Research
My research interests are in algebraic geometry, with a particular focus on tropical geometry and moduli spaces, and, in general, on combinatorial algebraic geometry. I enjoy building bridges to other areas of modern algebra, geometry, and topology. These areas include non-Archimedean geometry, logarithmic geometry, the geometry of flat surfaces, Brill-Noether theory, enumerative geometry, Hurwitz theory, geometric group theory (in the form of Bass-Serre theory, the geometry of Outer space, and in the study of buildings), the geometry of vector bundles and principal bundles, as well as buildings and matroids.
My preprints are all on the arXiv. Feel free to check out my Google Scholar profile.
Collaborators
Dan Abramovich, Luca Battistella, Barbara Bolognese, Madeline Brandt, Renzo Cavalieri, Melody Chan, Qile Chen, Man-Wai Cheung, Lorenzo Fantini, Tyler Foster, Andreas Gross, Alana Huszar, Kevin Kühn, Arne Kuhrs, Alex Küronya, Yoav Len, Sara Lamboglia, Bo Lin, Steffen Marcus, Margarida Melo, Samouil Molcho, Martin Möller, Jennifer Park, Dhruv Ranganathan, Pedro Souza, Mattia Talpo, Alejandro Vargas, Filippo Viviani, Annette Werner, Jonathan Wise, Dmitry Zakharov.
Preprints
Buildings, valuated matroids, and tropical linear spaces (joint with L. Battistella, K. Kühn, A. Kuhrs, A. Vargas)
$P=W$ phenomena on abelian varieties (joint with B. Bolognese and A. Küronya)
Tropicalization of toric prevarieties (joint with P. Souza and A. Küronya)
Tropical double ramification loci (joint with D. Zakharov).
In Press
Abelian tropical covers (joint with Y. Len and D. Zakharov). To appear in Math. Proc. Camb. Philos. Soc.
Publications
Principal bundles on metric graphs: the $\mathrm{GL}_n$ case (joint with A. Gross and D. Zakharov). Adv. Math., 411 (2022) Part A.
Tropicalization of the universal Jacobian (joint with M. Melo, S. Molcho, and F. Viviani). Épijournal de Géométrie Algébrique, Volume 6 (2022).
Symmetric powers of algebraic and tropical curves: a non-Archimedean perspective (joint with M. Brandt). Trans. Amer. Math. Soc. Ser. B, 9 (2022), 586--618.
Divisorial motivic zeta functions for marked stable curves (joint with M. Brandt). Michigan Math. J., 71 (2022), 271--282.
A non-Archimedean analogue of Teichmüller space and its tropicalization. Selecta Math., 27 (2021), no. 3, article 29.
Skeletons of Prym varieties and Brill-Noether theory (joint with Y. Len). Algebra Number Theory, 15 (2021), no. 3, 785-820.
Realizability of tropical canonical divisors (joint with M. Möller and A. Werner). J. Eur. Math. Soc. (JEMS), 23 (2021), no. 1, 185–217.
A moduli stack of tropical curves (joint with R. Cavalieri, M. Chan, and J. Wise). Forum Math. Sigma, Volume 8 (2020), e23.
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.
Non-Archimedean geometry of Artin fans. Adv. Math., Volume 345 (2019), Pages 346-381.
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.
Tropicalization is a non-Archimedean analytic stack quotient. Math. Res. Lett. 24 (2017) No. 4, 1205-1237.
Functorial tropicalization of logarithmic schemes: The case of constant coefficients. Proc. Lond. Math. Soc. (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. J. Lond. Math. Soc. (2015) 92 (2): 427-450.
Tropical compactification in log-regular varieties. Math. Z., June 2015, Volume 280, Issue 1-2, pp 195-210.
Conference Proceedings
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.
Outreach
From the dollar game to the Riemann-Roch Theorem (joint with Sara Lamboglia). Snapshots of modern mathematics from Oberwolfach, Nr. 1/2021.
Non-refereed publications, theses, and reports
Towards a logarithmic compactification of strata of abelian differentials. Oberwolfach Reports Volume 16, Issue 2, 2019, pp. 1681--1682.
What is the fundamental group of a tropical curve? Oberwolfach Reports, Volume 16, Issue 2, 2019, pp. 1270--1271.
Skeletons of Prym varieties and Brill-Noether theory. Oberwolfach Reports Volume 16, Issue 1, 2019, pp. 534--536.
Around tropical curves. Oberwolfach Reports, Volume 15, Issue 3, 2018, pp. 2583–2650.
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