Books
Y.A. Rubinstein, B. Shiffman, Eds.,
Advances in Complex Geometry ,
American Mathematical Society, 2019.
M.A. Peterson, Y.A. Rubinstein, Eds.,
Directions for Mathematics Research Experience for Undergraduates , World Scientific, 2016.
(book
review in Notices of the American Mathematical Society ,
book
review in The Mathematical Gazette )
Meetings
Frontiers of Geometric Analysis

In honor of Gang Tian 's 65th birthday,
Santa Cruz,
June 37, 2024.
Convex and Complex: Perspectives on Positivity in Geometry

In honor of Bo Berndtsson 's 70th birthday,
Cetraro,
October 31November 4, 2022.
34th Geometry Festival 
University of Maryland,
April 57, 2019.
Metro Area Differential Geometry Seminar (MADGuyS)

Howard University, April 27, 2019.
UMD Geometry Week

University of Maryland,
March 1216, 2018.
Maryland Analysis and Geometry Atelier

University of Maryland,
August 2125, 2017.
Videos of lectures
Pacific Rim
Workshop on Geometric Analysis
 University of British Columbia, July 2428, 2017.
New Directions for Mathematics Research Experiences for Undergraduates
 Mt. Holyoke College, June 2122, 2013.
28th Geometry Festival: Calabifest 
In celebration of the 90th birthday of Eugenio Calabi ,
University of Maryland,
April 1214, 2013.
Complex Geometry and Microlocal Analysis  special session at
the AMS Eastern Spring meeting, Boston College, April 67, 2013.
Bay Area Differential Geometry Seminar (BADG)
 Stanford, June 2, 2012.
Postdocs
Jakob Hultgren
Tamás Darvas
Jesus Martinez Garcia
Students
Vlasios Mastrantonis
Chenzi Jin
Yuxiang Ji
Mirna Pinsky
Kewei Zhang
Matthew Dellatorre
Ryan Hunter
Michael Lindsey
Frederick TszHo Fong
Research Experience for Undergraduates (REU)
2016: Coedited the book
Directions for Mathematics Research Experience for Undergraduates .
The book has been reviewed in
Notices of the American Mathematical Society
and in The Mathematical Gazette .
2016: Coauthored an
article on new models for REUs and
the culture of Mathematics.
2013: Coorganized the conference
New Directions for Mathematics Research Experiences for Undergraduates
at Mt. Holyoke College. Here is an
article summarizing the
conference.
2012: Cofounded
SURIM
and led an REU on Optimal
Transportation, resulting in a
theorem , a
senior thesis project , and the
Kennedy
Honors Thesis Prize
and the SIAM
Student Paper prize for
my student Michael Lindsey.
REU students
Vishesh Jain
Michael Lindsey
Lyuboslav Panchev
2016: Coedited the book
Directions for Mathematics Research Experience for Undergraduates .
The book has been reviewed in
Notices of the American Mathematical Society
and in The Mathematical Gazette .
Minicourses & Lecture notes
A crash course to large deviation principles and optimal transport 
UMD graduate course on Topics in Geometric Analysis Fall 2017.
Here are the
lecture notes
. (Published version in: Birational Geometry, KahlerEinstein Metrics and Degenerations (I. Cheltsov et al., Eds), Springer Proceedings in Mathematics & Statistics 409, Springer, 2023, pp. 795837.)
An introduction to Kahler geometry: Tian's properness
conjectures  SCGAS
Winter School
 UC Irvine, January 2529, 2016.
Here are the
lecture notes . (Published version
in: Geometric Analysis (J. Chen et al., Eds.),
Progress in Mathematics, Vol. 333, Birkhauser, 2020,
pp. 381443.)
Solutions to an equation that do not
solve the equation  Getting
Started with PDEs
 IIAS, Hebrew University, September 39, 2015.
Smooth and singular KahlerEinstein metrics 
Advanced School on PDEs in Geometry and Physics
 Hefei, China, June 30July 11, 2014.
Here are the
lecture notes .
(Published version in:
Geometric and Spectral Analysis
(P. Albin et al., Eds.),
Contemp. Math. 630, AMS and Centre Recherches Mathematiques, 2014,
pp. 45138.)
