Lionel Levine 1043 Evans Hall University of California Berkeley, CA 94720 (510) 965-8421 levinemath.berkeley.edu

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Recent and upcoming talks:

• AMS Western Sectional Meeting, Vancouver, October 2008.
  
  
• CRM Workshop on Laplacian Growth and Related Topics, Montreal, August 2008.
  
  
• FPSAC, Valparaiso, Chile, June 23, 2008.
  
  
• "Random Interactions" Seminar, Tata Institute for Fundamental Research, Mumbai, India, April 9, 2008.
  
  
• Brown "Xtreme Geometry" Seminar, February 21, 2008.
  
  
• Free boundary problems arising from combinatorial and probabilistic growth models, MIT PDE Seminar, February 15, 2008.
  
  
• The limiting shape of internal DLA with multiple sources, Microsoft Research, January 30, 2008.
  
  
• The limiting shape of internal DLA with multiple sources, VU Amsterdam, January 23, 2008.
  
  
• Internal diffusion-limited erosion, EURANDOM, Eindhoven, Netherlands, January 18, 2008.
  
  
• Scaling limits for internal aggregation models with multiple sources, Berkeley Probability Seminar, November 14, 2007.
  
  
• Chip-firing and rotor-routing on trees, MIT Combinatorics Seminar, November 2, 2007.
  
  

Papers and preprints (click here for brief descriptions):

• (with Y. Peres) Scaling limits for internal aggregation models with multiple sources, submitted.
  
  
• (with A. Holroyd, K. Meszaros, Y. Peres, J. Propp and D. Wilson) Chip-firing and rotor-routing on directed graphs, submitted.
  
  
• (with I. Landau) The rotor-router model on regular trees, to appear in J. Combin. Theory Ser. A.
  
  
• (with Y. Peres) Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile, to appear in Potential Analysis.
  
  
• The sandpile group of a tree, to appear in the European Journal of Combinatorics.
  
  
• (with Y. Peres) Spherical asymptotics for the rotor-router model in Z^d, Indiana Univ. Math. J. 57 (2008), no. 1, 431--450.
  
  
  • Here's our less technical summary, The rotor-router shape is spherical, 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:

• Math 54: Linear Algebra and Differential Equations, Spring 2005
  
  
• Math 1A: Calculus, 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Unpublished and expository papers: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: