Lionel Levine's Home Page

Lionel Levine

Room 2-332
Department of Mathematics
Massachusetts Institute of Technology

levine

math . mit . edu

Quick links: Top | Recent and upcoming talks | Papers and preprints | Teaching | Picture Gallery

I'm a Moore Instructor and NSF Postdoctoral Fellow in the math department at MIT. During January-June 2010 I will be a Visiting Researcher at the Theory Group, Microsoft Research.

I finished my Ph.D. at Berkeley in December 2007, where I was a student of Yuval Peres. My thesis used ideas from free boundary problems in PDE to prove limiting shape theorems for some combinatorial and probabilistic growth models.

Henry Cohn, Todd Kemp, Scott Sheffield, Dan Stroock and I organize the MIT Probability Seminar.

Recent and upcoming talks:

University of Washington / PIMS mathematics colloquium, January 29, 2010.
UPenn combinatorics and probability seminar, November 17, 2009.
Brown discrete math seminar, November 10, 2009.
Séminaire de combinatoire et d'informatique mathématique du LaCIM, Montreal, October 16, 2009.
Dartmouth Computer Science Colloquium, October 7, 2009.
Growth rates and explosions in sandpiles, 27th Brazilian Colloquium of Mathematics, Rio de Janeiro, July 2009.
Chip-firing and a devil's staircase, FPSAC, Hagenberg, Austria, July 2009.
Obstacle problems and lattice growth models, KTH Stockholm, June 2009.
Older talks:
- Video of a talk I gave at Microsoft in 2008 on the scaling limit of internal DLA (requires windows).
- All talks since late 2007.

Papers and preprints (click here for brief descriptions):

Sandpile groups and spanning trees of directed line graphs, submitted.
(with W. Kager) Diamond Aggregation, Mathematical Proceedings of the Cambridge Philosophical Society, to appear.
(with A. Fey and Y. Peres) Growth rates and explosions in sandpiles, Journal of Statistical Physics, to appear.
Parallel chip-firing on the complete graph: devil's staircase and Poincaré rotation number, submitted.
(with Y. Peres) Scaling limits for internal aggregation models with multiple sources, to appear in J. d'Analyse Math.
(with A. Holroyd, K. Meszaros, Y. Peres, J. Propp and D. Wilson) Chip-firing and rotor-routing on directed graphs, in In and Out of Equilibrium 2, Progress in Probability, vol. 60 (2008).
(with I. Landau) The rotor-router model on regular trees, J. Combin. Theory Ser. A 116 (2009), 421--433.
(with Y. Peres) Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile, Potential Analysis 30 (2009), 1--27.
The sandpile group of a tree, European Journal of Combinatorics 30 (2009) 1026--1035.
- An extended abstract summarizing the results of the previous three papers appears in the FPSAC 2008 proceedings.
(with Y. Peres) Spherical asymptotics for the rotor-router model in Z^d, Indiana Univ. Math. J. 57 (2008), no. 1, 431--450.
- A less technical summary of this paper appeared in the Mathematical Intelligencer 27 (2005) no. 3, 9--11
(with C. Hillar) Polynomial recurrences and cyclic resultants, Proceedings of the AMS 135 (2007), 1607--1618.
Fractal sequences and restricted Nim, Ars Combinatoria 80 (2006), 113--127.

Teaching:

Summer & Fall 2009: As part of MIT's UROP program, I am mentoring undergraduates Giuliano Giacaglia and Linda Zayas-Palmer in a research problem about bijections between spanning trees and recurrent configurations of the sandpile model. Starting this fall, Jim Propp will help supervise the project, and we plan to work on some recent conjectures of his about rotor-routing.
18.177: Stochastic Processes. In Spring 2009 I taught four guest lectures in Scott Sheffield's graduate topics class, covering random walks and electrical networks, uniform spanning trees, Wilson's algorithm, loop-erased walk and convergence to SLE.
18.095: Mathematics Lecture Series, January 2009.
Math 54: Linear Algebra and Differential Equations, UC Berkeley, Spring 2005
Math 1A: Calculus, UC Berkeley, Fall 2004

Unpublished and expository papers:

Internal erosion and the exponent 3/4 describes how an unusual exponent arises from a very simple erosion process in one dimension. The proof we give is due to Kingman and Volkov (2003), who thought of this not as an erosion process but as a model of a gunfight (!)
Orlik-Solomon Algebras of Hyperplane Arrangements, an expository paper proving the basic theorem of Orlik-Solomon and Brieskorn on the cohomology ring of the complement of a complex arrangement, along with some remarks about the associated combinatorics of the intersection lattice.
My senior thesis on the rotor-router model was advised by Jim Propp. Here are some slides for talks I gave on this material in 2002 and 2003.
Confounding factors for Hamilton's rule, the final paper for an anthropology class I took in 2002. I found that the rule is surprisingly sensitive to changes in Hamilton's original hypotheses, which casts some doubt on the evolutionary stability of kin selection.
Hall's marriage theorem and Hamiltonian cycles in graphs, the final paper from a spring 2001 reading course with Richard Stanley.
A graph on n=24 vertices having no Hamiltonian cycle, in which every set of k<22 vertices is adjacent to at least k+3 vertices:
Some basic results on Sturmian words, written before I knew that's what they were called. These results are all known in some form. Theorem 1 and the surprising corollary to Theorem 2 go back to Morse and Hedlund (1940). I don't know if Theorem 2 appears anywhere in exactly this form.

The beginning of the factor tree of the Sturmian word of slope sqrt(2)/2 and intercept zero:

Odds and Ends:

A few links to friends' and family's sites.
Occasional attempts to fit something nontrivial into 140 characters or less.
Photos from my India trip in 2008. (Central Asia & China will be added eventually!)
Job search materials from 2007.