Scott Sheffield
sheffield@math.mit.edu
(617) 253-4350
Massachusetts Institute of Technology
77 Massachusetts Avenue, 2-180
Cambridge, MA 02139-4307
Education
Ph.D., Mathematics, Stanford University, April 2003
A.M., Mathematics, Harvard University, June 1998
A.B., Mathematics, Harvard University, June 1998
Major research awards
2011
2010
2009
2007
2006
Loève Prize
ICM Invited Lecture
Presidential Early Career Award for Scientists and Engineers (PECASE)
Sloan Research Fellowship
Rollo Davidson Prize
Major research grants
2012-2017 NSF Award: DMS 1209044
2008-2013 NSF PIRE Continuing Grant: OISE 0730136 with Charles Newman,
Daniel Stein, Srinivasa Vardhan, and Gerard Ben Arous
2007-2012 NSF CAREER Award: DMS 0645585
2011-2012 BSF Award: 2010357 with Ariel Yadin
2010-2011 MISTI Global Seed Funds with Bertrand Duplantier
2004-2007 NSF Postdoctoral Fellowship: DMS 0403182
Primary Employment
MIT Department of Mathematics
Fall 2008–present
Courant Institute; NYU Mathematics Department
Fall 2007–Spring 2008
Institute for Advanced Study
Fall 2006–Spring 2007
Courant Institute; NYU Mathematics Department
Fall 2005–Spring 2006
U.C. Berkeley Mathematics Department
Summer 2004–Summer 2005
Microsoft Research
Summer 2002–Summer 2004
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Professor
Associate Professor
NSF/IAS Postdoctoral Fellow
Assistant Professor
NSF Postdoctoral Fellow
Postdoctoral Researcher
Professional Activities
Associate editor:
Co-organizer:
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Referee:
Annales de l’Institut Henri Poincaré (until 2011), Probability Theory and Related Fields, and International Mathematics Research Notices.
“Random Geometry” semester at Newton Institute in Cambridge,
UK. January to July, 2015.
Charles River Lectures, October 2012 and October 2013.
MIT probability seminar, September 2008 to present.
MSRI workshop on “Statistical Mechanics and Conformal Invariance”, March 26-30, 2012.
2008-09 CRM thematic program on Random Surfaces, Random
Functions, Interfaces
Summer 2007 IAS/PCMI Summer Session on Statistical Physics
Fall 2006 KITP Semester on Stochastic Geometry and Field Theory
Research papers in many journals including Acta Mathematica, Annals of Mathematics, Annals of Probability, Communications in
Mathematical Physics, Duke Journal of Mathematics, Journal of
the American Mathematical Society, and Memoirs of the American
Mathematical Society
Ph.D. students
Completed: Oren Louidor (Technion), Tom Alberts (Caltech/University of Utah),
Sungwook Kim (Bank of America)
Current: Sam Watson (MIT), Asad Lodhia (MIT), Xin Sun (MIT), Menglu Wang (MIT)
Selected Talks
6/2014
5/2013
4/2013
4/2013
2/2013
2/2013
3/2012
3/2012
4/2012
5/2012
2/2012
5/2011
5/2011
1/2011
8/2010
7/2010
6/2010
5/2010
5/2010
5/2010
3/2010
Plenary Lecture at 37th Conference on Stochastic Processes (SPA),
Buenos Aires, Argentina
Conference in honor of Jean-Michel Bismut at Paris-Sud, France
Workshop on Conformal Invariance at Simons Center, Stony Brook, NY
Probability and Statistics Day at UC San Diego
Minicourse on Imaginary Geometry at Les Diablaret, Switzerland
Minicourse on Game Theory and PDEs at University of Pittsburg
Stanford Mathematics Colloquium
Loève Prize Lecture at UC Berkeley
Courant Institute Probability Seminar, New York
Advances in Percolation Meeting at University of Michigan
American Physical Society Joint Meeting in Boston
Game Theory and PDE Conference at University of Kentucky
Minicourse at Oberwolfach Conference on Random Geometry
