Research Interests

My research interests are in the tropical geometry of moduli spaces and its applications to classical questions in arithmetic and algebraic geometry. Given an algebraic geometric moduli problem, the basic idea is to first study the geometry of its tropical analogue, a polyhedral shadow of the original situation, and then to use techniques from non-Archimedean geometry and from logarithmic geometry to lift the insights gained this way back from the combinatorics  to the  world of algebraic geometry. Here is a list of buzzwords that will surely spark my interest:

  • Tropical and non-Archimedean geometry
  • Logarithmic and toric geometry
  • Moduli spaces and stacks
  • Picard varieties and Brill-Noether theory

Preprints

  1. Realizability of tropical canonical divisors (joint with Martin Möller and Annette Werner)
  2. Clutching and gluing in tropical and logarithmic geometry (joint with Alana Huszar and Steffen Marcus)
  3. A moduli stack of tropical curves (joint with Renzo Cavalieri, Melody Chan, and Jonathan Wise)
  4. Non-Archimedean geometry of Artin fans

In press

  1. Logarithmic Picard groups, chip firing, and the combinatorial rank (joint with T. Foster, D. Ranganathan, and M. Talpo), Mathematische Zeitschrift, to appear.
  2. Towards a tropical Hodge bundle (joint with Bo Lin), Fields 2016 CAG book, to appear.
  3. Tropicalization is a non-Archimedean analytic stack quotient, Mathematical Research Letters, to appear.

Publications

  1. Functorial tropicalization of logarithmic schemes: The case of constant coefficients, Proceedings of the London Mathematical Society (2017) 114 (6), 1081-1113.
  2. Skeletons and fans of logarithmic structures (joint with D. Abramovich, Q. Chen, S. Marcus, and J. Wise), Non-Archimedean and Tropical Geometry, Simons Symposia (2016), Editors: Matt Baker and Sam Payne, pp. 287-336.
  3. Faithful realizability of tropical curves (joint with M. Cheung, L. Fantini, and J. Park), Int. Math. Res. Notices (2016) 2016 (15): 4706-4727.
  4. Tropical geometry of moduli spaces of weighted stable curves, Journal of the London Mathematical Society (2015) 92 (2): 427-450.
  5. Tropical compactification in log-regular varieties, Mathematische Zeitschrift, June 2015, Volume 280, Issue 1-2, pp 195-210.

Non-refereed publications, theses, and reports

  1. 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), Research Perspectives CRM Barcelona, to appear.
  2. Logarithmic structures, Artin fans, and tropical compactifications, Oberwolfach Reports Volume 12, Issue 4, 2015, pp. 3271–3331.
  3. Tropical geometry of logarithmic schemes, my PhD-Thesis

Collaborators

Dan Abramovich, Renzo Cavalieri, Melody Chan, Qile Chen, Man-Wai Cheung, Lorenzo Fantini, Tyler Foster, Alana Huszar, Bo Lin, Steffen Marcus, Martin Möller, Jennifer Park, Dhruv Ranganathan, Mattia Talpo, Annette WernerJonathan Wise