its tropicalization.

  2. Tropical double ramification loci (joint with D. Zakharov).

  3. Abelian tropical covers (joint with Y. Len and D. Zakharov).

  4. Skeletons of Prym varieties and Brill-Noether theory (joint with Y. Len).

  5. Symmetric powers of algebraic and tropical curves: a non-Archimedean perspective (joint with M. Brandt).

In Press

  1. Divisorial motivic zeta functions for marked stable curves (joint with M. Brandt). Michigan Math. J., to appear.

  2. Realizability of tropical canonical divisors (joint with M. Möller and A. Werner). J. Eur. Math. Soc. (JEMS), to appear.

Publications

  1. A moduli stack of tropical curves (joint with R. Cavalieri, M. Chan, and J. Wise). Forum Math. Sigma, Volume 8 (2020), e23.

  2. Logarithmic Picard groups, chip firing, and the combinatorial rank (joint with T. Foster, D. Ranganathan, and M. Talpo). Math. Z., Volume 291 (2019), Issue 1–2, Pages 313–327.

  3. Non-Archimedean geometry of Artin fans. Adv. Math., Volume 345 (2019), Pages 346-381.

  4. Clutching and gluing in tropical and logarithmic geometry (joint with A. Huszar and S. Marcus). J. Pure Appl. Algebra, Volume 223 (2019), Issue 5, Pages 2036-2061.

  5. Tropicalization is a non-Archimedean analytic stack quotient. Math. Res. Lett. 24 (2017) No. 4, 1205-1237.

  6. Functorial tropicalization of logarithmic schemes: The case of constant coefficients. Proc. Lond. Math. Soc. (2017) 114 (6), 1081-1113.

  7. Faithful realizability of tropical curves (joint with M. Cheung, L. Fantini, and J. Park). Int. Math. Res. Notices (2016) 2016 (15): 4706-4727.

  8. Tropical geometry of moduli spaces of weighted stable curves. J. Lond. Math. Soc. (2015) 92 (2): 427-450.

  9. Tropical compactification in log-regular varieties. Math. Z., June 2015, Volume 280, Issue 1-2, pp 195-210.

Proceedings

  1. 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.

  2. 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.



Collaborators

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

  1. Skeletons of Prym varieties and Brill-Noether theory, Oberwolfach Reports Volume 16, Issue 1, 2019, pp. 534--536.

  2. Around tropical curves. Oberwolfach Reports, Volume 15, Issue 3, 2018, pp. 2583–2650.

  3. 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.

  4. Logarithmic structures, Artin fans, and tropical compactifications, Oberwolfach Reports Volume 12, Issue 4, 2015, pp. 3271–3331.

  5. Tropical geometry of logarithmic schemes, my PhD-Thesis