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 2013:
Convex Bodies in High Dimension - List of Exercises
Previous Advanced Courses Given:
Isoperimetric Inequalities and Applications - List of Exercises
- with Franck Barthe, Transference Principles for Log-Sobolev and Spectral-Gap with Applications to Conservative Spin Systems, to appear in Comm. Math. Physics. [arXiv link]
Isoperimetric Inequalities and Model Spaces for
Curvature-Dimension-Diameter Condition, submitted. [arXiv link]
Publications (according to topic):
- 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
- 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 “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, 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]