Research & Publications



    My current research interests center around combinatorial algebraic geometry with a view towards applications in moduli theory, classical and virtual enumerative geometry, and the theory linear series. Over the last few years, I have tried to answer questions in these areas by using tropical techniques, blended with logarithmic geometry. 

    One of my current focuses is the geometry of general curves of a prescribed gonality, and specifically, their Brill-Noether theory. For instance, in a recent paper, Dave and I used ideas from logarithmic Gromov-Witten theory to establish a generalization of Griffiths and Harris's Brill-Noether theorem, which calculates the dimension of the space of embeddings of a general curve into projective space, as a subvariety of given degree. I am presently interested in finer aspects of the space of embeddings of a k-gonal curve, such as the Hilbert functions of these embeddings. 

    Another recent direction has concerned the geometry of moduli spaces of embedded curves and enumerative geometry. Jonathan, Keli, and I recently constructed non-singular moduli spaces of elliptic curves in toric varieties, reinterpreting and generalizing work of Vakil and Zinger. This has opened the door to a number of questions concerning the (honest/non-virtual) enumerative geometry of elliptic curves in toric varieties. I am presently developing tropical tools to study the characteristic numbers problem, an old question in enumerative geometry about the enumeration of curves with contact order constraints.

    For a more detailed idea of the kind of mathematics that I work on feel free to take a look at my papers, listed below, and accessible on arXivMathSciNetand GoogleScholar.

Publications
  1. Topology of tropical moduli of weighted stable curves. With A. Cerbu, S. Marcus, L. Peilen, and A. Salmon. Preprint.
  2. Motivic Hilbert zeta functions of curves are rational. With D. Bejleri & R. Vakil. Preprint.
  3. Moduli of stable maps in genus one & logarithmic geometry. With K. Santos-Parker & J. Wise. Preprint.
  4. Curve counting on toric surfaces: tropical geometry & the Fock space. With R. Cavalieri, P. Johnson, & H. Markwig. Submitted.
  5. Brill-Noether theory for curves of a fixed gonality. With D. Jensen. Submitted.
  6. Logarithmic Picard groups, chip firing, and the combinatorial rank. With T. Foster, M. Talpo, & M. Ulirsch. Submitted.
  7. Skeletons of stable maps II: superabundant geometries. In Research in the Mathematical Sciences.
  8. A note on Brill-Noether existence for graphs of low genus. With S. Atanasov. In Michigan Mathematical Journal.
  9. Skeletons, degenerations, & Gromov-Witten theory. 2016 Doctoral Dissertation, Yale University. 
  10. A graphical interface for the Gromov-Witten theory of curves. With R. Cavalieri, P. Johnson, & H. Markwig. In Proceedings of the 2015 Algebraic Geometry Summer Institute.
  11. Tropical Hurwitz numbers. With H. Markwig. Appendix to A First Course in Hurwitz theory by Cavalieri and Miles. 
  12. Enumerative geometry of elliptic curves on toric surfaces. With Y. Len. In Israel Journal of Mathematics.
  13. Degenerations of toric varieties over valuation rings. With T. Foster. In Bulletin of the London Mathematical Society
  14. Skeletons of stable maps I: rational curves in toric varieties. In Journal of the London Mathematical Society
  15. Superabundant curves and the Artin fan. In International Mathematics Research Notices.
  16. Hahn analytification and connectivity of higher rank tropical varieties. With T. Foster. In Manuscripta Mathematica.
  17. Toric graph associahedra and compactifications of M0,n. With R. Ferreira da Rosa & D. Jensen. In Journal of Algebraic Combinatorics
  18. Realization of groups with pairing as Jacobians of finite graphs. With L. Gaudet, D. Jensen, N. Wawrykow, & T. Weisman. Submitted.
  19. Tropical compactification and the Gromov-Witten theory of P1. With R. Cavalieri & H. Markwig. In Selecta Mathematica
  20. Moduli spaces of rational weighted stable curves and tropical geometry. With R. Cavalieri, S. Hampe, & H. Markwig. In Forum of Mathematics (Sigma)
  21. Tropicalizing the space of admissible covers. With R. Cavalieri & H. Markwig. In Mathematische Annalen
  22. Brill-Noether theory of maximally symmetric graphs. With T. Leake. In European Journal of Combinatorics.
  23. Gromov-Witten theory of P1xP1xP1. With D. Karp. In Journal of Pure and Applied Algebra 
  24. Toric Symmetry of CP3. With D. Karp, P. Riggins, & U. Whitcher. In Advances in Theoretical and Mathematical Physics.
  25. Gromov-Witten theory of blowups of toric threefolds (2012). Undergraduate thesis, Harvey Mudd College. 87 pages.
In Preparation 

    My ongoing projects include the following. A forthcoming paper with Cerbu, Marcus, Peilen, and Salmon studies the topological aspects of the moduli space of tropical weighted stable curves, building on the results of the paper above with Cavalieri, Hampe, and Markwig. Bejleri, Vakil, and I have established the rationality of the Hilbert zeta function of an arbitrary reduced curve. Brandt, Jones, Lee, and I have proved tropical versions of some classical incidence geometry theorems, including analogues of the Sylvester-Gallai and Motzkin-Rabin theorems. I am presently working on the tropical degeneration formula for the enumerative genus 1 invariants of toric varieties, building on work with Santos-Parker and Wise. Finally, an ongoing project gives a new tropical perspective on the characteristic numbers problem for toric varieties using tropical descendant theory. 

Unpublished Notes
  1. Analytification and tropicalization over Z. I establish Sam's inverse limit over the integers. I never found an application.
  2. Toric Severi degrees are logarithmic Gromov-Witten invariants. Sam asked if the logarithmic GW invariants of toric surfaces were enumerative.
People

I have been fortunate to work with some awesome people! Here's a list of my co-authors and research advisors. Stanislav Atanasov, Renzo Cavalieri, Rodrigo Ferreira da Rosa, Tyler Foster, Lou Gaudet, Simon Hampe, Dave JensenPaul JohnsonDagan Karp, Timothy Leake, Yoav LenHannah MarkwigMike OrrisonSam PaynePaul Riggins, Keli Santos-Parker, Mattia Talpo, Martin Ulirsch, Nick Wawrykow, Teddy WeismanUrsula Whitcher, Jonathan Wise.

I've also had the pleasure of working with some great undergraduates: Timothy Leake (Summer '13, combinatorial Brill-Noether theory), Rodrigo Ferreira da Rosa (Summer '14, M0,n and graph associahedra), Andrew Deveau, Jenna Kainic, and Dan Mitropolsky (Summer '14, Gonality of random graphs), Louis Gaudet, Nicholas Wawrykow, and Teddy Weisman (Summer '14, Jacobians of finite graphs), Derek Boyer, Andre Moura, and Scott Weady (Summer '16, Tropical line arrangements), Stanislav Atanasov (Summer '16, Combinatorial Brill-Noether theory), Johnny Gao (Fall '16-Spring '17, Betti numbers of toric graph associahedra), Milo Brandt, Michelle Jones, and Catherine Lee (Summer '17, Incidence geometry of tropical lines), and Alois Cerbu, Luke Peilen, and Andrew Salmon (Summer '17, topology of tropical moduli spaces).

A lot of my learning happens with my academic "family", Dan Corey, Shaked Koplewitz, Yoav Len, Tif Shen, and Jeremy Usatine and "extended family", Kenny Ascher, Dori Bejleri, and Martin Ulirsch.


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