I am an Associate Professor of Mathematics at Brown University. From 2012 to 2015 I was an NSF Postdoctoral Fellow and Lecturer in the mathematics department at Harvard. I got my PhD from UC Berkeley in 2012, advised by Bernd Sturmfels. I am interested in combinatorial algebraic geometry, particularly: algebraic curves and their moduli, graph theory, and tropical geometry.

I am supported by a 2018 Sloan Research Fellowship, an NSF CAREER grant DMS-1844768, and the 2020 AWM-Microsoft Research Prize. I am also currently supported by an NSF FRG (Focused Research Group) award with June Huh, Nick Proudfoot, and Sam Payne. I was previously supported by NSF grant DMS-1701924 and a Henry Merritt Wriston Fellowship from Brown University.

I am a 2022 Fellow of the American Mathematical Society. In 2023, I am on sabbatical, supported by a Simons Foundation Fellowship.

Here is my CV. Here are my profiles on Google Scholar and MathSciNet.

Research

1. Homology representations of compactified configurations on graphs applied to M_{2,n}, with C. Bibby, N. Gadish, and C. Yun.
   Preprint.
2. On the top-weight rational cohomology of A_g, with M. Brandt, J. Bruce, M. Melo, G. Moreland, C. Wolfe.
   Geometry and Topology, to appear.
3. Relative Richardson varieties, with N. Pflueger.
   Math. Proc. Cambridge Philos. Soc., to appear.
4. The S_n-equivariant top weight Euler characteristic of M_{g,n}, with C. Faber, S. Galatius, and S. Payne.
   Amer. J. Math., to appear.
   Notes and exercises for a lecture series on this work
5. Moduli spaces of curves: classical and tropical. Notices Amer. Math. Soc. 68 (2021), no. 10, 1700–1713.
6. Combinatorial relations on skew Schur and skew stable Grothendieck polynomials, with N. Pflueger.
   Algebraic combinatorics 4 (1) (2021), 175-188.
7. Tropical curves, graph complexes, and top weight cohomology of M_g, with S. Galatius and S. Payne.
   This subsumes a part of the preprint The tropicalization of the moduli space of curves II: Topology and applications.
   J. Amer. Math. Soc. 34 (2) (2021), 565-594.
8. Topology of the tropical moduli spaces M_{2,n}
   Beiträge zur Algebra und Geometrie 63 (2022), no. 1, 69–93.
   Supplementary sage code for Section 6
9. Topology of moduli spaces of tropical curves with marked points, with S. Galatius and S. Payne.
   This is the second in a series of papers that subsumes the preprint The tropicalization of the moduli space of curves II: Topology and applications.
   Facets of algebraic geometry. Vol. I, 77–131, London Math. Soc. Lecture Note Ser., 472, Cambridge Univ. Press, Cambridge, 2022.
10. Euler characteristics of Brill-Noether varieties, with N. Pflueger.
    Transactions of the AMS, 374 (2021), 1513-1533.
11. A moduli stack of tropical curves, with R. Cavalieri, M. Ulirsch, and J. Wise.
    Forum of Mathematics Sigma, 8, E23 (2020). doi:10.1017/fms.2020.16
The Gieseker-Petri theorem and imposed ramification, with B. Osserman and N. Pflueger.
Bulletin of the LMS 51 (6) (2019), 945-960.
12. Lectures on tropical curves and their moduli spaces
    Moduli of curves, Lect. Notes Unione Mat. Ital., vol. 21, Springer, Cham, 2017, pp. 1–26.
13. Genera of Brill-Noether curves and staircase paths in Young tableaux, with A. López Martín, N. Pflueger, M. Teixidor i Bigas.
    Transactions of the AMS 370 (2018), no. 5, 3405–3439.
14. The expected jaggedness of order ideals, with S. Haddadan, S. Hopkins, L. Moci.
    Forum of Mathematics, Sigma (2017), Vol. 5, e9, 27 pages.
    arxiv version
15. Theta characteristics of tropical K_4-curves, with P. Jiradilok.
    Combinatorial Algebraic Geometry Fields Inst. Commun., vol. 80, Fields Inst. Res. Math. Sci., Toronto, ON, 2017, pp. 65–86.
16. Sandpiles, spanning trees, and plane duality, with D. Glass, M. Macauley, D. Perkinson, C. Werner, Q. Yang
    SIAM J. Discrete Math 29-1 (2015), 461-471.
17. Fano schemes of determinants and permanents, with N. Ilten.
    Algebra and Number theory 9-3 (2015), 629--679.
18. Rotor-routing and spanning trees on planar graphs, with T. Church and J. Grochow.
    Int. Math. Res. Not. 11 (2015), 3225-3244.
19. Tropical Teichmüller and Siegel spaces, with M. Melo and F. Viviani.
    Algebraic and combinatorial aspects of tropical geometry, Contemporary Mathematics 589 (2013), 45-85.
20. Tropical curves and metric graphs
    Ph.D. dissertation, 107 pages, 2012.
21. Elliptic curves in honeycomb form, with B. Sturmfels.
    Algebraic and combinatorial aspects of tropical geometry, Contemporary Mathematics 589 (2013) 87-107.
    Supplementary materials and computations
22. Tropical hyperelliptic curves
    Journal of Algebraic Combinatorics 37 (2013), no. 2, 331-359.
    Slides
    