Publications
(with C. Jin)
Tian's stabilization problem for toric Fanos ,
preprint, 2024, arxiv:2403.17262.
(with V. Mastrantonis)
Twodimensional Blocki, L^pMahler, and Bourgain conjectures ,
preprint, 2024, arxiv:2401.10992.
(with B. Berndtsson, V. Mastrantonis)
L^ppolarity, Mahler volumes, and the isotropic constant ,
preprint, 2023, arxiv:2304.14363.
(with C. Jin)
Chebyshev potentials, FubiniStudy metrics, and geometry of the space of Kahler metrics ,
Bull. London Math. Soc. (2023).
(with V. Mastrantonis)
The Nazarov proof of the nonsymmetric BourgainMilman inequality ,
preprint, 2022, arxiv:2206.06188. To appear in Indiana Univ. Math. J.
On large deviation principles and the MongeAmpère equation (following Berman, Hultgren) , in: Birational Geometry, KahlerEinstein Metrics and Degenerations (I. Cheltsov et al., Eds), Springer Proceedings in Mathematics & Statistics 409, Springer, 2023, pp. 795837.
(with Y. Ji, K. Zhang)
EguchiHanson metrics arising from KahlerEinstein edge metrics ,
preprint, 2021, arxiv:2111.00652. To appear in J. Topology & Analysis.
(with K. Zhang)
Angle deformation of KahlerEinstein edge metrics on Hirzebruch
surfaces ,
Pure Appl. Math. Quart. 18, special issue in honor
of Bernard Shiffman (2022), 343366.
(with G. Tian, K. Zhang)
Basis divisors and balanced metrics ,
Crelle J. 778 (2021), 171218.
(with P. Cascini, J. MartinezGarcia)
On the body of ample angles of asymptotically log Fano varieties ,
Rendiconti Circolo Mat. Palermo (2022).
Classification of strongly asymptotically log del Pezzo
flags and surfaces ,
Int. J. Math. 33 (2022).
Highdimensional convex sets arising in algebraic geometry ,
in: Geometric Aspects of Functional Analysis (B. Klartag, E. Milman,
Eds.),
Lecture Notes in Mathematics 2266 (2020), pp. 301323.
(with K. Zhang)
Small angle limits of Hamilton's footballs ,
Bull. London Math. Soc. 52 (2020) 189199.
(with T. Buttsworth, A. Pulemotov, W. Ziller)
On the Ricci iteration for homogeneous metrics on spheres and projective
spaces ,
Transformation Groups 26 (2021), 145164.
(with I.A. Cheltsov, K. Zhang)
Basis log canonical thresholds, local intersection estimates, and
asymptotically log del Pezzo surfaces ,
Selecta Math. 25 (2019), 136.
Tian's
properness conjectures: an introduction to Kahler geometry ,
in: Geometric Analysis (J. Chen et al., Eds.),
Progress in Mathematics, Vol. 333, Birkhauser, 2020,
pp. 381443.
(with T. Darvas, C.H. Lu)
Quantization in geometric pluripotential theory ,
Comm. Pure Appl. Math. 73 (2020), 11001138.
(with T. Darvas)
Convergence of the KahlerRicci iteration ,
Analysis and PDE 12 (2019), 721735.
(with B. Berndtsson, D. CorderoErausquin, B. Klartag)
Complex Legendre duality ,
Amer. J. Math. 142 (2020), 323339.
(with B. Berndtsson, D. CorderoErausquin, B. Klartag)
Complex interpolation of Rnorms, duality and foliations ,
J. Eur. Math. Soc. (JEMS) 22 (2020), 477–505.
(with T. Darvas)
A minimum principle for Lagrangian graphs ,
Comm. Anal. Geom. 27 (2019), 857876.
(with A. Pulemotov)
Ricci iteration on homogeneous spaces ,
Trans. Amer. Math. Soc. 371 (2019), 62576287.
(with M. Lindsey)
Optimal transport via a MongeAmpère optimization problem
,
SIAM J. Math. Anal. 49 (2017), 30733124.
(with I.A. Cheltsov)
On flops and canonical metrics ,
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018), 283–311.
