

Volumes in High Dimensions
נפחים בממדים גבוהים
Spring semester, 2014
Lecturer
Bo'az Klartag
Room 229, Schreiber building
Phone: 036406957
Email:
Classes
Wednesday, 1013, Ornstein 111
Syllabus
The concentration phenomenon on the highdimensional sphere and the highdimensional cube.
Approximately Gaussian Marginals and the "thin shell" theorem of Sudakov/DiaconisFreedman.
Logconcavity and Thin Shell estimates, the PrekopaLeindler inequality, Poincare inequalities through the Bochner technique.
The isotropic constant, proof of the BourgainMilman 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 finitedimensional normed spaces. With an appendix by M. Gromov. Lecture Notes in Mathematics, 1200. SpringerVerlag, 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 BarOn.
Class notes by Silouanos Brazitikos of a similar, but more condensed,
course I taught last year.
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