23. Combinatorics of the tropical Torelli map
    Algebra and Number Theory 6 (2012), no. 6, 1133-1169.
    Slides     Pictures of tropical M_g
24. Three notions of tropical rank for symmetric matrices, with D. Cartwright.
    Combinatorica 32 (2012), no. 1, 55-84.
    Slides     Extended abstract
25. The 4x4 minors of a 5xn matrix are a tropical basis, with A. N. Jensen and E. Rubei.
    Linear Algebra and its Applications 435 (2011), no. 7, 1598-1611.
    Slides
26. The distinguishing number of the augmented cube and hypercube powers
    Discrete Mathematics 308 (2008) no. 11, 2330-2336.
27. Improved pebbling bounds, with A. Godbole.
    Discrete Mathematics 308 (2008) no. 11, 2301-2306.
28. The distinguishing number of the direct product and wreath product action
    Journal of Algebraic Combinatorics 24 (2006), no. 3, 331-345.
29. The maximum distinguishing number of a group
    Electronic Journal of Combinatorics 13 (2006), no. 1, R70.
30. Stably computable properties of network graphs, with D. Angluin, J. Aspnes, M. J. Fischer, H. Jiang, R. Peralta
    Proceedings of the International Conference on Distributed Computing in Sensor Systems, Lecture Notes in Computer Science, 3560 (2005) 63-74.

Expository notes, videos

Course Notes

• Introduction to Higher Mathematics: Combinatorics
  Freely downloadable PDF textbook, including lecture notes and exercises, for an 4-week introduction to combinatorics and graph theory at the undergraduate level. With addtions by Joe Silverman
• Course notes for Linear Algebra With Theory (Math 540) at Brown, taught mostly from Sheldon Axler's Linear algebra done right.
• Course notes for second semester graduate algebra (Math 2520) at Brown. Category theory, commutative algebra, and homological algebra, for Math 2520 at Brown, loosely following Atiyah and Macdonald.

Expository articles and surveys

• Moduli spaces of curves: classical and tropical, 2021. Colloquium-style survey article for Notices of the AMS
• Graph Complexes and the top-weight Euler characteristic of M_{g,n}, 2020. Notes and exercises for a lecture series on this paper, aimed at graduate students
• Lectures on tropical curves and their moduli spaces, 2017. Notes for a lecture series at the CIMPA-CIMAT-ICTP School on Moduli of Curves, Guanajuato, Mexico.
  Videos here.
• A survey of cluster algebras, 2010. Ancient! Final paper for a graduate class taught by Lauren Williams
• A survey of the cycle double cover conjecture, 2007. Ancient! Written in preparation for a workshop on the cycle double conjecture.

Some videos of lectures

• The top-weight cohomology of A_g, at BIRS Cohomology of Arithmetic Groups: Duality, Stability, and Computations, October 2021
• Tropical moduli spaces and the cohomology of M_{g,n}, at MSRI Recent Progress in Moduli Theory, May 2019
• Lectures on tropical curves and their moduli spaces. Five lectures at the CIMPA-CIMAT-ICTP School on Moduli of Curves, Guanajuato, Mexico, 2016 (notes)

Conferences/events/online events

• ICERM Reunion event: Combinatorial Algebraic Geometry, July 31-August 18, 2023
• FPSAC 2023, UC Davis, July 17-21, 2023
• AGNES summer school: Intersection theory on moduli spaces, July 11-14, 2023
• Oberwolfach Recent trends in algebraic geometry, June 18-June 23, 2023
• Arithmetic, Birational Geometry, and Moduli Spaces, Brown University, June 12-June 16, 2023
• Stanford algebraic geometry seminar, June 2, 2023
• Oberwolfach Tropical methods in Geometry, May 15-19, 2023
• Princeton University Horizons Lectures, April 26-27, 2023
• Yale colloquium, April 5, 2023
• Louisiana State colloquium and AWM student talk, March 30-31, 2023
• Harvard algebraic geometry seminar, March 28, 2023
• Georgia Tech colloquium, March 9, 2023

Graduate students, current and former

• Siddarth Kannan, joint with Dhruv Ranganathan
• Shiyue Li
• Stefano Serpente, joint with Margarida Melo
• Claudia He Yun

Teaching

• Math 2710O, Topics in Combinatorics, Fall 2022
• Math 2720T, Graphs, matroids, and moduli spaces, Spring 2022
• Math 540, Honors linear algebra, Summer 2021
• Math 2520, Algebra, Spring 2021
• Math 2520, Algebra, Spring 2020
• Math 750, Introduction to higher mathematics: combinatorics, Fall 2019
• Math 2050, Algebraic geometry, Fall 2019
• Math 2520, Algebra, Spring 2019
• Math 760, Introduction to higher mathematics: algebra, Spring 2019
• Math 2050, Algebraic geometry, Fall 2017
• Math 750, Introduction to higher mathematics: combinatorics, Fall 2017
• Math 1530, Abstract algebra, Spring 2017
• Math 1040, Fundamental problems in geometry, Spring 2016
• Math 0540, Honors Linear Algebra, Fall 2015
• Math 137, Algebraic geometry, Spring 2014 (Harvard)
• Math Ma, Introduction to functions and calculus, Fall 2013 (Harvard)
• Math 285y, Tropical geometry, Spring 2013 (Harvard)
• Math 233a, Theory of schemes, Fall 2012 (Harvard)

Miscellaneous

• This is my award citation response for the 2020 AWM-Microsoft Award in Algebra and Number Theory.
• I support the Brown Math Department statement condemning racial injustices.
• In 2016 we started the Horizons seminar at Brown. It is now partially supported by my CAREER grant. Please join us!
• I co-organize the Brown Combinatorics Seminar with Caroline Klivans.
• I'm a co-organizer of BATMOBYLE.
• I support the open access journal Algebraic combinatorics. See here and here.
• I am an alum of Budapest semesters in mathematics and REUs at East Tennessee State and Duluth MN. Check them out.
• I am an alum of the Kneisel Hall chamber music festival and the Perlman music program. Check them out too! Fun fact: Almost every summer since 2005, I schedule all the rehearsals and coachings at Kneisel Hall using graph coloring algorithms.