AMS Invited Address at Join Mathematics Meeting in New Orleans
ICM Invited Lecture in Hyderabad, India
Minicourse at Clay Institute Summer School in Búzios, Brazil
Full Course (15 lectures) at PCMI Summer School in Seattle
Conference on Conformal Maps in Ascona, Switzerland
Barrett Lectures, University of Tennessee
NYU Probability Seminar
Rice University Mathematics Colloquium
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Selected Talks
1/2010
12/2009
10/2009
7/2009
7/2009
4/2009
2/2009
1/2009
12/2008
10/2008
8/2007
10/2006
10/2006
8/2006
4/2006
1/2006
6/2006
5/2006
2/2006
1/2006
10/2005
10/2005
9/2005
8/2005
8/2005
6/2005
5/2005
4/2005
11/2004
11/2004
11/2004
10/2004
9/2004
9/2004
9/2004
7/2004
2/2004
2/2004
2/2004
2/2004
2/2004
1/2004
11/2003
11/2003
11/2003
7/2003
7/2003
5/2003
3/2003
1/2003
10/2002
10/2002
5/2002
PECASE Ceremony
Oded Schramm Memorial Conference in Jerusalem
Probability Seminar at Boston University
Minicourse on Tug of War at UBC, Vancouver, Canada
Minicourse on SLE at Cornell Summer School
Blyth Lectures (3-lecture series) at University of Toronto
Minicourse at Les Diablaret, Switzerland
“Random functions/surfaces/interfaces” Workshop at CRM, Montreal
Harvard Probability Seminar
Harvard Analysis Seminar
32nd SPA at University of Illinois at Urbana Champaign
Midwestern Probability Colloquium Tutorial, Northwestern University
University of Chicago Applied Math./PDE Seminar
IMS Annual Meeting, Rio de Janeiro, Brazil
Sherman Conference, University of Indiana
AIM ARCC workshop on Random Analytic Functions
Conference on Stochastic Processes in Math. Physics; Florence, Italy
Conference on SLE and Loop Measure, Cornell
Courant Institute Probability Seminar, New York
Stanford Mathematics Colloquium
Conference in honor of Peter Lax and Louis Nirenberg, NYU
Princeton Mathematical Physics Seminar
Minicourse at Fields Institute, Toronto
IMA Workshop in Minneapolis
Brigham Young University Colloquium
BIRS Conference in Banff
Berkeley Probability Seminar
MSRI Postdoctoral Seminar
Berkeley Probability Seminar
Berkeley Interdisciplinary Stochastic Processes Colloquium
Stanford Analysis Seminar
U.C. Davis Colloquium
University of Utah Probability Seminar
Brigham Young University Mathematics Colloquium
Berkeley Analysis Seminar
6th World Congress for Bernoulli Soc./67th Ann. Meeting of IMS in Barcelona
Courant Institute Probability Seminar.
Princeton Ergodic Theory Seminar and Probability Seminar
Princeton Mathematical Physics Seminar
Cornell Probability Seminar
Cornell Mathematics Colloquium
University of Washington Mathematics Colloquium
UCLA Probability Seminar
University of Wisconsin Probability Seminar
University of Washington Probability Seminar
ICMS Conformal Invariance Conference in Edinburgh
ICMS Conformal Invariance Conference in Edinburgh
Stanford Probability Seminar
IHP Random Growth Conference in Paris
Joint Mathematics Meeting in Baltimore
Northwest Probability Seminar in Seattle
Stanford Probability Seminar
Stanford Probability Seminar
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Publications and arXiv preprints
• Book-length papers and series:
1. Random surfaces. Scott Sheffield. Astérisque 2006, No. 304 (177 pages).
Proved several conjectures (along with vast generalizations) of Cohn, Elkies, Kenyon, and Propp
about Gibbs measure classifications and random surface local statistics; classified possible slopes
of smooth phases (a.k.a. crystal facets) for general class of two dimensional random surfaces.
