## Lionel LevineAssistant ProfessorDepartment of Mathematics Cornell University Email: my last name at math.cornell.edu Office: 438 Malott Hall |

I study abelian networks, an invention of the statistical physicist Deepak Dhar, who called them "abelian distributed processors." In one point of view, abelian networks are discrete dynamical systems with a certain strong convergence property. From another point of view, they are a means of associating algebraic invariants to graphs. In yet a third viewpoint, they are a model of asynchronous computation: nodes of the network pass messages along edges to perform a computation requiring no central control over timing.

Some other things I've worked on since my Ph.D. are the scaling limit of the abelian sandpile in Z^2 where an Apollonian circle packing makes a surprise appearance, the devil's staircase for parallel chip-firing, refuting the density conjecture for sandpiles, logarithmic fluctuations for internal DLA, fast simulation of growth models, a generalization of Knuth's formula for spanning trees, and word equations in uniquely divisible groups.

My Ph.D. thesis, advised by Yuval Peres at Berkeley, used ideas from free boundary problems in PDE to prove limiting shape theorems for growth models in probability and combinatorics.

I thank the National Science Foundation and the Sloan Foundation for supporting my research.

- Alexander E. Holroyd, Lionel Levine and Peter Winkler

Abelian logic gates

*Draft of November 1, 2015*

- Daniel C. Jerison, Lionel Levine and John Pike

Mixing time and eigenvalues of the abelian sandpile Markov chain

*Draft of November 1, 2015*

- Matthew Farrell and Lionel Levine

Multi-Eulerian tours of directed graphs

*Submitted*

- Matthew Farrell and Lionel Levine

CoEulerian graphs

*Proceedings of the American Mathematical Society, to appear*

- Lionel Levine, Mathav Murugan, Yuval Peres and
Baris
Ugurcan

The divisible sandpile at critical density

*Annales Henri Poincare, to appear*

- Shirshendu Ganguly, Lionel Levine, Yuval Peres and James Propp

Formation of an interface by competitive erosion

*Submitted*

- Benjamin Bond and Lionel Levine

Abelian networks III. The critical group

*Journal of Algebraic Combinatorics, to appear*

- Benjamin Bond and Lionel Levine

Abelian networks II. Halting on all inputs

*Selecta Mathematica, to appear*

- Laura Florescu, Lionel Levine and Yuval Peres

The range of a rotor walk

*Submitted*

- Elisabetta Candellero, Shirshendu Ganguly, Christopher Hoffman and Lionel Levine

Oil and water: a two-type internal aggregation model

*Submitted*

- Benjamin Bond and Lionel Levine

Abelian networks I. Foundations and examples

*Submitted*

- Lionel Levine, Wesley Pegden and Charles K. Smart

Apollonian structure of integer superharmonic matrices

*Submitted*

- Lionel Levine, Wesley Pegden and Charles K. Smart

Apollonian structure in the abelian sandpile

*Geometric and Functional Analysis, to appear*

- Lionel Levine

Threshold state and a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle

*Communications in Mathematical Physics (2015) 335(2):1003–-1017*

- Louis J. Billera, Lionel Levine and Karola Mészáros

How to decompose a permutation into a pair of labeled Dyck paths by playing a game

*Proceedings of the American Mathematical Society (2015) 143:1865-–1873*

- David Jerison, Lionel Levine and Scott Sheffield

Internal DLA and the Gaussian free field

*Duke Mathematical Journal (2014) 163(2):267–-308*

- Lionel Levine and Yuval Peres

The looping constant of Z^d

*Random Structures & Algorithms (2014) 45:1--13*

- Laura Florescu, Shirshendu Ganguly, Lionel Levine and Yuval Peres

Escape rates for rotor walks in Z^d

*SIAM Journal on Discrete Mathematics (2014) 28(1):323--334*

- Lionel Levine, Scott Sheffield and Katherine E. Stange

A duality principle for selection games

*Proceedings of the American Mathematical Society (2013) 141(12): 4349--4356*

- 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

Internal DLA in higher dimensions

*Electronic Journal of Probability (2013) 18(98): 1--14*

- 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*

- Lionel Levine and Katherine E. Stange

How to make the most of a shared meal: plan the last bite first.

*American Mathematical Monthly 119 (2012) no. 7, 550--565*

- Lionel Levine and James Propp

WHAT IS a sandpile?

*Notices of the American Mathematical Society 57, (2010) no. 8, 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*

- Circles in the sand

- Introduction to abelian networks

- Logarithmic fluctuations from circularity

- An algebraic analogue of a formula of Knuth

- Chip-firing and a devil's staircase

- Obstacle problems and lattice growth models

- All talks since late 2007.

- Math 6710: Probability Theory I, Fall 2014.

- Math 4740: Stochastic processes, Spring 2014.

- Math 4740: Stochastic processes, Spring 2013.

- Math 7770: Topics in probability: Laplacian growth, Fall 2012.

- 18.312 Algebraic
Combinatorics, Spring 2011.

- (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:

- 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.