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of the European RTN Project "Phenomena in High Dimensions"

PHD - Work Package I:

PHD - Work Package II:

PHD - Work Package III:

PHD - Work Package IV:

PHD - Work Package V:

PHD - Work Package VI:

Asymptotic Geometric Analysis:

(Research field of WK I)

Ball, K.M.   Convex geometry and Functional analysis. In: Handbook of the Geometry of Banach spaces (Lindenstrauss-Johnson eds), Elsevier (2001), 161-194.
Ball, K.M.   An elementary introduction to modern convex geometry. In: Flavors of Geometry, MSRI Publications, Volume 31, Cambridge Univ. Press, Cambridge, (1997), 1-58.
Davidson, K.R. and Szarek, S.J.   Local operator theory, random matrices and Banach spaces. In: Handbook of the Geometry of Banach spaces (Lindenstrauss-Johnson eds), Elsevier (2001), 317-366.
Figiel, T., Lindenstrauss, J. and Milman, V.D.   The dimension of almost spherical sections of convex bodies. Acta Math. 139 (1977), 53-94.
Giannopoulos, A.A. and Milman, V.D.   Euclidean structure in finite dimensional normed spaces. In: Handbook of the Geometry of Banach spaces (Lindenstrauss-Johnson eds), Elsevier (2001), 707-779.
Giannopoulos, A.A. and Milman, V.D.   Asymptotic convex geometry: short overview. Different faces of geometry, 87-162, Int. Math. Ser. (N.Y.), Kluwer/Plenum, NewYork, 2004.
Gluskin, E.D.   Probability in the geometry of Banach spaces. Proc. Int. Congr. Berkeley, Volume 2 (1986), 924-938.
Johnson, W.B. and Schechtman, G.   Finite dimensional subspaces of $L_p$. In: Handbook of the Geometry of Banach spaces (Lindenstrauss-Johnson eds), Elsevier (2001), 837-870.
Lindenstrauss, J. and Milman, V.D.   The Local Theory of Normed Spaces and its Applications to Convexity. In: Handbook of Convex Geometry (edited by Gruber, P.M. and Wills, J.M.), Elsevier (1993), 1149-1220.
Mankiewicz, P. and Tomczak-Jaegermann, N.   Quotients of finite-dimensional Banach spaces; random phenomena. In: Handbook of the Geometry of Banach spaces (Lindenstrauss-Johnson eds), Elsevier (2003), 1201-1246.
Maurey, B.   Type, cotype and $K$-convexity. In: Handbook of the Geometry of Banach spaces (Lindenstrauss-Johnson eds), Elsevier (2003), 1299-1332.
Maurey, B.   Inégalité de Brunn-Minkowski-Lusternik, et autres inégalités géometriques et fonctionnelles. Séminaire Bourbaki. Volume 2003/2004. Asterisque No. 299 (2005), Exp. No. 928, vii, 95-113.
Milman, V.D.   The concentration phenomenon and linear structure of finite-dimensional normed spaces. Proceedings of the ICM, Berkeley (1986), 961-975.
Milman, V.D.   Dvoretzky's theorem- Thirty years later. Geom. Functional Anal. 2 (1992), 455-479.
Milman, V.D. and Pajor, A.   Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n -dimensional space. Lecture Notes in Mathematics, 1376, Springer, Berlin (1989), 64-104.
Milman, V.D. and Schechtman, G.   Asymptotic Theory of Finite Dimensional Normed Spaces. Lecture Notes in Mathematics, 1200, (1986), Springer, Berlin.
Pisier, G.   Probabilistic methods in the geometry of Banach spaces. Lecture Notes in Mathematics, 1206, (1986), 167-241.
Pisier, G.   The Volume of Convex Bodies and Banach Space Geometry. Cambridge Tracts in Mathematics, 94, (1989).
Schechtman, G.   Concentration results and applications. In: Handbook of the Geometry of Banach spaces (Lindenstrauss-Johnson eds), Elsevier (2003), 1603-1634.
Schneider, R.   Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge (1993).
Tomczak-Jaegermann, N.   Banach-Mazur Distances and Finite Dimensional Operator Ideals. Pitman Monographs, 38, (1989), Pitman, London.


