Emanuel Milman

E-mail:	emanuel.milman 'at' gmail.com or emilman 'at' tx.technion.ac.il
Phone: Fax:	+972-4-8294196 +972-4-8293388
Postal Address:	Department of Mathematics Technion - Israel Institute of Technology Haifa 32000 Israel
Office:	Amado 630

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 space Rⁿ, Banach spaces (possibly infinite dimensional), Riemannian manifolds with non-negative (Ricci) curvature, and other generalizations. Since its conception at the intersection 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 research interests include isoperimetric, functional and concentration inequalities, the interplay between geometry and spectral properties of Riemannian manifolds, geometry of isoperimetric minimizing surfaces, Geometric Measure Theory, diffusion semi-group and heat-kernel estimates in convex manifolds, optimal transport, distribution of volume in convex bodies, classical Convex Geometry, “local theory” of Banach Spaces, convexity 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.

Preprints and Manuscripts:

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]
Sharp Isoperimetric Inequalities and Model Spaces for Curvature-Dimension-Diameter Condition, submitted. [arXiv link]

Publications (according to topic):

Isoperimetric Inequalities

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 link]
Properties of Isoperimetric, Functional and Transport-Entropy Inequalities Via Concentration, Probab. Theory Relat. Fields 152, 475–507, 2012. [arXiv link]
A 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 link]
Isoperimetric and Concentration Inequalities - Equivalence under Curvature Lower Bound, Duke Math. J. 154 (2), 207-239, 2010. [arXiv link]
Concentration and isoperimetry are equivalent assuming curvature lower bound, C. R. Math. Acad. Sci. Paris. 347, 73-76, 2009. [pdf]
On the role of convexity in functional and isoperimetric inequalities, Proc. London Math. Soc. 99 (3), 32-66, 2009. [arXiv link]
On the role of convexity in isoperimetry, spectral gap and concentration, Invent. Math. 177 (1), 1-43, 2009. [arXiv link]
Uniform tail-decay of Lipschitz functions implies Cheeger’s isoperimetric inequality under convexity assumptions, C. R. Math. Acad. Sci. Paris 346, 989-994, 2008. [pdf]
with Sasha Sodin, An isoperimetric inequality for uniformly log-concave measures and uniformly convex bodies, J. Func. Anal. 254 (5), 1235-1268, 2008. [arXiv link]

Distribution of Volume in Convex Bodies

with Bo'az Klartag, Inner Regularization of Log-Concave Measures and Small-Ball Estimates, Lecture Notes in Math. 2050, GAFA Seminar Notes 2006–2010, 267-278, 2012. [arXiv link]
with Bo'az Klartag, Centroid Bodies and the Logarithmic Laplace Transform - A Unified Approach, J. Func. Anal. 262 (1), 10-34, 2012. [arXiv link]
with Olivier Guedon, Interpolating Thin-Shell and Sharp Large-Deviation Estimates For Isotropic Log-Concave Measures, Geom. Funct. Anal. 21 (5), 1043-1068, 2011. [arXiv link]
On Gaussian marginals of uniformly convex bodies, J. Theor. Prob. 22 (1), 256-278, 2009. [arXiv link]
with Bo’az Klartag, On volume distribution in 2-convex bodies, Israel J. Math. 164, 221-249, 2008. [arXiv link]
Dual mixed volumes and the Slicing Problem, Adv. Math. 207 (2), 566-598, 2006. [arXiv link]

Contracting Maps

with Young-Heon Kim, A Generalization of Caffarelli's Contraction Theorem via (reverse) Heat Flow, Math. Annal 354 (3), 827-862, 2012. [arXiv link]

Low-Dimensional Sections of Star Bodies

Generalized Intersection Bodies are not equivalent, Adv. Math. 217 (6), 2822-2840, 2008. [arXiv link]
Generalized Intersection Bodies, J. Func. Anal. 240 (2), 530-567, 2006. [arXiv link]
A comment on the low-dimensional Busemann-Petty problem, Lecture Notes in Math. 1910, GAFA Seminar Notes 2004–5, 245-253, 2007. [arXiv link]

Covering Numbers and Metric Entropy

A remark on two duality relations, Integral Equations and Operator Theory 57 (2), 217-228, 2007. [arXiv link]

Game Theory (M.Sc. Thesis)

Approachable sets of vector payoffs in stochastic games, Games and Economic Behavior 56, 135-147, 2006.
The semi-algebraic theory of stochastic games, Math. Oper. Research 27, 401-418, 2002.

Conferences Organized:

ISF Workshop on “Interactions between Asymptotic Geometric Analysis and Mathematical Physics”
Eilat and Technion, Israel, May 3-7 and May 8-10 2012.
Co-organized with Shiri Artstein-Avidan (Tel-Aviv University) and Shahar Mendelson (Technion).
International Workshop on “Concentration, Functional Inequalities and Isoperimetry”
(merged with Special Session of the AMS 2009 Fall Southeastern Meeting with same title),
Boca Raton , Florida , October 29 - November 1, 2009.
Co-organized with Christian Houdré (GeTech) and Mario Milman (FAU).

Co-editor of the workshop proceedings in volume 545 of Contemporary Mathematics, together with Christian Houdré (GeTech), Michel Ledoux (Toulouse) and Mario Milman (FAU).

Invited Talks:

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 2011.
Colloquium, Tel-Aviv University, May 2011.
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, July 2010.
Analysis Seminar, McGill-Concordia, April 2010.
Workshop on "Volume Inequalities", Banff, March 2010.
Conference on the State of Geometry and Functional Analysis, Tel-Aviv University, June 2009.
Discrete Math Seminar, Rutgers University, April 2009.
Colloquium, Lehigh University, April 2009.
Members Seminar, Institute for Advanced Study, February 2009.
Functional Analysis Seminar, Technion, January 2009.
Colloquium, Polytechnic Institute of NYU, December 2008.
Probability Seminar, City University of New York, October 2008.
Geometric Functional Analysis and Probability Seminar, Weizmann Institute of Science, August 2008.
Fourth annual conference on "Phenomena in High Dimensions", Sevilla, June 2008.
Colloquium, Technion, Haifa, June 2008.
Fifth annual conference on "Probability in High Dimensions", CIRM at Luminy, May 2008.
Colloquium, Case Western University, May 2008.
PDE Seminar, Ohio-State University, May 2008.
Differential Geometry and Topology Seminar, Courant Institute, April 2008.
Conference on Convex Geometry, University of Missouri – Columbia, March 2008.
PDE Seminar, MIT, February 2008.
Colloquium, Polytechnic University, November 2007.
Workshop on "Algorithmic Convex Geometry", American Institute of Mathematics, November 2007.
Workshop on "Geometrization of Probability", Fields Institute in Ottawa, September 2007.
Workshop on "Fourier Analytic Methods in Convex Geometry", American Institute of Mathematics, August 2007. [Slides]
Third annual conference on "Phenomena in High Dimensions", Samos, June 2007. [Slides]
Educational Workshop on "Convex Geometry - Analytic aspects", Cortona, June 2007. [Slides]
Israel Mathematical Union annual meeting, Ben-Gurion University, May 2007.
Geometric and Functional Analysis Seminar, Tel-Aviv University, April 2007.
Educational workshop on "Convexity and Probability", Freudenstadt, September 2006. [Slides]
Second annual conference on "Phenomena in High Dimensions", Institute Henri Poincare, Paris, June 2006. [Slides]
First annual conference on "Convex Geometry and High Dimensional Phenomena", Vienna University of Technology, July 2005. [Slides]
Game Theory and Mathematical Economics Seminar, Hebrew University, June 2002. [Slides]