Abstract
The term “Geometric Analysis” is a recent one but it has quickly become fashionable and is used too often and for very different mathematics. So, we added the adjective “asymptotic” to be more specific, and we will first explain the subject of this talk.
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References
[books]
N. Alon, J.H. Spencer, The Probabilistic Method, Wiley Interscience, 1992.
Yu.D. Burago, V.A. Zalgaller, Geometric Inequalities, Grundlehren der mathematischen Wissenschaften 285, Springer Verlag, 1980.
M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, based on “Structures métriques des variétés Riemanniennes” (L. LaFontaine, P. Pansu, eds.), English translation by Sean M. Bates, Birkhäuser, Boston-Basel-Berlin, 1999 (with Appendices by M. Katz, P. Pansu and S. Semmes).
M. Ledoux, M. Talagrand, Probability in Banach Spaces, Ergeb. Math. Grenzgeb. 3 Folge, vol. 23, Springer, Berlin, 1991.
P. Lévy, Problèmes concrets d’analyse fonctionelle, Gauthier-Villars, Paris, 1951.
V. Milman, G. Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces, Springer Lecture Notes in Math. 1200, 1986.
R. Motwain, P. Raghavan, Randomized Algorithms, Cambridge Univ. Press, 1995.
G. Pisier, Factorization of Linear Operators and the Geometry of Banach Spaces, CBMS, vol. 60, American Math. Soc., Providence, RI, 1986.
G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math. 94, 1989.
R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications 44, Cambridge University Press, 1993.
N. Tomczak-Jaegermann, Banach-Mazur Distance and Finite-Dimensional Operator Ideal, Pitman Monographs 38, Pitman, London, 1989.
[Surveys]
B. Bollobas, Volume estimates and rapid mixing, in “Flavors in Geometry” (S. Levy, ed.), MSRI Pub. 31, Cambridge University Press (1997), 151–182.
A.A. Giannopoulos, V. Milman, Euclidean structures in finite dimensional spaces, Handbook on the Geometry of Banach Spaces, to appear.
M. Gromov, Spaces and question, Proceedings of Visions in Mathematics — Towards 2000, Israel 1999, GAFA, GAFA2000, Special Volume, issue 1 (2000), 118–161..
J. Lindenstrauss, V. Milman, The local theory of normed spaces and its applications to Convexity, in “Handbook of Convex Geometry” (P.M. Gruber, J.M. Wills, eds.) 1149–1220 (1993).
B. Maurey, Quelques progrés dans la compréhension de la dimension infinie, in “Espaces de Banach classiques et quantiques”, Societé Matmatiques de France, Journee Annuelle (1994), 1–29.
V.D. Milman, The concentration phenomenon and linear structure of finite dimensional normed spaces, Proceedings I.C.M, Berkeley (1986).
V. Milman, The heritage of P. Lèvy in geometric functional analysis, Asterisque 157/8, 73–141 (1988).
V. Milman, Surprising geometric phenomena in high-dimensional convexity theory, Proc. ECM2, vol. II, Birkhäuser Progress in Math. 196, 73–91(1996).
V. Milman, Randomness and pattern in convex geometric analysis, Proceedings of ICM-98, Berlin, v. 2 (1998), 665–677.
V. Milman, A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n dimensional space, GAFA-Seminar87–88, Springer Lecture Notes in Math. 1376 (1989), 64–104.
T. Odell, in “Proceedings of Analysis and Logic Meeting, Mons, Belgium, August 1997” (C. Finet, C. Michaux, eds), London Math. Soc. Lecture Notes, Cambridge Press, to appear.
M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, IHES Publ. Math. 81 (1995), 73–205.
M. Talagrand, A new look at independence, Ann. Probab. 24:1 (1996),1–34.
[Articles]
N. Alon, V.D. Milman, λ1, isoperimetric inequalities for graphs and superconcentrators, J. Combinatorial Theory, Ser. B 38:1 (1985), 73–88.
M. Anttila, K. Ball, I. Perissinaki, The central limit problem for convex bodies, Trans. Amer. Math. Soc., to appear
J. Bastero, J. Bernués, A. Peña, An extension os Milman’s reverse Brunn-Minkowski inequality, GAFA 5:3 (1995), 572–581.
J. Bourgain, J. Lindenstrauss, V. Milman, Mikowski sums and symmetrization, GAFA-Seminar notes 86–87, Springer Lecture Notes in Math. 1317 (1988), 283–289.
J. Bourgain, J. Lindenstrauss, V. Milman, Estimates related to Steiner symmetrizations, Springer Lecture Notes in Math. 1367 (1989), 264–273.
J. Bourgain, V.D. Milman, Distances between normed spaces, their subspaces and quotient spaces, Integral Equations and Operator Theory 9(1986), 31–46.
J. Bourgain, A. Pajor, S.J. Szarek, N. Tomczak-Jaegermann, On the duality problem for entropy numbers of operators, in “Geometric Aspects of Functional Analysis (1987–88) (J. Lindenstrauss, V.D. Milman, eds.), Springer Lecture Notes in Math 1376 (1989), 50–63.
S. Dar, Remarks on Bourgain’s problem on slicing of convex bodies, in “Seminar notes of Israel Seminar on Geometric Aspects of Functions Analysis”, Birkhauser, Operator Theory: Advances and Applications 77 (1995),61–66.
T. Figiel, J. Lindenstrauss, V.D. Milman, The dimension of almost spherical sections of convex bodies, Acta Math. 139:1–2 (1977), 59–94.
A.A. Giannopoulos, V. Milman, Concentration property on probability spaces, Advances in Math. 156 (2000).
B.S. Kashin, Sections of some finite-dimensional sets and classes of smooth functions (in Russian), Izv. Akad. SSR Ser. Mat. 41 (1977), 334–351.
B. Klartag, Remarks on Minkowski symmetrizations, GAFA Seminar Notes 1996–2000, Springer Lecture Notes in Mathematics 1745 (2000), 109–117.
V.D. Milman, A new proof of the theorem of Dvoretzky on sections of convex bodies, Functional Analysis and its Applications 5:4 (1971), 28–37.
V. Milman, Proportional quotients of finite dimensional normed spaces, Springer Lecture Notes in Math. 1573 (1994), 3–5.
V.D. Milman, A few observations on the connections between Local Theory and some other fields, GAFA-Seminar Notes 86–87, Springer Lecture Notes in Math. 1317 (1988), 283–289.
V. Milman, A. Pajor, Entropy and asymptotic geometry of non-symmetric convex bodies. Advances in Math., 152 (2000), 314–335.
V. Milman, G. Schechtman, Global vs. local asymptotic theories of finite dimensional normed spaces, Duke Math. J. 90: (1997), 73–93.
V. Milman, G. Szarek, A geometric lemma and duality of entropy numbers, GAFA Seminar Notes 1996–2000, Springer Lecture Notes in Math. 1745 (2000), 191–222.
J. von Neumann, Approximative properties of matrices of high order rank, Portugal. Math. 3 (1942), 1–62.
V.G. Pestov, Amenable representations and dynamics of the unit sphere in an infinite-dimensional Hilbert space, GAFA, Geom. funct. anal. 10 (2000), 1171–1201.
V.G. Pestov, Ramsey-Milman phenomenon, Urysohn metric spaces and extremely amenable groups, preprint.
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Milman, V. (2000). Topics in Asymptotic Geometric Analysis. In: Alon, N., Bourgain, J., Connes, A., Gromov, M., Milman, V. (eds) Visions in Mathematics. Modern Birkhäuser Classics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0425-3_8
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