emanuel.milman 'at' gmail.com or
Technion - Israel Institute of Technology
I am an Associate Professor at the Technion's Mathematics Department.
My main research
interests lie in the various aspects of Asymptotic Convex Geometric
Analysis. This is
the study of geometric structures satisfying appropriate convexity
conditions from a geometric and analytic point of view, with an
emphasis on the asymptotic dependence (or independence) of various
parameters on the underlying dimension.
Examples of such structures include bounded convex domains in Euclidean
Banach spaces (possibly infinite dimensional), Riemannian manifolds
with non-negative (Ricci)
curvature, and other generalizations. Since its conception at the
of classical Convex Geometry and the local theory of Banach
spaces, the field of Asymptotic Convex Geometric Analysis has been
evolving constantly, and has uncovered connections to many other
fields, such as
Probability Theory, PDE, Riemannian Geometry, Harmonic Analysis,
Mathematical Physics, Combinatorics, Graph Theory and Learning Theory.
Some of my related
interests include isoperimetric, functional and concentration
inequalities, the interplay between
spectral properties of Riemannian manifolds, geometry of isoperimetric
Geometric Measure Theory, diffusion semi-group and heat-kernel
convex manifolds, optimal transport, distribution of volume in convex
bodies, classical Convex Geometry, “local
theory” of Banach
in graphs, metric entropy and
covering numbers, empirical processes, general phenomena in high
dimensions, Integral Geometry.
Teaching Spring 2016:
High-Dimensional Convex Geometric Analysis - List of Exercises
Previous Advanced Courses Given:
Isoperimetry, Sobolev and Concentration inequalities through the lens of Convexity - List of Exercises
Convex Bodies in High Dimension - List of Exercises
Isoperimetric Inequalities and Applications - List of Exercises
- with Alexander Kolesnikov, Riemannian metrics on convex sets with applications to Poincaré and log-Sobolev inequalities, to appear in Calc. Var. & PDE. [arXiv link]
- Spectral Estimates, Contractions and Hypercontractivity, to appear in J. Spectr. Theory. [arXiv link]
- Harmonic Measures on the Sphere via Curvature-Dimension, to appear in Annales de la Faculté des Sciences de Toulouse. [arXiv link]
- with Alexander Kolesnikov, Sharp Poincare-type inequality for the Gaussian measure on the boundary of convex sets, to appear in GAFA Seminar Notes. [arXiv link]
traditional Curvature-Dimension I: new model spaces for isoperimetric
and concentration inequalities in negative dimension, to appear in Trans. Amer. Math. Soc. [arXiv link]
- with Alexander Kolesnikov, Brascamp–Lieb type inequalities on weighted Riemannian manifolds with boundary, to appear in J. Geom. Anal. [arXiv link]
- with Alexander Kolesnikov, Poincare and Brunn–Minkowski inequalities on the boundary of weighted Riemannian manifolds, submitted. [arXiv link]
- with Technion - Texas A&M Working Group, Generalized Sudakov via Dimension Reduction - A Program, manuscript. [local link]
Publications (according to topic):
And Functional Inequalities
Isoperimetric Inequalities and Model Spaces for
Curvature-Dimension-Diameter Condition, J. Eur. Math. Soc. 17 (5), 1041–1078, 2015. [arXiv link]
- with Alexander Kolesnikov, Isoperimetric Inequalities on Weighted Manifolds with Boundary, Doklady Akademii Nauk 464 (2), 136–140, 2015 (in Russian).
