Volumes in High Dimensions, Tel Aviv University, 2010


 
 

Volumes in High Dimensions

נפחים בממדים גבוהים

Spring semester, 2014


Lecturer

Bo'az Klartag
Room 229, Schreiber building
Phone: 03-6406957
E-mail:    


Classes

Wednesday, 10-13, Ornstein 111


Syllabus

The concentration phenomenon on the high-dimensional sphere and the high-dimensional cube. Approximately Gaussian Marginals and the "thin shell" theorem of Sudakov/Diaconis-Freedman. Log-concavity and Thin Shell estimates, the Prekopa-Leindler inequality, Poincare inequalities through the Bochner technique. The isotropic constant, proof of the Bourgain-Milman inequality.

Prerequisites

Real Analysis, Introduction to Hilbert Spaces, Probability (for Mathematicians or for Sciences)


Final grade

The formal assignments are:

  • Solve and submit these exercises. My plan is to add exercises every week or so during the semester.


    Related literature

  • Brazitikos, S., Giannopoulos, A., Valettas, P., Vritsiou, B.-H., Geometry of Isotropic Convex Bodies. Mathematical Surveys and Monographs, 196. American Mathematical Society, Providence, RI, 2014.

  • Ledoux, M., The concentration of measure phenomenon. Mathematical Surveys and Monographs, 89. American Mathematical Society, Providence, RI, 2001.

  • Milman, V.D., Schechtman, G., Asymptotic theory of finite-dimensional normed spaces. With an appendix by M. Gromov. Lecture Notes in Mathematics, 1200. Springer-Verlag, Berlin, 1986.

  • Pisier, G., The volume of convex bodies and Banach space geometry. Cambridge Tracts in Mathematics, 94. Cambridge University Press, Cambridge, 1989.


  • Comments

  • Here is a proof that the level sets of the distance function are null sets (which is not needed for the isoperimetric inequality, but still good to know).

  • Class notes in Hebrew by Amir Livne Bar-On.

  • Class notes by Silouanos Brazitikos of a similar, but more condensed, course I taught last year.


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