2. Contour lines of the two-dimensional discrete Gaussian free field. Oded Schramm
and Scott Sheffield. Acta Mathematica 2009 (130 pages).
Proved that the chordal level lines of the discrete Gaussian free field have scaling limits that are
variants of SLE(4).
3. Conformal loop ensembles: the Markovian characterization and the loop-soup construction, Scott Sheffield and Wendelin Werner. Annals of Math. 2012 (91 pages).
Characterized conformally invariant random collection of loops (“conformal loop ensembles”)
in terms of a Markov property similar to the one used to characterize conformally invariant
random paths (“Schramm Loewner evolutions”). Obtained construction of CLE loops as outer
boundaries of a Poissonian random collection of Brownian loops with intensity given by the
“central charge” parameter.
4. Imaginary geometry I: Interacting SLEs. Jason Miller and Scott Sheffield. arXiv
2012 (154 pages).
Constructed and thoroughly studied “imaginary” analog of Liouville quantum gravity, recognizing
corresponding geodesics as simple SLE curves coupled with the Gaussian free field, and providing
interpretation of non-simple SLE curves (“counterflow lines”) as ordered “light cones” obtained
as unions of all angle-varying curves with angles in some range.
5. Imaginary geometry II: reversibility of SLEκ (ρ1 ; ρ2 ) for κ ∈ (0, 4), Jason Miller and
Scott Sheffield. arXiv 2012 (89 pages).
Further developed imaginary geometry machinery and applied it to settle conjecture of Dubédat
by establishing“time reversal symmetry” of SLEκ (ρ1 ; ρ2 ) curves.
6. Imaginary geometry III: reversibility of SLEκ for κ ∈ (0, 8), Jason Miller and Scott
Sheffield. arXiv 2012 (35 pages).
Applied imaginary geometry machinery to settle long-standing conjecture of Oded Schramm by
establishing “time reversal symmetry” of SLEκ for κ ∈ (4, 8).
7. Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling
trees, Jason Miller and Scott Sheffield. arXiv 2013 (174 pages).
Developed robust theory of joint laws of imaginary geometry geodesics started at interior points of
domain, interactions between these paths, trees of merging geodesics, and the space-filling curves
tracing the boundaries of these space-filling trees. Applied imaginary geometry machinery to
settle additional time-reversal symmetry conjectures and establish local finiteness of conformal
loop ensembles.
• Quantum gravity
1. Liouville Quantum Gravity and KPZ. Bertrand Duplantier and Scott Sheffield.
Inventiones Mathematicae 185:333–393 (2011).
Formulated and proved a mathematical statement of the Knizhnik-Polyakov-Zamolodchikov formula, which relates fractal scaling dimensions in Euclidean geometry to fractal scaling dimensions of analogous fractals defined on quantum surfaces.
2. Duality and KPZ in Liouville Quantum Gravity. Bertrand Duplantier and Scott
Sheffield. Phys. Rev. Lett. 102 (2009).
Constructed a “dual” version of quantum gravity corresponding to random surfaces with many
bottlenecks, and a corresponding dual formulation of KPZ.
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3. Schramm Loewner Evolution and Liouville Quantum Gravity. Bertrand Duplantier
and Scott Sheffield. Phys. Rev. Lett. (2012).
Described the conformal symmetries of the “quantum zipper”, which involves the conformal
welding of distinct quantum surfaces along their boundaries.
4. Conformal weldings of random surfaces: SLE and the quantum gravity zipper.
Scott Sheffield. arXiv (2010).
Showed that if one glues together two “canonical random surfaces” (constructed via Liouville
quantum gravity) the interface between the two has the law of a “canonical random curve”
(Schramm Loewner evolution) on the resulting combined surface.
5. Quantum gravity and inventory accumulation. Scott Sheffield arXiv (2011).
Established a scaling limit result for the exploration tree of a discretized quantum surface decorated with a statistical physics model (the FK cluster model). Proof uses new bijection between
decorated surfaces and “hamburger/cheeseburger” inventory accumulation histories.
