Scott Sheffield

Scott Sheﬃeld
sheﬀ@math.nyu.edu
Courant Institute
(212)998-3262
251 Mercer Street
http://math.nyu.edu/faculty/sheﬀ
New York, NY
Education
Ph.D., Mathematics, Stanford University, April 2003
A.B., Mathematics, Harvard University, June 1998
A.M., Mathematics, Harvard University, June 1998
Recent Awards
2006 Rollo Davidson Award for “work on spatial models of probability theory
and especially their relationship to stochastic (Schramm) Loewner evolutions”
NSF Postdoctoral Fellowship: DMS 0403182
NSF CAREER award: DMS 0645585, Feb. 1, 2007 – Jan. 31, 2012
2007 Sloan Research Fellowship
NSF PIRE Continuing Grant: OISE 0730136 with Charles Newman (PI),
Daniel Stein, Srinivasa Varadhan, Gerard Ben Arous.
Primary Employment
Courant Institute; NYU Mathematics Department
Spring 2007–present
Associate Professor
Institute for Advanced Study, Princeton
Fall 2006–Spring 2007
NSF/IAS Postdoctoral Fellow
Courant Institute; NYU Mathematics Department
Fall 2005–Spring 2006
Assistant Professor
U.C. Berkeley Mathematics Department
Summer 2004–Summer 2005
NSF Postdoctoral Fellow
Microsoft Research
Summer 2002–Summer 2004
Postdoctoral Researcher
1

Professional Activities
Organizer:
Summer 2008 World Congress in Probability and
Statistical Physics Invited Session
Instructor:
Summer 2008 ICTP Summer School
Instructor:
Spring 2008 Warwick Spring Meeting
Co-organizer:
Fall 2008 CRM program on Random Surfaces, Random
Functions, and Interfaces
Co-organizer/instructor: Fall 2007 Oberwolfach Seminar on Conformal Invariance
in Mathematical Physics
Co-organizer:
Summer 2007 IAS/PCMI Summer Session on Statistical
Physics
Co-organizer:
Fall 2006 KITP Semester on Stochastic Geometry and
Field Theory
Co-organizer:
2005–06 Courant Institute Probability Seminar
Co-organizer:
2004–05 Berkeley Probability Seminar
Committee member:
Fall 2005 Courant Math. Finance Masters Committee
Committee member:
1997–98 Harvard Undergraduate Curriculum Committee
Referee:
Research papers in Acta Mathematica, Annals of Mathe-
matics, Annals of Probability, Communications in Math-
ematical Physics, Duke Journal of Mathematics, Journal
of the American Mathematical Society, Memoirs of the
American Mathematical Society, several other journals
Teaching
Primary instructor:
Two NYU courses (Math. Finance for undergraduates
and Limit Theorems for graduate students)
Advisor:
Multiple mathematics Ph.D. students at NYU
Advisor:
NYU undergraduate summer researcher Julia Spencer
Teaching/course assistant:
Six Stanford courses, ﬁve Harvard courses
Collaborators and Advisors
•
Collaborators: Amir Dembo, Stanford University; Richard Kenyon, University of
British Columbia; Yevgeniy Kovchegov, Oregon State; Assaf Naor, Microsoft Research;
Peter M¨orters, University of Bath, Andrei Okounkov, Princeton University; Yuval
Peres, U.C. Berkeley; Oded Schramm, Microsoft Research; David Wilson, Microsoft
Research; Wendelin Werner, Universit´e Paris-Sud
•
Graduate and Postdoctoral Advisors: Christian Borgs, Microsoft Research; Jen-
nifer Chayes, Microsoft Research; Amir Dembo, Stanford University; Yuval Peres, U.C.
Berkeley; Oded Schramm, Microsoft Research; Thomas Spencer, Institute of Advanced
Studies, Princeton
2

