Curriculum Vitae of Ewain Gwynne
Massachusetts Institute of Technology
Department of Mathematics, E17-301W
ewain@mit.edu
Education
Ph.D., Mathematics (expected)
Massachusetts Institute of Technology
Adviser: Scott Sheffield
Started Sep. 2013
B.A., Mathematics & Mathematical Methods in the Social Sciences
Northwestern University
Honors in Mathematics, Summa Cum Laude (G.P.A. 4.0/4.0)
Sep. 2009—Jun. 2013
Articles
1. Dimension transformation formula for conformal maps into the complement of an SLE curve
(with Nina Holden and Jason Miller). ArXiv e-prints, 2016.
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We prove a formula relating the Hausdorff dimension of a set Y ⊂
and its image under a conformal map from the
upper half plane to a complementary connected component of an SLEκ curve.
2. Joint scaling limit of a bipolar-oriented triangulation and its dual in the peanosphere sense
(with Nina Holden and Xin Sun). ArXiv e-prints, 2016.
We study the pair of contour functions of the the east and west-going trees on an infinite-volume random bipolaroriented triangulation and the pair of contour functions of the north and south-going trees on its dual map. We prove
that the joint law of these two pairs of contour functions converges in the scaling limit to the two correlated Brownian
p
motions which encode (via the peanosphere construction of Duplantier, Miller, and Sheffield, 2014) a 4/3-Liouville
quantum gravity surface decorated by two SLE12 curves coupled together in the sense of imaginary geometry.
3. An almost sure KPZ relation for SLE and Brownian motion (with Nina Holden and Jason Miller).
ArXiv e-prints, 2015.
Using the peanosphere (or “mating of trees”) construction of Duplantier, Miller, and Sheffield (2014), we prove a
KPZ type formula which reduces the problem of computing the Hausdorff dimension of any set associated with an
SLEκ or CLEκ to the problem of computing the Hausdorff dimension of a certain set associated with a correlated
two-dimensional Brownian motion. As corollaries, we obtain short proofs of the Hausdorff dimensions of several sets
associated with SLE.
4. Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map III: finite
volume case (with Xin Sun). ArXiv e-prints, 2015.
We study the finite-volume version of Sheffield’s (2011) inventory accumulation model which encodes a random planar
map decorated with a collection of loops sampled from the critical FK model. We prove that the two-dimensional walk
associated with the finite-volume version of the model converges in the scaling limit to a correlated Brownian motion
Ż conditioned to stay in the first quadrant for two units of time and satisfy Ż(0) = 0. We also prove finite-volume
analogues of the results of Gwynne, Mao, and Sun (2015) concerning the convergence of functionals of the FK loops.
5. Brownian motion correlation in the Peanosphere for κ > 8 (with Nina Holden, Jason Miller, and
Xin Sun). ArXiv e-prints, 2015.
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We prove that in the case when κ > 8 (equivalently γ ∈ (0, 2)) the correlation of the two-dimensional Brownian
motion in the peanosphere (or “mating of trees”) construction of SLEκ on an independent Liouville quantum gravity
surface is given by − cos(4π/κ).
6. Asymptotic behavior of the Eden model with positively homogeneous edge weights (with
Sébastien Bubeck). ArXiv e-prints, 2015.
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We consider a variant of the Eden model on d in which edges are added to the cluster with probability proportional
to the values of a function f which is positively homogeneous of degree α ∈ , rather than uniformly. We prove that
the clusters have a deterministic limit shape if α < 1 and are a.s. contained in a Euclidean cone of opening angle < π
for certain choices of f if α > 1.
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7. Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map II: local
estimates and empty reduced word exponent (with Xin Sun). ArXiv e-prints, 2015.
We prove various local limit theorems for Sheffield’s (2011) inventory accumulation model which encodes the critical
FK model on a random planar map. We use our estimates to obtain the exponent for the probability that a word of
length 2n sampled from this model reduces to the empty word.
8. Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map I: cone times
(with Cheng Mao and Xin Sun). ArXiv e-prints, 2015.
We study an inventory accumulation model introduced by Sheffield (2011) which encodes a random planar map
decorated by a collection of loops sampled from the critical FK model. We show that the times corresponding to
complementary connected components of FK loops in the infinite-volume version of the model converge in the scaling
limit to the π/2-cone times of a correlated Brownian motion. When combined with results of Duplantier, Miller, and
Sheffield (2014), our results imply the convergence of many interesting functionals of the FK loops (e.g. their boundary
lengths and areas) to the corresponding functionals of CLEκ loops on a Liouville quantum cone.
9. Almost sure multifractal spectrum of SLE (with Jason Miller and Xin Sun). ArXiv e-prints, 2014.
We compute the almost sure multifractal spectrum and bulk integral means spectrum of the Schramm-Loewner evolution (SLE) curve using various couplings of SLE with a Gaussian free field.
10. On Beckner’s Inequality for Gaussian Measures (with Elton Hsu). Elemente der Mathematik, 2015.
We give a simple proof of an extension of a functional inequality for Gaussian measures originally due to William
Beckner, using the heat semigroup.
11. Functional Inequalities for Gaussian and Log-Concave Probability Measures. Undergraduate
Thesis, advised by Elton Hsu. Northwestern University Undergraduate Research Journal, 2013.
We prove a number of Sobolev-type inequalities for Gaussian and other log-concave probability measures, including
generalizations of the logarithmic Sobolev inequality, Beckner’s inequality, and the Brascamp-Lieb inequality. A
mixture of approximately equal parts original results or new proofs of existing results; and exposition of known
mathematics.
