Course Identification

Familiarity with undergraduate probability, real analysis (say, Lebesgue measure) and Hilbert spaces.

Part I: High dimension 

• Estimates for the central limit theorem for i.i.d random variables.
• The isoperimetric inequality on the sphere, concentration of measure.
• Maximal volume ellipsoid (John) and Dvoretzky's theorem: Any high-dimensional convex body contains approximately-spherical sections.
• Thin shell theorem, Gaussian marginals with geometric assumptions on the random variables in place of independence.
   

Part II: Convexity  

• Brunn-Minkowski inequality, concentration for uniformly convex sets.
• Volume-ratio and Kashin's theorem (approximately-spherical sections of almost full dimension).
• Santalo inequality, Legendre transform, Brascamp-Lieb inequality.
• The isotropic constant and the Bourgain-Milman inequality.
• Milman ellipsoid, quotient of subspace, reverse Brunn-Minkowksi.

Upon successful completion of this course students will be able to

Demonstrate intuition and knowledge of geometry in high-dimensions, concentration of measure and interactions with convexity.

Recommended books from the 1980s:

• Milman, Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces, 1986.

• Pisier, The Volume of Convex Bodies and Banach Space Geometry, 1989.

Recommended recent books on the subject:

• Vershynin, High-Dimensional Probability, 2018.

• Aubrun, Szarek, Alice and Bob meet Banach, 2017.

• Artstein-Avidan, Giannopoulos, Milman, Asymptotic Geometric Analysis, 2015.

• Brazitikos, Giannopoulos, Valettas, Vritsiou, Geometry of Isotropic Convex Bodies, 2014.

• Boucheron, Lugosi, Massart, Concentration Inequalities, 2013.

• Ledoux, The Concentration of Measure Phenomenon, 2001.