Emanuel Milman
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E-mail: |
emilman 'at' math.ias.edu or |
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Phone: |
+1-609-734-8388 |
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Postal Address: |
School of Mathematics |
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Office: |
Simonyi 204 |
I am a second-year post-doctoral member at the Institute for Advanced Study. I graduated
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. Since its conception at the intersection of classical
Convex Geometry and the "Local Theory" of Banach
spaces, this field has been a source of rich and fruitful research, with
connections to many other fields, such as Probability Theory, PDE, Harmonic
Analysis, Differential Geometry, Mathematical Physics, Combinatorics
and Graph Theory.
My PhD Thesis has its origins set in the study of several fundamental
questions on the asymptotic distribution of volume inside convex bodies in
n-dimensional Euclidean space, when n tends to infinity. The questions
which arise in this setting are naturally related to the study of
isoperimetric inequalities, metric entropy and central-limit theorems for
convex domains and even Integral Geometry.
My additional research interests include Convex Geometry, the Neumann and Dirichlet Laplacians on convex
domains, isoperimetric minimizing surfaces, optimal transportation and the heat
diffusion semi-group in convex domains, asymptotic theory of finite-dimensional
normed spaces ("Local Theory"), general
phenomena in high dimensions, empirical processes, Radon transforms and
Harmonic Analysis on Grassmann manifolds, and their
connection to Asymptotic Convex Geometric Analysis.
Publications, Preprints and Manuscripts:
- Uniform tail-decay of Lipschitz functions implies Cheeger’s isoperimetric inequality under convexity assumptions, to appear in Comptes Rendus Mathematique. [pdf]
- On the role of Convexity in Functional and Isoperimetric Inequalities, 42 pages, to appear in the Proceedings of the London Mathematical Society. [arXiv link]
- On the role of Convexity in Isoperimetry, Spectral Gap and Concentration, 50 pages, submitted. [arXiv link]
Ph.D. :
- 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]
- Generalized Intersection Bodies are not equivalent, Advances in Mathematics 217 (6), 2822-2840, 2008. [arXiv link]
- A remark on two duality relations, Integral Equations and Operator Theory 57 (2), 217-228, 2007. [arXiv link]
- On Gaussian marginals of uniformly convex bodies, Journal of Theoretical Probability, 25 pages, to appear. [arXiv link]
- with Bo’az Klartag, On volume distribution in 2-convex bodies, Israel Journal of Mathematics 164, 221-249, 2008. [arXiv link]
- A comment on the low-dimensional Busemann-Petty problem, Lecture Notes in Math. 1910, GAFA Seminar Notes 2004–5, 245-253. [arXiv link]
- Generalized Intersection Bodies, Journal of Functional Analysis 240 (2), 530-567, 2006. [arXiv link]
- Dual mixed volumes and the Slicing Problem, Advances in Mathematics 207 (2), 566-598, 2006. [arXiv link]
M.Sc. :
- 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.
Invited Talks:
- Probability Seminar, City
- 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,
- Fifth annual conference on
“Probability in High Dimensions”, CIRM at Luminy,
May 2008.
- Colloquium,
- PDE Seminar,
- Differential Geometry and
Topology Seminar, Courant Institute, April 2008.
- Conference on Convex
Geometry,
- Quantitative stability of
spectral-gap and isoperimetry under
perturbations of convex domains, PDE Seminar, MIT, February 2008.
- The role of convexity
in isoperimetry, spectral-gap and concentration,
Colloquium,
- The role of convexity
in isoperimetry, spectral-gap and concentration,
Workshop on "Algorithmic Convex Geometry", American
- Weak Poincare
inequalities and Isoperimetry in Convex Domains,
Workshop on "Geometrization of
Probability", Fields Institute in
- Generalized Intersection Bodies, the Low-Dimensional Busemann-Petty Problem, and what's in between, Workshop on "Fourier Analytic Methods in Convex Geometry", American Institute of Mathematics, August 2007. [Slides]
- Isoperimetric
Inequalities for Uniformly Log-Concave Measures and Uniformly Convex
Bodies, Third annual conference on "Phenomena in High
Dimensions",
- Generalized Intersection Bodies Are Not Equivalent, Educational Workshop on "Convex Geometry - Analytic aspects", Cortona, June 2007. [Slides]
- Isoperimetric Inequalities for Uniformly Log-Concave Measures and Uniformly Convex Bodies, Israel Mathematical Union annual meeting, Ben-Gurion Universiy, May 2007.
- Isoperimetric
Inequalities for Uniformly Log-Concave Measures and Uniformly Convex
Bodies, Geometric and Functional Analysis Seminar,
- Generalized Intersection Bodies and the low dimensional Busemann-Petty Problem, educational workshop on "Convexity and Probability", Freudenstadt, September 2006. [Slides]
- Gaussian Marginals of Uniformly Convex Bodies, Second
annual conference on "Phenomena in High Dimensions", Institute
Henri Poincare,
- Covering Numbers and the Duality of Entropy, Geometric Functional Analysis and Probability Seminar, Weizmann Institute of Science, April 2006.
- Dual Mixed Volumes and the Slicing Problem, First annual conference on "Convex Geometry and High Dimensional Phenomena", Vienna University of Technology, July 2005. [Slides]
- The Semi-Algebraic
Theory of Stochastic Games, Game Theory and Mathematical
Economics Seminar,