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Asymptotic Geometric Analysis [MA5024]

Sommersemester 2015

Prof. Dr. Michael M. Wolf

Dozent: Prof. Dr. Michael M. Wolf
Übungsleitung: Michael Kech
Mitwirkende:  
Vorlesung: Thursday, 10:15-11:45, room 03.08.011 Anmeldung
Zentralübung:   Anmeldung
Tutorübungen:   Anmeldung

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Content

Mixing concepts from geometry and analysis we will study high dimensional linear structures with a strong focus on asymptotic (in the dimension of the space) and probabilistic results. The course will cover:

Lecture notes

File Date Content Comments/further reading
Lecture 1 16.04.15 Introduction, normed spaces and convex bodies, duality and polars  
Lecture 2 30.04.15 Dual norm, nuclear norm, Lewis' theorem  
Lecture 3 07.05.15 Extremal volume ellipsoids, John's theorem, Banach-Mazur distance  
Lecture 4 13.05.15 Banach-Mazur metric, Brunn-Minkowski inequalities, isoperimetric inequality Survey Pfeil on Brunn-Minkowski
Lecture 5 20.05.15 Concentration of measure on the sphere, Levy's Lemma for median and expectation value Book Pfeil on measure concentration
Lecture 6 27.05.15 Haar measures on compact metric spaces, Johnson-Lindenstrauss flattening Lemma remarks added (04.06.)
Lecture 7 03.06.15 Enflo's Hamming cube bound, Frechet embedding, Bourgain's minimal distortion embedding  
Lecture 8 11.06.15 Bourgain embedding into l_p, epsilon nets, Dvoretzky criterion for existence of almost Euclidean subspaces  
Lecture 9 18.06.15 Spherical sections of ellipsoids, Dvoretzky-Rogers Lemma, Dvoretzky's thm. part I Notes Pfeil on Dvoretzky's thm.
Lecture 10 25.06.15 Dvoretzky's thm. part II, Dvoretzky dimension and duality comments added (02.07.)
Lecture 11 02.07.15 Dvoretzky's thm. with projections, global version of Dvoretzky's thm.  

Exercises

File Solution Date Content
Excercise 1 Solution 1 29.04.2015 Polars and ellipsoids
Excercise 2 Solution 2 13.05.2015 Banach-Mazur distance, Brunn-Minkovski inequality
Excercise 3 Solution 3 27.05.2015 Concentration of Measure, Johnson-Lindenstrauss
Excercise 4 http://kam.mff.cuni.cz/~matousek/stml-53-matousek-1.pdf Pfeil Exercise2: page 19-22, Exercise3: page 145-149 11.06.2015 Isometric embeddings
Excercise 5 Solution 5 2.07.2015 Dvoretzky theorem

Literature

The following books have overlap with the lecture but also go considerably beyond it. More literature will be provided during the course.