6. Critical Gaussian Multiplicative Chaos: Convergence of the Derivative Martingale.
Bertrand Duplantier, Rémi Rhodes, Scott Sheffield, and Vincent Vargas. arXiv
(2012).
Constructed the Liouville quantum gravity random measure corresponding to the critical parameter value γ = 2 (c.f. previous works which required γ < 2).
7. Renormalization of Critical Gaussian Multiplicative Chaos and KPZ formula. Bertrand
Duplantier, Rémi Rhodes, Scott Sheffield, and Vincent Vargas. arXiv (2012).
Extended the KPZ formula to the γ = 2 setting.
• Gaussian free fields and SLE
1. Strong path convergence from Loewner driving function convergence. Scott Sheffield,
Nike Sun. Annals of Probability 40:578–610 (2012).
Showed that a weaker (and often more easily established) form of convergence to SLE implies a
much stronger form of convergence if the weaker form is established in both directions.
2. Schramm’s proof of Watts’ formula. Scott Sheffield, David B. Wilson. Annals of
Probability (2011).
Gave a detailed and illustrated exposition of Schramm’s beautiful unpublished proof of Watt’s
formula, as part of a memorial volume for Oded Schramm.
3. Hausdorff dimension of the SLE curve intersected with the real line Tom Alberts,
Scott Sheffield. Electronic Journal of Probability 13:1166-1188 (2008).
Computed Hausdorff dimension of an SLEκ intersected with the domain boundary when κ ∈ (4, 8)
4. The natural parametrization for the Schramm-Loewner evolution. Gregory Lawler,
Scott Sheffield. Annals of Probability, 39:1896–1937 (2011).
Used Doob-Meyer decomposition to construct and prove uniqueness of the parameterization of
SLE with the scaling properties of a fractional-dimensional length measure.
5. The Covariant Measure of SLE on the Boundary. Tom Alberts, Scott Sheffield.
Probab. Theory and Rel. Fields 149:331–371 (2011).
Described an analogous fractal measure for the intersection of an SLE with the boundary.
6. Harmonic explorer and its convergence to SLE(4). Oded Schramm and Scott
Sheffield. Annals of Probability 33(6):2127–2148 (2005).
Proved that the harmonic explorer (a natural random interface constructed from simple random
walks) converges to SLE(4) as mesh size gets finer.
7. Gaussian free fields for mathematicians. Scott Sheffield. Probability Theory and Related
Fields 139:521–541 (2007).
Surveyed mathematical literature on the Euclidean bosonic massless free field and studied new
exploration-process-based couplings of Gaussian free fields and Brownian motions.
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8. Exploration trees and conformal loop ensembles. Scott Sheffield. Duke Mathematical
Journal 147:79–129 (2009).
Introduced and gave a rigorous construction of objects called “conformal loop ensembles” (CLEs),
which are canonical random collections of loops that look locally like SLE curves and arise as
scaling limits of various loop models from statistical physics.
9. Conformal radii for conformal loop ensembles. Oded Schramm, Scott Sheffield,
David Wilson. Communications in Mathematical Physics 1149–1159 (2011).
Computed the fractal dimension of the “carpet” of points not surrounded by a CLE loop, thereby
confirming a physics literature prediction made by Cardy and Ziff.
10. A contour line of the continuum Gaussian free field. Oded Schramm, Scott Sheffield.
Probability Theory and Related Fields. (2012).
Showed that although the Gaussian free field is a random distribution (not a function) it has
well-defined level sets. Developed more general theory of “local” sets of Gaussian free fields.
• Internal Diffusion Limited Aggregation (DLA)
1. Logarithmic fluctuations for internal DLA. David Jerison, Lionel Levine, Scott
Sheffield. JAMS 25:271–301 (2012).
Resolved an old conjecture of Greg Lawler by showing that the internal DLA growth model in
two dimensions produces shapes remarkably close to discs: the fluctuations from roundness are
asymptotically at most logarithmic in the diameter of the set.