Isometric Convex Geometry:

(Research field of WK II)


Barvinok, A.   A Course in Convexity. Amer. Math. Soc., Providence, RI, 2002.
Bonnesen, T. and Fenchel, W.   Theorie der konvexen Körper. Springer, Berlin, 1934.
English translation: BCS Associates, Moscow, Idaho (1987).
Burago, Yu.D. and Zalgaller, V.A.   Geometric Inequalities. Springer, Berlin, 1980.
Hadwiger, H.   Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer, Berlin (1957).
Gardner, R.J.   Geometric Tomography. Cambridge Univ. Press, Cambridge, 1995.
Groemer, H.   Geometric Applications of Fourier Series and Spherical Harmonics. Cambridge Univ. Press, Cambridge, 1996.
Grünbaum, B.   Convex Polytopes. Interscience, Wiley, London 1967 and a second edition (prepared by Kaibel, V., Klee, V., Ziegler, G.M.): Springer, New York (2003).
Koldobsky, A.   Fourier Analysis in Convex Geometry. Amer. Math. Soc., Providence, RI, 2005.
Leichtweiß, K.   Affine Geometry of Convex Bodies. Johann Ambrosius Barth, Heidelberg, 1998.
Leichtweiß, K.   Konvexe Mengen. VEB Deutscher Verlag der Wissenschaften, Berlin, 1980.
Schneider, R.   Convex Bodies: the Brunn-Minkowski Theory. Cambridge Univ. Press, Cambridge, 1993.
Thompson, A.C.   Minkowski Geometry. Cambridge Univ. Press, Cambridge, 1996.
Webster, R.   Convexity. Oxford Univ. Press, Oxford, 1994.
Ziegler, G.M.   Lectures on Polytopes. Springer, New York, 1995 and a second edition in 1998.
Zong, Ch.   Strange Phenomena in Convex and Discrete Geometry. Springer, New York, 1996.


Collections of survey articles:

Convexity.   Ed. Klee, V.; Proc. Symp. Pure Math., Volume VII, Amer. Math. Soc., Providence, RI, 1963.
Contributions to Geometry (Part I: Geometric convexity).   Eds. Tölke, J., Wills, J.M.; Birkhäuser, Basel, 1979.
Convexity and Its Applications.   Eds. Gruber, P.M., Wills, J.M.; Birkhäuser, Basel, 1983.
Handbook of Convex Geometry, vols. A and B.   Eds. Gruber, P.M., Wills, J.M.: North-Holland, Amsterdam, 1993.


Selected later survey articles:

Ball, K.M.   An elementary introduction to modern convex geometry. In: Flavors of Geometry (S. Levy, ed.), MSRI Publications, Volume 31, Cambridge University Press, Cambridge, (1997), 1-58.
Barthe, F.   Autour de l'inégalité de Brunn-Minkowski. Ann. Fac. Sci. Toulouse (VI), Math. 12 (2003), 127-178.
Gardner, R.J.   The Brunn-Minkowski inequality. Bull. Amer. Math. Soc. 39 (2002), 355-405.
Zong, Ch.   What is known about cubes. Bull. Amer. Math. Soc. 42 (2005), 181-211.


Asymptotic Combinatorics:

(Research field of WK III)

Alon, N. and Spencer, J.   The Probabilistic Method, 2nd Ed., Wiley, 2000.
Bollobàs, B.   Extremal Graph Theory, Paperback Ed., Dover, 2004.
Bollobàs, B.   Random Graphs, 2nd Ed., Cambridge Univ. Press, 2001.
Janson, S., Luczak, T. and Ricinski, A.   Random graphs, Wiley, 2000.
Jukna, S.   Extremal Combinatorics, Springer 2001.
Krivelevich, M. and Sudakov, B.   Pseudo-random graphs, preprint, available at: http://


Randomised Computation and Complexity:

(Research field of WK IV)