12. On a Quaternionic Analogue of the Cross Ratio (with Matvei Libine). Advances in Applied Clifford
Algebras, 2012.
We prove several basic properties of fractional linear transformations on the quaternions using an analogue of the
cross ratio, including that such transformations preserve the class of higher-dimensional lines and planes; and minimal
conditions under which such transformations are uniquely determined.
13. The Poisson Integral Formula and Representations of SU(1,1). Rose-Hulman Undergraduate
Math Journal, 2011.
We prove the Poisson integral formula for harmonic functions on the disk using representation theory.
Works in progress or in preparation
1. Convergence of the topology of critical FK planar maps to that of CLEκ on an independent
Liouville quantum surface (with Jason Miller).
We introduce a topological structure called a weighted lamination which represents a collection of loops and a measure
C
in
viewed modulo ambient homeomorphisms. We prove that the weighted lamination associated with a critical
FK planar map converges in the scaling limit to the weighted lamination associated with a conformal loop ensemble
(CLEκ ) on an independent Liouville quantum gravity surface. In preparation.
2. Conformal invariance of whole-plane CLEκ for κ ∈ (4, 8) (with Jason Miller).
We show that the law of CLEκ on the whole plane is invariant under Möbius transformations for κ ∈ (4, 8). Our
argument is uses invariance of the law of FK planar maps under uniform re-rooting together with scaling limit results
for these maps. In preparation.
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3. Active spanning trees with bending energy on planar maps and SLE-decorated Liouville
quantum gravity for κ > 8 (with Adrien Kassel, Jason Miller, and David Wilson).
We study a model on spanning-tree decorated planar maps which generalizes critical Fortuin-Kasteleyn planar maps.
Using a generalization of Sheffield’s hamburger-cheeseburger bijection, we prove that our spanning tree decorated
planar maps converge in the scaling limit in the peanosphere sense result toward SLEκ -decorated Liouville quantum
gravity for κ > 8. In preparation.
4. Phases of Quantum Loewner Evolution (with Jason Miller, Scott Sheffield, and Xin Sun).
We prove that Quantum Loewner evolution, a stochastic growth process introduced by Miller and Sheffield (2013),
has phase transitions analogous to the phase transitions of the SLE processes from which it is constructed. Work in
progress.
Presentations
1. Asymptotic behavior of the Eden model with positively homogeneous edge weights Feb. 2016
MIT probability seminar
2. An almost sure KPZ relation for SLE and Brownian motion
Jan. 2016
University of Chicago probability seminar
3. Scaling limit of the topological structure of critical Fortuin-Kasteleyn planar maps Nov. 2015
Michigan State University probability seminar
4. Scaling limit of the topological structure of critical Fortuin-Kasteleyn planar maps Oct. 2015
Northwestern University analysis seminar
5. Asymptotic behavior of the Eden model with positively homogeneous edge weights
2015
Aug.
Microsoft Research, Redmond, WA.
6. Almost sure multifractal spectrum of SLE
Jan. 2015
Conformally invariant scaling limits conference at the Isaac Newton Institute, Cambridge UK.
7. Phases of Quantum Loewner evolution
Nov. 2014
MIT Pure Math graduate seminar.
8. Random surfaces, Gaussian free fields, and Liouville quantum gravity
Feb. 2014
MIT Pure Math graduate seminar.
9. A Quaternionic Analogue of the Cross Ratio
Jul. 2011
Indiana REU conference.
Employment
1. Microsoft Research theory group intern
Summer 2015
I conducted research in probability theory with Sébastien Bubeck and David Wilson.
2. Northwestern University undergraduate teaching assistant
2011-2013
Each quarter during my Junior and Senior years as an undergraduate, I served as a teaching assistant for a mathematics course. My responsibilities, which were identical to those of a graduate teaching assistant, included teaching a
discussion section, holding office hours, and grading tests and worksheets. I have been a TA for four sections of integral
calculus, one section of single variable differential calculus, and one section of multivariable differential calculus.
3. Tutor.com online math tutor
2010-2013
For my last three years as an undergraduate, I worked for 30 hours per week helping students throughout the country
with subjects that ranged from elementary math to multivariable calculus and statistics.
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4. Private tutoring
2010-2013
I tutored individual undergraduate students in math and statistics.
Awards
1. MIT Presidential Fellowship
2013
2. National Defense Science and Engineering Graduate Fellowship (NDSEG)
2013
3. Offered NSF Graduate Research Fellowship
2013
Declined; accepted NDSEG instead.
4. Putnam Exam Honorable Mention
2013
Score of 50, ranked 49th.
5. Robert R. Welland Prize for Achievement in Mathematics by a Northwestern University Senior
2013
6. Phi Beta Kappa Prize
2013
7. Fletcher Undergraduate Research Prize
2012
8. Oliver Marcy Scholarship
2012
9. Barry M. Goldwater Scholarship
2012
10. Elected to Phi Beta Kappa
2012
11. NU Summer Undergraduate Research Grant
2012
12. National Merit Scholarship
2009
Teaching and Departmental Service
1. Teaching Assistant
2016
I was a TA for 18.03 (ordinary differential equations) in Spring 2016.
2. Integration Bee
2014-2016
I was a co-organizer for the MIT integration bee, an event wherein undergraduate students compete to evaluate
integrals and win prizes, in 2014 and 2016. I also contributed integrals in 2014, 2015, and 2016.
3. IAP Directed Reading Program
2014
I mentored an MIT undergraduate student studying probability during MIT’s Independent Activities Period (the
month of January). We read sections of Durrett’s Probability: Theory and Examples and Mörters and Peres’ Brownian
Motion.
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