- with Alexander Kolesnikov, Remarks on KLS conjecture and Hardy-type inequalities, Lecture Notes in Math. 2116, GAFA Seminar Notes 2011–2013, 273-292, 2014. [arXiv link]
- with Liran Rotem, Complemented Brunn-Minkowski Inequalities and Isoperimetry for Homogeneous and Non-Homogeneous Measures, Adv. Math. 262, 867-908, 2014. [arXiv link]
- with Franck Barthe, Transference Principles for Log-Sobolev and Spectral-Gap with Applications to Conservative Spin Systems, Comm. Math. Physics 323, 575–625, 2013. [arXiv link]
- Model Spaces for Sharp Isoperimetric Inequalities, C. R. Math. Acad. Sci. Paris 350, 897-902, 2012. [pdf]
- A Proof of Bobkov's Spectral Bound For Convex Domains via Gaussian Fitting and Free Energy Estimation, Centre de Recherches Mathématiques, CRM Proceedings and Lecture Notes, Vol. 56, 181-196, 2013. [arXiv link]
- Isoperimetric Bounds on
Convex Manifolds, "Concentration, Functional Inequalities and
Isoperimetry", Contemporary Mathematics 545, Amer. Math. Soc.,
195-208, 2011. [arXiv
of Isoperimetric, Functional and Transport-Entropy Inequalities Via
Concentration, Probab. Theory Relat. Fields 152, 475–507, 2012. [arXiv
converse to the Maz'ya inequality for capacities under curvature lower
bound, Springer's International Mathematical Series
Vol. 11, Around
the Research of
Vladimir Maz'ya I. Function Spaces, 321-348. [arXiv
Concentration Inequalities - Equivalence under Curvature Lower
Bound, Duke Math. J. 154 (2), 207-239, 2010. [arXiv
and isoperimetry are equivalent assuming curvature lower bound,
C. R. Math. Acad. Sci. Paris. 347, 73-76, 2009. [pdf]
the role of convexity in functional and isoperimetric inequalities,
Proc. London Math. Soc. 99 (3), 32-66, 2009. [arXiv
the role of convexity in isoperimetry, spectral gap and concentration,
Invent. Math. 177 (1), 1-43, 2009. [arXiv link]
tail-decay of Lipschitz functions implies Cheeger’s
isoperimetric inequality under
C. R. Math. Acad. Sci. Paris 346,
989-994, 2008. [pdf]
Sasha Sodin, An
isoperimetric inequality for uniformly log-concave
measures and uniformly convex bodies,
J. Func. Anal. 254 (5), 1235-1268, 2008. [arXiv
Distribution of Volume in Convex Bodies
- On the mean-width of isotropic convex bodies and their associated Lp-centroid bodies, Int. Math. Res. Not. 11, 3408–3423, 2015. [arXiv link]
- with Apostolos Giannopoulos, M-estimates for isotropic convex bodies and their Lq-centroid bodies, , Lecture Notes in Math. 2116, GAFA Seminar Notes 2011–2013, 159-182, 2014. [arXiv link]
- with Bo'az
Regularization of Log-Concave Measures and Small-Ball Estimates, Lecture Notes in Math. 2050, GAFA Seminar Notes 2006–2010,
2012. [arXiv link]
- with Bo'az
Bodies and the Logarithmic Laplace Transform - A Unified Approach, J.
Func. Anal. 262 (1), 10-34, 2012. [arXiv
- with Olivier Guedon, Interpolating
Thin-Shell and Sharp Large-Deviation Estimates For Isotropic
Log-Concave Measures, Geom. Funct. Anal. 21 (5), 1043-1068,
Gaussian marginals of uniformly convex bodies,
J. Theor. Prob. 22 (1), 256-278, 2009. [arXiv
- with Bo’az
volume distribution in 2-convex bodies,
Israel J. Math. 164, 221-249, 2008. [arXiv
mixed volumes and the
Adv. Math. 207 (2), 566-598,
- with Young-Heon Kim,
Generalization of Caffarelli's Contraction Theorem via (reverse) Heat
Flow, Math. Annal. 354 (3), 827-862, 2012. [arXiv
Sections of Star Bodies
Intersection Bodies are not equivalent,
Adv. Math. 217 (6), 2822-2840,
Intersection Bodies, J. Func. Anal. 240 (2), 530-567, 2006. [arXiv
comment on the low-dimensional Busemann-Petty problem,
Lecture Notes in Math. 1910, GAFA Seminar Notes 2004–5,
Theory (M.Sc. Thesis)
remark on two duality relations,
Integral Equations and Operator Theory 57 (2), 217-228, 2007. [arXiv
sets of vector
payoffs in stochastic games,
Games and Economic Behavior 56,
semi-algebraic theory of
Math. Oper. Research 27,
- Workshop on "Analytic Aspects of Convexity", Rome, October 2016.
- Barcelona Analysis Conference BAC16, September 2016.
- Israeli Mathematical Union annual meeting, Dead Sea, June 2016.
- Joram Memorial Seminar on “Ricci Flow and Geometrization”, Hebrew University, May 2016. [Slides]
- Workshop on "Asymptotic Geometric Analysis", Oberwolfach, February 2016.
- Colloquium at Tel-Aviv University, Nov 2015.
- Conference on "Analytic and Probabilistic Techniques in Modern Convex Geometry", Columbia, Missouri, November 2015.
- Conference on "Convexity, Probability and Discrete Structures, a Geometric Viewpoint", Paris-Est, October 2015.
- Colloquium at Ben-Gurion University, May 2015.
- Colloquium at the Hebrew University, May 2015.
- Workshop on "Information Theory and Concentration Phenomena", IMA, Minneapolis, April 2015.
- Mini-Course, Workshop on "New developments in Optimal Transport, Geometry and Analysis", HIM, Bonn, February 2015.
- Conference in Honour of the 60th Birthday of Dominique Bakry, Toulouse. December 2014.
- Workshop on "Geometry and Probability", Kyoto, September 2014.
