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| Lionel Levine | |
|---|---|
| Name | Lionel Levine |
| Fields | Mathematics, Probability Theory, Combinatorics |
| Workplaces | Brown University, Cornell University |
| Alma mater | Massachusetts Institute of Technology, Princeton University |
| Doctoral advisor | William Thurston |
| Known for | Rotor-router model, Abelian sandpile model, chip-firing |
Lionel Levine is an American mathematician specializing in probability theory, combinatorics, and dynamical systems. He is noted for rigorous contributions to the study of discrete growth models such as the rotor-router model and the abelian sandpile model, and for work connecting statistical mechanics to graph theory and potential theory. Levine has held academic positions at prominent institutions and collaborated with researchers across mathematics and computer science.
Levine earned undergraduate and graduate degrees from Massachusetts Institute of Technology and completed his Ph.D. at Princeton University under the supervision of William Thurston. During his formative years he engaged with topics spanning topology, differential geometry, and probability theory, interacting with scholars associated with Institute for Advanced Study and programs at Mathematical Sciences Research Institute. His doctoral work built on influences from research threads linked to Thurston's geometrization conjecture and collaborations touching on complex dynamics.
Levine held faculty appointments at Cornell University where he contributed to departments bridging mathematics and computer science, and later at Brown University where he served on the Mathematics Department faculty. He has taught courses intersecting probability theory, combinatorics, and graph theory and supervised doctoral students who pursued research in statistical mechanics, random processes, and discrete mathematics. Levine has been an invited speaker at conferences organized by American Mathematical Society, Society for Industrial and Applied Mathematics, and international meetings such as International Congress of Mathematicians satellite events.
Levine's research centers on discrete stochastic and deterministic processes on graphs, notably the rotor-router model (also called the Propp machine) and the abelian sandpile model (chip-firing). He established rigorous results on scaling limits, internal diffusion limited aggregation by deterministic analogues, and connections between discrete models and potential theory on Euclidean space, lattices, and general graphs. Collaborations with researchers including James Propp, Yuval Peres, Scott Sheffield, and David Wilson produced results linking rotor walks, internal DLA, and harmonic measure. Levine developed frameworks relating chip-firing to the Sandpile group (also called the critical group) of a graph and connected these to algebraic combinatorics topics such as spanning trees via the Matrix-Tree Theorem.
His work produced explicit characterizations of limiting shapes for growth models, using techniques from potential theory, probability theory, and discrete analytic functions. Levine proved aggregation and stabilization results for abelian networks, clarified self-organized criticality phenomena in the sandpile model, and advanced understanding of recurrent configurations through algebraic invariants tied to graph Laplacian operators. He contributed to the theory of rotor-router aggregation demonstrating deterministic analogs of random aggregation phenomena and to ergodicity properties of rotor walks on both transient and recurrent graphs such as Z^d lattices and regular trees. Levine's results have influenced research in statistical mechanics, theoretical computer science, and algebraic graph theory.
Levine's contributions have been recognized through invited lectures and prizes affiliated with organizations such as the American Mathematical Society and the Society for Industrial and Applied Mathematics. He has been invited to speak at venues including conferences organized by Probability Theory and Related Fields, Combinatorics symposia, and workshops hosted by the Banff International Research Station and the Mathematical Sciences Research Institute. His research has been supported by grants from agencies such as the National Science Foundation and fellowships that support early-career investigators in mathematics.
- Levine, L.; Propp, J. "What is a sandpile?" in collections of essays and lecture notes from MSRI programs on statistical mechanics. - Levine, L.; Peres, Y. "Scaling limits for internal DLA and related growth models." Annals of Probability and proceedings of Probability Theory and Related Fields conferences. - Levine, L.; Pegden, W.; Smart, C. "Deterministic and stochastic growth models: rotor-router aggregation and divisible sandpiles." Publications appearing in journals linked to combinatorics and probability theory. - Levine, L.; Wilson, D. "Rotor walks and reversible Markov chains." Articles published in journals addressing random processes and graph theory. - Levine, L. "The sandpile group of a graph and connections to the Matrix-Tree Theorem." Research articles in algebraic combinatorics venues.
Category:Living people Category:American mathematicians Category:Probabilists