Aldous, D. and Fill, J.   Reversible Markov Chains and Random walks on graphs. Cf. Aldous' homepage: http://www.stat.Berkeley.EDU/users/aldous/book.html.
Garey, M.R. and Johnson, D.S.   Computers and intractability. A guide to the theory of NP-completeness. A Series of Books in the Mathematical Sciences. W. H. Freeman and Co., San Francisco, Calif., 1979.
Grötschel, M., Lovàsz, L. and Schrijver, A.   Geometric Algorithms and Combinatorial Optimization. Second edition. Algorithms and Combinatorics, 2. Springer-Verlag, Berlin, 1993.
Jerrum, M.   Mathematical foundations of the Markov chain Monte Carlo method. Probabilistic methods for algorithmic discrete mathematics, 116-165, Algorithms Combin., 16, Springer, Berlin, 1998.
Kannan, R.   Markov chains and polynomial time algorithms. 35th Annual Symposium on Foundations of Computer Science (Santa Fe, NM, 1994), 656-671, IEEE Comput. Soc. Press, Los Alamitos, CA, 1994. 68Q25 (68R05).
Lovàsz, L.   An algorithmic theory of numbers, graphs and convexity. SIAM, Philadelphia, Pa., 1986.
Lovàsz, L. and Winkler, P.   Mixing times. In: Microsurveys in discrete probability (Princeton, NJ, 1997), 85-133, DIMACS, Ser. Discrete Math. Theoret. Comput. Sci., 41.
Simonovits, M.   How to compute the volume in high dimension? ISMP, 2003 (Copenhagen). Math. Program. 97, no. 1-2, Ser. B, (2003), 337-374.


High-Dimensional Phenomena in Mathematical Physics:

(Research field of WK V)

Amit, D. J.   Modeling Brain Function. The World of Attractor Neural Networks. Cambridge University Press, 1989.
Bessis, D., Itzykson, C. and Zuber, J.-B.   Quantum Field Theory Techniques in Graphical Enumeration. Adv. Appl. Math. 1, (1980), 109-157.
Chueshov, I.   Monotone random systems theory and applications Lecture Notes in Mathematics, 1779. Springer-Verlag, Berlin, 2002.
Guhr, T., Mueller-Groeling, A. and Weidenmueller, H.A.   Random matrix theories in quantum physics: common concepts, Phys. Rept. 299, (1998), 189-425.
Mehta, M.L.   Random Matrices, New York, Academic Press, 1991.
Mézard, M., Parisi, G., Virasiro, M.   Spin glass theory and beyond. World Scientific, Singapore, 1987.
Pastur, L.   Random Matrices as Paradigm. In: Mathematical Physics 2000, 216-266, A. Grigoryan, T. Kibble, B. Zegarlinskii (Eds.), World Scientific, Singapore, 2000.
Ruelle, D.   Statistical Mechanics Rigorous Results. W. A. Benjamin, 1969.
Talagrand, M.   Spin glasses: a challenge for mathematicians. Cavity and mean field models. Springer-Verlag, Berlin, 2003.


Isoperimetric Principles in Geometry and Probability:

(Research field of WK VI)

Ball, K.M.   Convex geometry and functional analysis. In: Handbook of the geometry of Banach spaces. Volume 1. Johnson, W. B. (ed.) et al., Amsterdam: Elsevier, (2001), 161-194.
Ledoux, M.   The concentration of measure phenomenon. Mathematical Surveys and Monographs, 89, American Mathematical Society, Providence, RI, 2001.
Ledoux, M. and Talagrand, M.   Probability in Banach spaces, Springer, Berlin, 1991.
Mehta, M.   Random matrices. Third edition. Pure and Applied Mathematics (Amsterdam), 142. Elsevier/Academic Press, Amsterdam, 2004.
Pastur, L.   Matrices aléatoires: statistique asymptotique des valeurs propres. Séminaire de Probabilités, XXXVI. Lecture Notes in Math., 1801, (2003), 135-164. Springer, Berlin, 2003.
Talagrand, M.   The generic chaining. Upper and lower bounds of stochastic processes. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005.
Villani, C.   Topics in optimal transportation. Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003.