Lionel Levine

Events:

• Generalizations of chip-firing and the critical group, July 8-12, 2013.
  
• Cornell probability summer school, July 15-26, 2013.
  
• The Cornell probability seminar meets Mondays during the academic year at 4pm in 406 Malott.
  

Research:

Papers and preprints:

• Lionel Levine, Wesley Pegden and Charles K. Smart
  Apollonian structure in the abelian sandpile (Draft of August 21, 2012)
  
  
• Ben Bond and Lionel Levine
  Abelian networks I. Foundations and examples (Draft of June 21, 2013)
  
  
• Louis J. Billera, Lionel Levine and Karola Mészáros
  How to decompose a permutation into a pair of labeled Dyck paths by playing a game
  Submitted
  
  
• David Jerison, Lionel Levine and Scott Sheffield
  Internal DLA and the Gaussian free field
  Duke Mathematical Journal, to appear.
  
  
• David Jerison, Lionel Levine and Scott Sheffield
  Internal DLA in higher dimensions
  Submitted
  
  
• Laura Florescu, Shirshendu Ganguly, Lionel Levine and Yuval Peres
  Escape rates for rotor walks in Z^d
  Submitted
  
  
• Lionel Levine and Yuval Peres
  The looping constant of Z^d
  Random Structures & Algorithms, to appear.
  
  
• Lionel Levine, Scott Sheffield and Katherine E. Stange
  A duality principle for selection games
  Proceedings of the American Mathematical Society, to appear.
  
  
• Christopher J. Hillar, Lionel Levine and Darren Rhea
  Equations solvable by radicals in a uniquely divisible group
  Bulletin of the London Mathematical Society (2013) 45(1): 61--79
  
  
• Tobias Friedrich and Lionel Levine
  Fast simulation of large-scale growth models
  Random Structures & Algorithms (2013) 42: 185–-213
  
  
• David Jerison, Lionel Levine and Scott Sheffield
  Logarithmic fluctuations for internal DLA
  Journal of the American Mathematical Society (2012) 25: 271--301
  
  
• Giuliano Giacaglia, Lionel Levine, James Propp and Linda Zayas-Palmer
  Local-to-global principles for the hitting sequence of a rotor walk
  Electronic Journal of Combinatorics (2012) 19: P5
  
  
• Lionel Levine
  Sandpile groups and spanning trees of directed line graphs
  Journal of Combinatorial Theory A (2011) 118: 350-–364
  
  
• Lionel Levine
  Parallel chip-firing on the complete graph: devil's staircase and Poincaré rotation number
  Ergodic Theory and Dynamical Systems (2011) 31: 891--910
  
  
• Wouter Kager and Lionel Levine
  Rotor-router aggregation on the layered square lattice
  Electronic Journal of Combinatorics (2010) 17: R152
  
  
• Anne Fey, Lionel Levine and David B. Wilson
  Driving sandpiles to criticality and beyond
  Physical Review Letters (2010) 104: 145703
  
  
• Anne Fey, Lionel Levine and David B. Wilson
  The approach to criticality in sandpiles
  Physical Review E (2010) 82: 031121
  
  
• Wouter Kager and Lionel Levine
  Diamond Aggregation
  Mathematical Proceedings of the Cambridge Philosophical Society (2010) 149: 351--372
  
  
• Anne Fey, Lionel Levine and Yuval Peres
  Growth rates and explosions in sandpiles
  Journal of Statistical Physics (2010) 138: 143--159
  
  
• Lionel Levine and Yuval Peres
  Scaling limits for internal aggregation models with multiple sources
  Journal d'Analyse Mathematique (2010) 111: 151--219
  
  
• Itamar Landau and Lionel Levine
  The rotor-router model on regular trees
  Journal of Combinatorial Theory A (2009) 116: 421--433
  
  
• Lionel Levine and Yuval Peres
  Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile
  Potential Analysis 30 (2009), 1--27
  
  
• Lionel Levine
  The sandpile group of a tree
  European Journal of Combinatorics 30 (2009) 1026--1035
  
  
• Alexander E. Holroyd, Lionel Levine, Karola Mészáros, Yuval Peres, James Propp and David B. Wilson
  Chip-firing and rotor-routing on directed graphs
  in In and Out of Equilibrium II, Progress in Probability vol. 60 (Birkhauser, 2008)
  
  
• Lionel Levine and Yuval Peres
  Spherical asymptotics for the rotor-router model in Z^d
  Indiana University Mathematics Journal 57 (2008), no. 1, 431--450
  
  
• Christopher J. Hillar and Lionel Levine
  Polynomial recurrences and cyclic resultants
  Proceedings of the American Mathematical Society 135 (2007), 1607--1618
  
  
• Lionel Levine
  Fractal sequences and restricted Nim
  Ars Combinatoria 80 (2006), 113--127
  

Expository notes:

• Lionel Levine and Katherine E. Stange
  How to make the most of a shared meal: plan the last bite first.
  American Mathematical Monthly, to appear.
  
  
• Lionel Levine and James Propp
  WHAT IS a sandpile?
  Notices of the American Mathematical Society 57, no. 8 (September 2010), 976--979
  
  
• Lionel Levine and Yuval Peres
  The rotor-router shape is spherical
  Mathematical Intelligencer 27 (2005) no. 3, 9--11
  
  
• Lionel Levine
  Fermat's little theorem: a proof by function iteration
  Mathematics Magazine 72, no. 4 (1999), 308--309

Selected talks:

• Abelian networks, Berkeley combinatorics seminar, November 2011.
  
  
• Logarithmic fluctuations from circularity, AMS Eastern sectional meeting, April 2011.
  
  
• An algebraic analogue of a formula of Knuth, FPSAC, San Francisco, August 2010.
  
  
• Growth rates and explosions in sandpiles, University of Washington / PIMS mathematics colloquium, January 2010.
  
  
• Chip-firing and a devil's staircase, FPSAC, Hagenberg, Austria, July 2009.
  
  
• Obstacle problems and lattice growth models, Workshop in PDE and potential theory, KTH Stockholm, June 2009.
  
  
• All talks since late 2007.

Teaching:

• Math 4740: Stochastic processes, Spring 2013.
  
  
• Math 7770: Topics in probability: Laplacian growth, Fall 2012.
  
  
• 18.312 Algebraic Combinatorics, Spring 2011.
  
  
• 18.01 Calculus, Spring 2011.
  
  
• 2009-2010: As part of the UROP program, I mentored MIT undergraduates Giuliano Giacaglia and Linda Zayas-Palmer in a research problem about bijections between spanning trees and recurrent configurations of the sandpile model. Jim Propp helped supervise the project starting in fall 2009, and we proved some conjectures of his about local-to-global properties of rotor-router walk. A paper is in the works. In fall 2010 Damien Jiang joined the group and we are working on characterizing the identity elements of sandpile groups of circulant graphs.
  
  
• 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

Some unpublished papers and notes:

• (with Y. Peres) 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.
  
  
  
  
• 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:

• This image has more pixels than the population of the earth! (See here for the story of how it was generated.)
  
  
• A few links to friends' and family's sites.
  
  
• Occasional attempts to fit something nontrivial into 140 characters or less.
  
  
• +Lionel Levine
  
  
• Photos from my India trip in 2008. (Central Asia & China will be added eventually!)
  
  
• Job search materials.