- Worskhop on "Isoperimetric problems between Analysis and Geometry", Pisa, June 2014.
- Asymptotic Geometric Analysis seminar, Tel-Aviv University, May 2014.
- Workshop on "Functional Inequalities", Toulouse, April 2014.
- Asymptotic Geometric Analysis seminar, Tel-Aviv University, January 2014.
- Israeli Mathematical Union annual meeting, Haifa University, June 2013.
- Conference on "Banach Spaces: Geometry and Analysis", Hebrew University, May 2013.
- Conference on "Complex Analysis and Dynamical Systems VI", Naharyia, May 2013.
- Workshop on "Heat Kernels, Stochastic Processes and Functional Inequalities", Oberwolfach, May 2013.
- Workshop on “Convex Geometry and its Applications", Oberwolfach, December 2012.
- Conference on “Optimal Transport and Applications", Pisa, November 2012.
- Analysis Seminar, Texas A&M, October 2012.
- Conference on “Convex Geometric Analysis”, Heraklion, September 2012.
- Conference on "Phenomena in high dimensions in
geometric analysis, random matrices, and computational geometry",
Roscoff, June 2012.
- Conference on "Interactions between Asynmptotic Geometric Analysis and Mathematical Physics", Eilat and Technion, May 2012.
- Probability Seminar, University of Geneva, April 2012.
- Plenary Talk, “Concentration inequalities and their applications”, CIRM, Luminy, January 2012.
- Plenary Talk, “Introductory workshop on quantitative geometry”, MSRI, August 2011 (cancelled).
- Geometric Analysis and Gravitation
Seminar at the Max-Planck-Institut für Gravitationsphysik
(Albert-Einsten-Insitut), Golm, July 2011.
- Mini-Course at Séminaire de
Mathématiques Supérieures on “Metric Measure
Spaces: Geometric and Analytic Aspects”, Montreal, July 2011.
- Seminar at "Discrete Analysis"
program, Cambridge, June 2011.
- Colloquium, Ben-Gurion University, May
- Colloquium, Tel-Aviv University, May
- Conference on "Asymptotic Geometric
Analysis and Convexity", Hagoshrim - Tel-Aviv, April 2011.
- Short Course, Workshop on "Functional
Inequalities and Discrete Spaces", Paris-Est, January 2011.
- Workshop on "Optimal Transport and
Geometric Probability", Fields Institute, November 2010.
- Workshop on "Asymptotic Geometric
Analysis", Fields Institute, September 2010.
- SAGE Workshop on "Optimal Transport
and Riemannian Geometry", University of Texas at Austin, September 2010.
- ICM Satellite Conference on
"Probability and Stochastic Processes", India, August 2010 (cancelled).
- Plenary Talk, Conference on
"Perspectives in High Dimensions", Cleveland, August 2010.
- Conference on "Asymptotic Geometric
Analysis", Euler International Mathematical Institute, St. Petersburg,
- Analysis Seminar, McGill-Concordia,
- Workshop on "Volume Inequalities",
Banff, March 2010.
on the State of
Geometry and Functional Analysis, Tel-Aviv University, June 2009.
Math Seminar, Rutgers
University, April 2009.
Lehigh University, April 2009.
Seminar, Institute for
Advanced Study, February 2009.
Analysis Seminar, Technion, January 2009.
Polytechnic Institute of
NYU, December 2008.
Seminar, City University of New York,
Functional Analysis and
Probability Seminar, Weizmann Institute of Science, August 2008.
annual conference on "Phenomena
in High Dimensions", Sevilla, June 2008.
Technion, Haifa, June 2008.
annual conference on "Probability in High Dimensions", CIRM at Luminy,
Case Western University, May 2008.
Seminar, Ohio-State University, May 2008.
Geometry and Topology
Seminar, Courant Institute, April 2008.
on Convex Geometry, University of Missouri
– Columbia, March 2008.
Convex Geometry", American Institute
of Mathematics, November 2007.
"Geometrization of Probability", Fields
Institute in Ottawa, September 2007.
Analytic Methods in Convex Geometry", American Institute of
Mathematics, August 2007. [Slides]
on "Phenomena in High Dimensions", Samos,
June 2007. [Slides]
"Convex Geometry - Analytic aspects", Cortona,
June 2007. [Slides]
Union annual meeting, Ben-Gurion University,
Analysis Seminar, Tel-Aviv University,
"Convexity and Probability", Freudenstadt,
September 2006. [Slides]
"Phenomena in High Dimensions", Institute Henri Poincare, Paris,
June 2006. [Slides]
on "Convex Geometry
and High Dimensional Phenomena", Vienna University of Technology, July
Theory and Mathematical Economics
Seminar, Hebrew University,
June 2002. [Slides]