2. Internal DLA in Higher Dimensions. David Jerison, Lionel Levine, Scott Sheffield.
Electronic Journal of Probability, to appear (2012).
Extended the fluctuation results to higher dimensions, and in fact showed that in dimension
greater than two the fluctuations are at most root logarithmic.
3. Internal DLA and the Gaussian free field. David Jerison, Lionel Levine, Scott
Sheffield. Duke Mathematical Journal, to appear (2012).
Described the scaling limit of the “average” fluctuations from roundness in terms of the Gaussian
free field. Showed that by some measures the discrete approximate-spheres produced by internal
DLA in high dimensions are even “rounder” than the discrete sets obtained by intersecting a
sphere with a lattice.
• Game theory, PDEs, and Lipschitz extension theory
1. Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces.
Assaf Naor, Yuval Peres, Oded Schramm, and Scott Sheffield. Duke Mathematical
Journal, 134(1):165–197 (2006).
Proved conjecture of Ball about Markov type and settled several conjectures on Lipschitz extensions and embeddings. Answered a question of Johnson and Lindenstrauss (1982) by showing
that for 1 < q < 2 < p < ∞, any Lipschitz mapping from a subset of Lp to Lq has a Lipschitz
extension defined on all of Lp .
2. Random-turn Hex and other selection games. Yuval Peres, Oded Schramm, Scott
Sheffield, and David Wilson. American Mathematical Monthly, 114:373–387 (2007).
Described optimal strategy for the variant of the game of Hex in which turn order is determined
by a sequence of fair coin tosses. Proved similar results for more general classes of games and
studied typical game trajectories when players play optimally.
3. Tug-of-war and the infinity Laplacian. Yuval Peres, Oded Schramm, Scott Sheffield,
and David Wilson. Journal of the American Mathematical Society, 22:167–210 (2009).
Proved that every Lipschitz function on a length space has a unique optimal extension—along
with several related results about PDE and metric space theory—using a spatial game called tug
of war.
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4. Absolutely minimal Lipschitz extension of tree-valued mappings. Assaf Naor, Scott
Sheffield. Math. Ann. 354:1049–1078 (2012).
Extended the optimal Lipschitz extension theory of real-valued maps to maps taking values in a
metric tree. The game of “tug of war” is replaced with an amusing variant called “politics” in
which players are allowed to change targets but are penalized for moving closer to the positions
of their opponents.
5. Tug of war with noise: a game theoretic view of the p-Laplacian. Yuval Peres, Scott
Sheffield. Duke Mathematical Journal 145:91–120 (2008).
Extended the relationship between tug of war and infinity harmonic functions to a setting in
which random “noise” is added to each tug-of-war step and the infinity Laplacian is replaced
with the p-Laplacian.
6. Tug-of-war and infinity Laplace equation with vanishing Neumann boundary condition. Tonći Antunović, Yuval Peres, Scott Sheffield, Stephanie Somersille. Comm.
Partial Differential Equations. 37:1839–1869 (2012).
Extended tug-of-war theory to lattice approximations of Euclidean domains with partially free
boundary conditions.
7. Vector-valued optimal Lipschitz extensions. Scott Sheffield, Charles K. Smart.
Comm. Pure Appl. Math. 65:128–154 (2012).
Partially extended optimal Lipschitz extension theory to functions taking values in higher dimensional Euclidean spaces, where rather different types of statements are true.
8. A duality principle for selection games. Lionel Levine, Scott Sheffield, Katherine
E. Stange. Proceedings of the AMS (to appear).
Established amusing result about a game in which players with publicly known preferences take
turns (according to a prescribed turn order) selecting objects from a finite set: the way the
objects are ultimately partitioned by a group of maximally selfish and shortsighted players (termed
“boorish louts”) is the same the way they are distributed by maximally altruistic and strategic
players (termed “gallant knights”) with the same preferences (if the turn order is reversed).
• Dimers, tilings, and trees:
1. Dimers and amoebae. Andrei Okounkov, Richard Kenyon and Scott Sheffield.
Annals of Mathematics, 163(3):1019–1056 (2006).
