Research Interests

Combinatorics, Number Theory, Algebraic Geometry, and Galois Theory. For a while now I have been studying analogies between finite graphs and Riemann surfaces, which has led to my current interest in chip-firing games on graphs and simplicial complexes. I also spend a lot of time thinking about mathematical physics, especially quantum mechanics.


Divisors and Sandpiles (with D. Perkinson), forthcoming from the AMS. An introduction to chip-firing, the Riemann-Roch Theorem for graphs, and the abelian sandpile model.

Symmetry and Quantum Mechanics, Monographs and Research Notes in Mathematics, Chapman & Hall / CRC Press, 2017.

Here is an excerpt containing the table of contents, preface, and material from the first two chapters.  Check out the section called “Plan of the Book” to get a sense of the overall story and an indication of the book’s somewhat novel structure.


Counting arithmetical structures on paths and cycles (with B. Braun, H. Corrales, L.D. García Puente, D. Glass, N. Kaplan, J.L. Martin, G. Musiker, C.E. Valencia), submitted. arXiv

Maximal harmonic group actions on finite graphs, Discrete Mathematics, 338, No. 5 (2015), 784-792.

Harmonic Galois theory for finite graphs, in “Galois-Teichmueller Theory and Arithmetic Geometry” (H. Nakamura, F. Pop, L. Schneps, A. Tamagawa eds.), Advanced Studies in Pure Mathematics, 63 (2012), 121-140.

Genus bounds for harmonic group actions on finite graphs, Int. Math. Res. Notices, 2011, No. 19 (2011), 4515-4533. arXiv

Galois covers of the open p-adic disc, manuscripta math., 131, No. 1-2 (2010), 43-61. The original publication is available at

The pro-p Hom-form of the birational anabelian conjecture (with F. Pop), J. Reine Angew. Math. (Crelle’s Journal), 628 (2009), 121-127.