(with J.P. Solomon)
The
degenerate special Lagrangian equation ,
Adv. Math. 310 (2017), 889939.
(with T. Darvas)
Tian's properness conjectures and Finsler geometry of the space of Kahler
metrics ,
J. Amer. Math. Soc. 30 (2017), 347387.
(with R. Vakil)
REUs with limited faculty involvement, "underrepresented" subjects in the
undergraduate curriculum, and the culture of Mathematics ,
in:
Directions for Mathematics Research Experience for Undergraduates
(M.A. Peterson, Y.A. Rubinstein, Eds.), World Scientific, 2016,
pp. 5372.
(with T. Darvas)
Kiselman's principle, the Dirichlet problem for the MongeAmpere equation, and
rooftop obstacle problems
(alternative link) ,
J. Math. Soc. Japan 68 (2016), 773–796.
Smooth
and singular KahlerEinstein metrics
in:
Geometric and Spectral Analysis
(P. Albin et al., Eds.),
Contemp. Math. 630, AMS and Centre Recherches Mathematiques, 2014,
pp. 45138.
(with A. Carlotto and O. Chodosh)
Slowly converging Yamabe flows ,
Geom. & Top. 19 (2015), 15231568.
(with O. Chodosh, V. Jain, M. Lindsey, and L. Panchev)
On discontinuity of planar optimal transport maps ,
J. Topology & Analysis 7 (2015), 239260.
(with S. ArtsteinAvidan)
Differential analysis of polarity: Polar HamiltonJacobi,
conservation laws, and Monge Ampère equations ,
J. d'Analyse Math. 132 (2017),
133156.
(with I.A. Cheltsov)
Asymptotically log Fano varieties ,
Adv. Math. 285 (2015), 12411300.
(with R. Mazzeo and N. Sesum)
Ricci flow on surfaces with conic singularities ,
Analysis and PDE 8 (2015), 839882.
(with R. Mazzeo)
The Ricci continuity method for the complex MongeAmpere equation,
with applications to KahlerEinstein edge metrics ,
C. R. Math. Acad. Sci. Paris 350 (2012), 693697.
(with S. Zelditch)
The Cauchy problem
for the homogeneous MongeAmpere equation, III. Lifespan ,
Crelle's J. 724 (2017), 105143.
(with R. Mazzeo and T. Jeffres)
KahlerEinstein metrics with edge singularities ,
(with an appendix by C. Li and Y.A. Rubinstein),
Annals of Math. 183 (2016), 95176.
(with B. Clarke)
Conformal deformations of the Ebin metric and a generalized Calabi metric
on the space of Riemannian metrics ,
Ann. IHP Anal. Non Lineaire 30 (2013), 251274.
(with B. Clarke)
Ricci flow and the metric
completion of the space of Kahler metrics ,
Amer. J. Math. 135 (2013), 14771505.
(with S. Zelditch)
The Cauchy problem for the
homogeneous MongeAmpere equation, II. Legendre transform ,
Adv. Math. 228 (2011), 29893025.
(with S. Zelditch)
The Cauchy problem for the
homogeneous MongeAmpere equation, I. Toeplitz
quantization , J. Differential Geom. 90 (2012), 303327.
(with I. Rodnianski and G. Staffilani)
On the global wellposedness of the onedimensional Schrodinger map
flow , Analysis and PDE 2 (2009), 187209.
(with S. Zelditch)
Bergman approximations of harmonic maps into the space of Kahler metrics on toric
varieties , J. Symp. Geom. 8 (2010), 239265.
Geometric quantization and dynamical constructions on the
space of Kahler metrics , Ph.D. Thesis, Massachusetts Institute of Technology, 2008.
Some discretizations of geometric evolution equations and the Ricci iteration
on the space of Kahler metrics , Adv. Math. 218 (2008), 15261565.
On the construction of Nadel multiplier ideal sheaves and the limiting behavior
of the Ricci flow ,
Trans. Amer. Math. Soc. 361 (2009), 58395850.
The Ricci iteration and its
applications , C. R. Acad. Sci. Paris 345 (2007), 445448.