Selected Talks
7/7-11/08
Course, ICTP, Trieste, Italy
5/12-13/08
Invited Address, Clay Research Conference, MIT
3/31-4/4/08 Minicourse, Warwick, UK
11/4-10/07
Course, Oberwolfach Seminar, Germany
10/6-7/07
Invited Address, Eastern Sectional AMS Meeting, Rutgers University
8/5-11/07
32nd conference on Stochastic Processes and their Applications at Uni-
versity of Illinois at Urbana Champaign
4/3/07
U. Michigan Colloquium
3/26-30/07
IPAM Random Shapes Conference
3/12/07
Cornell Probability Seminar
2/14/07
MIT Analysis Seminar
2/7/07
Princeton Colloquium
1/9/07
Caltech Colloquium
12/17/06
96th Statistical Mechanics Meeting, Rutgers University
12/08/06
Columbia Probability Seminar
11/15/06
IAS Member Seminar II
11/13/06
IAS Member Seminar I
11/09/06
U. Penn Probability Seminar
10/19/06
Midwestern Probability Colloquium Tutorial, Northwestern University
10/18/06
U. Chicago Applied Math./PDE Seminar
10/13/06
Courant institute Applied Math Seminar, New York
8/31/06
KITP Semester on Stochastic Geometry and Field Theory
8/4/06
IMS Annual Meeting, Rio de Janeiro, Brazil
6/22/06
Conference on Stochastic Processes in Mathematical Physics; Florence,
Italy
5/5/06
Conference on SLE and Loop Measures, Cornell
4/22/06
Sherman Conference, University of Indiana
2/3/06
Courant Probability Seminar, NYU
1/18/06
American Institute of Mathematics ARCC workshop on Random Ana-
lytic Functions
1/12/06
Stanford Colloquium
10/29/05
A conference in honor of Peter Lax and Louis Nirenberg (on the occa-
sion of their 80th birthdays), NYU
10/25/05
Princeton Mathematical Physics Seminar
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9/22–24/05 Three-lecture Minicourse at Fields Institute, Toronto
8/5/05
IMA Workshop in Minneapolis
8/3/05
Brigham Young University Colloquium
6/1/05
BIRS Conference in Banﬀ
5/4/05
Berkeley Probability Seminar
4/8/05
MSRI Postdoctoral Seminar
11/17/04
Berkeley Probability Seminar
11/16/04
Berkeley Interdisciplinary Stochastic Processes Colloquium
11/2/04
Stanford Analysis Seminar
10/5/04
U.C. Davis Colloquium
9/10/04
University of Utah Probability Seminar
9/9/04
Brigham Young University Colloquium
9/3/04
Berkeley Analysis Seminar
7/26–31/04 6th World Congress for Bernoulli Soc./67th Ann. Meeting of IMS in
Barcelona
2/13/04
Courant Institute Probability Seminar
2/12/04
Princeton Ergodic Theory Seminar and Probability Seminar
2/10/04
Princeton Mathematical Physics Seminar
2/6/04
Cornell Probability Seminar
2/5/04
Cornell Colloquium
1/20/04
University of Washington Colloquium
11/12/03
UCLA Probability Seminar
11/6/03
University of Wisconsin Probability Seminar
11/3/03
University of Washington Probability Seminar
7/17/03
ICMS Conformal Invariance Conference in Edinburgh
7/15/03
ICMS Conformal Invariance Conference in Edinburgh
5/19/03
Stanford Probability Seminar
3/21/03
IHP Random Growth Conference in Paris
1/15/03
Joint Mathematics Meeting in Baltimore
10/19/02
Northwest Probability Seminar in Seattle
10/14/02
Stanford Probability Seminar
5/20/02
Stanford Probability Seminar
4

Publications (available at arXiv.org)
•
Book-length research papers:
1. Random surfaces. Scott Sheﬃeld. Ast´erisque 2006, No. 304 (177 pages).
Proved several conjectures (along with extensive 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. (132 pages). Acta Mathematica. To appear.
Proved that the chordal level lines of the discrete Gaussian free ﬁeld have scaling
limits that are variants of SLE(4).
•
Gaussian free ﬁelds and Schramm-Loewner evolutions:
1. 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.
2. Gaussian free ﬁelds for mathematicians.
Scott Sheﬃeld.
Probability
Theory and Related Fields. To appear.
Surveyed mathematical literature on the Euclidean bosonic massless free ﬁeld and
studied new exploration-process-based couplings of Gaussian free ﬁelds and Brow-
nian motions.
3. Exploration trees and conformal loop ensembles. Scott Sheﬃeld. Sub-
mitted.
Constructed and studied the conformal loop ensembles CLE(κ), which are the
“loop ensemble” analogs of SLE.
4. Conformal radii in conformal loop ensembles. Oded Schramm, Scott
Sheﬃeld and David Wilson. Submitted.
Computed conformal radii distributions for conformal loop ensembles and the ex-
pectation dimension of the conformal gasket (which is the random closed set con-
sisting of points not surrounded by a loop). Results agree with physics literature
predictions by Kenyon and Wilson, by Cardy and Ziﬀ, and by Duplantier.
•
Dimer models, spanning trees, and tilings:
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 Leg-
endre 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.
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2. 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.
3. Dimers, tilings and trees. Richard Kenyon and Scott Sheﬃeld. Journal
of Combinatorial Theory Series B, 92(2):295–317, 2004.
Produced general weight-preserving bijection between perfect matchings of weighted,
bipartite planar graphs and spanning trees of so-called T-graphs.
4. Ribbon Tilings and Multidimensional Height Functions. Scott Sheﬃeld.
Trans. Amer. Math. Soc. 354 (2002), no. 12, 4789–4813.
Used multidimensional height function approach to give eﬃcient tilability algo-
rithm and to show that set of ribbon tilings of a simply connected grid graph is
connected under local moves, thereby resolving a conjecture of Pak.
•
Game theory, PDEs, and Lipschitz extension theory
1. Random-turn Hex and other selection games. Yuval Peres, Oded
Schramm, Scott Sheﬃeld, and David Wilson.
American Mathematical
Monthly. To appear.
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.
2. 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 Lin-
denstrauss (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.
3. Tug-of-war and the inﬁnity Laplacian. Yuval Peres, Oded Schramm,
Scott Sheﬃeld, and David Wilson. To appear in JAMS.
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.
4. Tug of war with noise: a game theoretic view of the p-Laplacian. Yuval
Peres, Scott Sheﬃeld. Submitted.
Used game theory to generalize to arbitrary p the well-known relationship between
Brownian motion and the p-Laplacian for p = 2. Proved new results about p-
capacity and p-harmonicity.
6

•
Percolation and large deviation theory
1. Linear speed large deviations for percolation clusters. Yevgeniy Kovchegov,
Scott Sheﬃeld. Electronic Journal of Probability. 8 (2003), 179–183.
Proved linear speed large deviations principle for cluster shapes in subcritical
Bernoulli bond percolation.
2. Large deviations of Markov chains indexed by random trees. Amir
Dembo, Peter Morters, Scott Sheﬃeld. Ann. Inst. H. Poincar´e Probab.
Statist. 41 (2005), no. 6, 971–996.
Deﬁned and studied entropy and speciﬁc free energy for processes on certain un-
derlying graphs which are themselves random. Proved large deviations principle
for tree-indexed Markov chains.
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