Publications
Books
Nonarchimedean and Tropical Geometry (edited with Sam Payne), Simons Symposia, Springer-Verlag, 2016.
Tropical and Nonarchimedean Geometry (edited with Omid Amini and Xander Faber), Contemporary Mathematics 605, American Mathematical Society, 2013.
Potential Theory and Dynamics on the Berkovich Projective Line, with Robert Rumely, AMS Mathematical Surveys and Monographs 159 (2010), 428 pages.
Preprints
Lift theorems for representations of matroids over pastures, with Oliver Lorscheid, submitted. Preprint available at arXiv:2107.00981
On some notions of rank for matrices over tracts, with Tianyi Zhang, submitted. Preprint available at arXiv:2202.02356
The Jacobian of a Sixth-Root-of-Unity Matroid, with Changxin Ding and Xu Zhuang, submitted. Preprint available at arXiv:2308.11760
Foundations of Matroids, Part 2, with Oliver Lorscheid and Tianyi Zhang, submitted. Preprint available at arXiv:2310.19951
The foundation of generalized parallel connections, 2-sums, and segment-cosegment exchanges of matroids, with Oliver Lorscheid, Zach Walsh, and Tianyi Zhang, submitted. Preprint available at arXiv:2404.10656
New building blocks for F_1-geometry: bands and band schemes, with Oliver Lorscheid and Tong Jin, submitted. Preprint available at arxiv:2402.09612
Research Papers
(freely available on the arXiv preprint server)
On a theorem of Lafforgue, with Oliver Lorscheid, to appear in International Math. Research Notices. Preprint available at arXiv:2309.01746
Reciprocity via Reciprocants, to appear in The American Mathematical Monthly. Preprint available at arXiv:22309.02512
Foundations of matroids, Part 1: Matroids without large uniform minors, with Oliver Lorscheid, to appear in Memoirs of the AMS. Preprint available at arxiv.org/abs/2008.00014
On the existence of balanced generalized de Bruijn sequences, with Bhumika Mittal, Haran Mouli, and Eric Tang, Discrete Mathematics 346 (2023), doi.org/10.1016/j.disc.2023.113487
Representability of orthogonal matroids over partial fields, with Tong Jin, Algebraic Combinatorics 6 (2023), no. 5, 1301-1311.
Fusion rules for pastures and tracts, with Tianyi Zhang, European J. Comb. 108 (2023), 15 pages.
The moduli space of matroids, with Oliver Lorscheid, Advances in Mathematics 390 (2021), 118 pages.
On the structure of hyperfields obtained as quotients of fields, with Tong Jin, Proc. Amer. Math. Soc. 149 (2021), 63-70.
Descartes' rule of signs, Newton polygons, and polynomials over hyperfields, with Oliver Lorscheid, Journal of Algebra, 569 (2021), 416-441.
Geometric bijections for regular matroids, zonotopes, and Ehrhart theory, with Spencer Backman and Chi Ho Yuen, Forum of Mathematics, Sigma 7 (2019), e45, 37 pages.
Matroids over partial hyperstructures, with Nathan Bowler, Advances in Mathematics 343 (2019), 821-863.
The Bernardi process and torsor structures on spanning trees, with Yao Wang, International Mathematics Research Notices 16 (2018), 5120-5147.
Weight functions on Berkovich curves, with Johannes Nicaise, Algebra & Number Theory 10 (2016), 2053-2079.
Nonarchimedean geometry, tropicalization, and metrics on curves, with Sam Payne and Joe Rabinoff, Algebraic Geometry 3 no. 1 (2016), 63-105.
Bitangents of tropical plane quartic curves, with Yoav Len, Ralph Morrison, Nathan Pflueger, and Qingchun Ren, Mathematische Zeitschrift 282 no. 3 (2016), 1017-1031.
Lifting harmonic morphisms II: Tropical curves and metrized complexes, with Omid Amini, Erwan Brugalle, and Joe Rabinoff, Algebra & Number Theory 9 no. 2 (2015), 267-315.
Lifting harmonic morphisms I: Metrized complexes and Berkovich skeleta, with Omid Amini, Erwan Brugalle, and Joe Rabinoff, Research in the Mathematical Sciences 2 no. 7 (2015).
Linear series on metrized complexes of algebraic curves, with Omid Amini, Mathematische Annalen 362 no. 1-2 (2015), 55-106.
The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves, with Joseph Rabinoff, International Math. Research Notices 16 (2015), 7436-7472.
Canonical representatives for divisor classes on tropical curves and the Matrix-Tree Theorem, with Yang An, Greg Kuperberg, and Farbod Shokrieh, Forum of Mathematics, Sigma 2 (2014), e24, 25 pages.
On the structure of non-Archimedean analytic curves, with Joseph Rabinoff and Sam Payne, Contemporary Mathematics 605 (2013), 93-121.
Special curves and postcritically-finite polynomials, with Laura DeMarco, Forum of Mathematics, Pi 1 (2013), e3, 35 pages.