Exhibited unexpected connection between algebraic geometry and statistical physics: the surface
tension of a class of perfect-matching-based random surfaces is the Legendre dual of the Ronkin
function of the spectral curve of the Kastelyn operator, which turns out to be a Harnack curve.
The amoeba of the curve is the phase diagram.
2. Dimers, tilings and trees. Richard Kenyon and Scott Sheffield. Journal of Combinatorial Theory Series B, 92(2):295–317 (2004).
Generalized the famous Temperley bijection (between domino tilings and spanning trees on grids)
to arbitrary planar graphs by producing weight-preserving bijection between perfect matchings of
weighted, bipartite planar graphs and spanning trees of so-called T-graphs.
3. Ribbon Tilings and Multidimensional Height Functions. Scott Sheffield. Transactions of the AMS 354:4789–4813 (2002).
Employed a multi-dimensional height function to show that any two tilings of a finite grid graph
by so-called “ribbon tiles” can be connected by a finite sequence of local moves, thereby resolving
a conjecture of Igor Pak.
4. Uniqueness of maximal entropy measure on essential spanning forests. Scott Sheffield.
Annals of Probability, Vol. 34, No. 3, 857–864 (2006).
Provided first correct proof that every quasitransitive amenable graph (in particular Zd ) admits
a unique maximal entropy measure on essential spanning forests.
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5. Computing and Sampling Restricted Vertex Degree Subgraphs and Hamiltonian
Cycles. Scott Sheffield. arXiv (2000).
Gave a polynomial time algorithm for determining whether induced subgraphs of Z2 with a fixed
number of holes admit Hamiltonian cycles.
6. Large deviations of Markov chains indexed by random trees. Amir Dembo, Peter
Mörters, Scott Sheffield. Ann. Inst. H. Poincaré Probab. Statist. 41(6):971–996 (2005).
Extended classical Markov chain large deviation theory to the setting in which the Markov chain
is indexed by a Galton-Watson tree instead of by Z.
• Other results from probability and mathematical physics
1. Deterministic approximations of random reflectors. Omer Angel, Krzysztof Burdzy,
Scott Sheffield. Transactions of the AMS (2013).
Showed (with a construction analogous to the famous “digital sundial”) that it is theoretically
possible within classical optics to construct a macroscopically flat but microscopically rough mirror such that almost all of the light reaching the mirror from a direction θ is reflected in a
prescribed outgoing direction f (θ) (or, more generally, according to a prescribed scattering measure on outgoing directions) subject only to the obvious flow-conservation restriction on f .
2. Absolute continuity of Brownian bridges under certain gauge transformations. Andrea Nahmod, Luc Rey-Bellet, Scott Sheffield, Gigliola Staffilani. Mathematical Research Letters 18:5 (2011).
Solved open problem about the nonlinear Schrödinger equation by proving absolutely continuity
of certain Gaussian Brownian bridge measures w.r.t. to gauge transformations of themselves.
3. Fixation for Distributed Clustering Processes. Marcelo Hilario, Oren Louidor,
Charles Newman, Leonardo Rolla, Scott Sheffield, Vladas Sidoravicius. Communications in Pure and Applied Math (2009).
Studied an amusing “rich steal from poor” model of resource distribution on lattice vertices and
settled a previously formulated conjecture about its long term stability.
4. Power law Polya’s urn and fractional Brownian motion. Alan Hammond, Scott
Sheffield. Probab. Theory Relat. Fields (2012).
Studied random walks that at every integer time n jump up or down by 1 (choosing between these
two possibilities by copying what was done at time n − m where m is chosen from a probability
distribution on Z+ with power law tail) and showed that such walks scale to fractional Brownian
motions.
5. Linear speed large deviations for percolation clusters Yevgeniy Kovchegov, Scott
Sheffield. Electron. Comm. Probab. 8: 179183 (2003).
Gave large deviations principle for shape of origin-containing cluster in subcritical percolation.
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