On energy
functionals, KahlerEinstein metrics, and the
MoserTrudingerOnofri neighborhood , J. Funct. Anal. 255, special issue
dedicated to Paul Malliavin (2008), 26412660.
(with M. Peterson)
Turbulence on a desktop ,
Comp. Sci. Eng. 3 (2001), 8694.
(with S. Gueron)
The minimal reversible coagulationfragmentation process having no factorized coagulation and
fragmentation rates, Markov Process. Related Fields 6 (2000), 257264.
(with H. Bailey)
A Variety of Triangle Inequalities ,
College Math. J. 31 (2000), 350355.
Errata list for my articles
Seminars
JHUUMD
Complex Geometry Seminar
Informal
Geometric Analysis Seminar
Distinguished Lecture Series in Geometric Analysis
Seminars in the Department.
Teaching
20222023
History
of Mathematics
(course 274, Fall 2022).
Topics in Geometric Analysis: the Mahler Conjectures
(course 848V, Fall 2022).
20212022
History
of Mathematics
(course 274, Fall 2021).
Topics in Differential Geometry
(course 848Y, Fall 2021).
20202021
Introduction
to Number Theory
(course 406, Spring 2021):
TTh 9:30am.
History
of Mathematics
(course 274, Fall 2020):
TTh 2pm.
Complex Surfaces
(course 848Y, Fall 2020):
TTh 12:30pm.
20192020
Topics in Differential Geometry
(course 20204211,
Fall 2020):
M 1:15, W 16:15, Goldsmith 208.
20182019
Introduction
to Number Theory
(course 406, Fall 2018):
MWF 12pm, MATH 0307.
Geometric
microlocal analysis
(course 848G, Fall 2018):
MWF 2pm, MATH 2300.
20172018
Differential
Equations for Scientists and Engineers (honors)
(course 246H, Spring 2018):
TuTh 12:30pm, MATH 1308.
Geometric Analysis
(course 742, Fall 2017):
TTh 2pm, MATH B0431.
20162017
Differential
Equations for Scientists and Engineers (honors)
(course 246H, Spring 2017):
TuTh 12:30pm, MATH 1308.
Differential
Equations for Scientists and Engineers (honors)
(course 246H, Fall 2016):
TuTh 12:30pm, MATH 0304.
20152016
Differential
Equations for Scientists and Engineers (honors)
(course 246H, Fall 2015):
MWF 10am, MATH 1311.
Topics in fully nonlinear
PDE
(course 858Y, Fall 2015):
MWF 12 noon, MATH 1313.
20142015
Geometric Analysis
(course 742, Fall 2014):
TTh 9:30am, MATH 0104.
Topics in Complex Analysis: Kahler
Manifolds
(course 868K, Fall 2014):
TTh 12:30am, EGR 3114.
20132014
Geometric Analysis
(course 742, Fall 2013):
TTh 11am, MATH 1308.
Differential Geometry of
Curves and
Surfaces I
(course 436, Fall 2013):
TTh 2pm, MATH B0421.
20122013
Topics in
Geometric Analysis: The Yamabe Problem
(course 748F, Spring 2013):
TTh 2pm.
Geometry
(course 430, Fall 2012):
TTh 2pm.
20112012
Undergraduate
research project: Optimal Transportation
(SURIM, Summer 2012).
Introduction
to Optimal Transportation (course 287, Winter 2012, at Stanford):
MWF 11am.
Linear
Algebra and Differential Calculus of Several Variables
(course 51, Autumn 2011, at Stanford):
MWF 11am and 1:15pm.
20092010
Partial
Differential Equations (course 256A, Spring 2010, at Stanford):
MWF 2:15pm.
Complex
Analysis, Geometry, and Topology (course 215A, Autumn 2009, at Stanford):
TTh 2:15pm.
20082009
Riemannian
Geometry (course 110.645, Fall 2008, at JHU).
Past Seminars
20092012 (Stanford):
Geometry Seminar.
20082009 (JHU):
Complex
Geometry Seminar.
Informal
graduate student seminar on Kahler geometry through examples.
20052006 (MIT):
Graduate
Student Seminar on Geometric
Stability in Differential
Geometry.