Chip firing games, potential theory on graphs, and spanning trees, with Farbod Shokrieh, J. Combinatorial Theory Series A 120, no. 1 (2013), 164-182.
Preperiodic points and unlikely intersections, with Laura DeMarco, Duke Math Journal 159, no. 1 (2011), 1-29.
Metric properties of the tropical Abel-Jacobi map, with Xander Faber, Journal of Algebraic Combinatorics 33, no. 3 (2011), 349-381.
Harmonic morphisms and hyperelliptic graphs, with Serguei Norine, International Math. Research Notices 15 (2009), 2914-2955.
A finiteness theorem for canonical heights attached to rational maps over function fields, J. Reine Angew. Math 626 (2009), 205-233.
Specialization of linear systems from curves to graphs, Algebra & Number Theory 2, no. 6 (2008), 613-653.
A finiteness property of torsion points, with Su-ion Ih and Robert Rumely, Algebra & Number Theory 2, no. 2 (2008), 217-248.
Riemann-Roch and Abel-Jacobi theory on a finite graph, with Serguei Norine, Advances in Mathematics 215 (2007), 766-788.
Harmonic analysis on metrized graphs, with Robert Rumely, Canad. J. Math. 59, no. 2 (2007), 225-275.
A lower bound for average values of dynamical Green's functions, Mathematical Research Letters 13, no. 2 (2006), 245-257.
Equidistribution of small points, tational dynamics, and potential theory, with Robert Rumely, Ann. Inst. Fourier (Grenoble) 56, no. 3 (2006), 625-688.
Global discrepancy and small points on elliptic curves, with Clayton Petsche, International Math. Research Notices 61 (2005), 3791-3834.
Finiteness results for modular curves of genus at least 2, with Enrique Gonzalez-Jimenez, Josep Gonzalez, and Bjorn Poonen, Amer. J. Math. 127, no. 6 (2005), 1325-1387.
Canonical heights, transfinite diameters, and polynomial dynamics, with Liang-Chung Hsia, J. Reine Angew. Math 585 (2005), 61-92.
Equidistribution of small subvarieties of Abelian varieties, with Su-ion Ih, New York J. Math. 10 (2004), 279-285.
A lower bound for the canonical height on Abelian varieties over Abelian extensions, with Joseph Silverman, Mathematical Research Letters 11 (2004), 377-396.
Galois theory and torsion points on curves, with Ken Ribet, Les XXIIèmes Journées Arithmétiques (Lille, France), Journal de Théorie des Nombres de Bordeaux 15 (2003), 11-32.
Lower bounds for the canonical height on elliptic curves over Abelian extensions, International Math. Research Notices 29 (2003), 1571-1589.
Automorphisms of X0*(p), with Yuji Hasegawa, Journal of Number Theory 100, no. 1 (2003), 72-87.
Torsion packets on curves, with Bjorn Poonen, Compositio Math. 127 (2001), 109-116.
Torsion points on modular curves, Inventiones Math. 140 (2000), 487-509.
Cartier points on curves, International Math. Research Notices 7 (2000), 353-370.
Kamienny's criterion and the method of Coleman and Chabauty, Proceedings of the A.M.S. 127, no. 10 (1999), 2851-2856.
Expository and Recreational Papers
Balancing centrifuges with number theory, Math Horizons 30, no. 1 (2022), 20-23.
John Horton Conway (1937-2020), Science 368 (22 May 2020), p.831.
Hodge theory in combinatorics, Bulletin of the AMS 55 (2018), 57-80.
Robert F. Coleman 1954-2014, with Barry Mazur and Ken Ribet, Research in the Mathematical Sciences 2 no. 1 (2015).
Degeneration of linear series from the tropical point of view and applications, with David Jensen, in Nonarchimedean and Tropical Geometry (edited with Sam Payne), Simons Symposia, Springer-Verlag, 2016.
An introduction to Berkovich analytic spaces and non-Archimedean potential theory on curves, p-adic Geometry (Lectures from the 2007 Arizona Winter School), AMS University Lecture Series, 2008.
Uncountable sets and an infinite real number game, Mathematics Magazine 80, no.5 (December 2007), 377--380.
Metrized graphs, electrical networks, and Fourier analysis, with Xander Faber, Contemporary Mathematics 415, Proceedings of the Joint Summer Research Conference on Quantum Graphs and Their Applications, Snowbird, Utah, 2006.
Selected Presentations
Slides for my talk on "Families of Matroids" at the 30 Years of Berkovich Spaces conference in Paris, France (pdf)
Slides for my talk at the BP Centennial Conference in Cambridge, MA (pdf)
Slides for my talk at the Second ERC Research Period on Diophantine Geometry in Cetraro, Italy (pdf)
Slides for my talk at the Szpiro Birthday Conference at the CUNY Graduate Center (pdf)
Slides for my talk at the 2009 Journées Arithmétiques in St. Etienne, France (pdf)
Course Notes
Lecture notes for a graduate course on Topics in Matroid Theory
Lecture notes for a graduate course on Non-Archimedean Geometry
Lecture notes for a graduate course in Algebraic Number Theory