Emanuel Milman
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E-mail: |
emilman
'at' math.toronto.edu or |
|
Phone: |
+1-416-946-5440 |
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Postal Address: |
Department
of Mathematics |
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Office: |
Bahen 6107 |
I am a
currently a Post-Doctoral Fellow at the University of
Toronto and a
Senior Lecturer (on leave) at the Technion.
Prior to
this, I was member at the Institute
for
Advanced Study between
2007-2009, after graduating from the Weizmann
Institute of Science in July
2007, under the supervision of Prof.
Gideon Schechtman.
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 classical Convex Geometry, the interplay between geometry and spectral properties of Riemannian manifolds, geometry of isoperimetric minimizing surfaces, isoperimetric, functional and concentration inequalities, Geometric Measure Theory, diffusion semi-group and heat-kernel estimates in convex manifolds, optimal transportation for the Monge-Amp`ere equation, distribution of volume in convex bodies, “local theory” of Banach Spaces, convexity in graphs, metric entropy and covering numbers, empirical processes, general phenomena in high dimensions, Radon transforms and Harmonic Analysis on Grassmann manifolds.
Publications, Preprints and Manuscripts (according to topic):
Contracting maps and PDE
- with Young-Heon Kim, A Generalization of Caffarelli's Contraction Theorem via (reverse) Heat Flow, submitted. [arXiv link]
Isoperimetric Inequalities
- Isoperimetric Bounds on Convex Manifolds, submitted. [arXiv link]
- Properties of Isoperimetric, Functional and Transport-Entropy Inequalities Via Concentration, submitted. [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, to appear in Duke Math. J. [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, Journal of Functional Analysis 254 (5), 1235-1268, 2008. [arXiv link]
Low-Dimensional Sections of Star Bodies
- Generalized Intersection Bodies are not equivalent, Advances in Mathematics 217 (6), 2822-2840, 2008. [arXiv link]
- Generalized Intersection Bodies, Journal of Functional Analysis 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]
Distribution of Volume in Convex Bodies
- On Gaussian marginals of uniformly convex bodies, Journal of Theoretical Probability 22 (1), 256-278, 2009. [arXiv link]
- with Bo’az Klartag, On volume distribution in 2-convex bodies, Israel Journal of Mathematics 164, 221-249, 2008. [arXiv link]
- Dual mixed volumes and the Slicing Problem, Advances in Mathematics 207 (2), 566-598, 2006. [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, Mathematics of Operations Research 27, 401-418, 2002.
Conferences Organized:
- International Workshop on
“Concentration, Functional Inequalities and
Isoperimetry”
(will merge 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 (Georgia Institute of Technology) and Mario Milman (Florida Atlantic University).
Invited Talks:
- 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.
- 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]