Volumes in High Dimensions
נפחים בממדים גבוהים
Spring semester, 2014
Room 229, Schreiber building
Wednesday, 10-13, Ornstein 111
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.
Real Analysis, Introduction to Hilbert Spaces, Probability (for Mathematicians or for Sciences)
The formal assignments are:
Solve and submit these exercises.
My plan is to add exercises every week or so during the semester.
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.
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.