Unit:
Κατεύθυνση Θεωρητικά ΜαθηματικάLibrary of the School of Science
Supervisors info:
Απόστολος Γιαννόπουλος, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ (επιβλέπων)
Δημήτριος Γατζούρας, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Σάμης Τρέβεζας, Λέκτορας, Τμήμα Μαθηματικών, ΕΚΠΑ
Original Title:
Εντροπία, υπερσυσταλτότητα και οι ανισότητες του Talagrand
Translated title:
Entropy, hypercontractivity and Talagrand's inequalities
Summary:
In this thesis we study four famous inequalities of Talagrand: the convex distance inequality, the L1-L2 variance inequality, the quadratic transportation cost inequality, and the inequality on the supremum of empirical processes. A main feature of the four Talagrand inequalities is that they are dimension-free: the constants do not depend on the size of the samples and the statements extend to infinite-dimensional systems. More precisely in this thesis:
• We present the motivation for each one of the inequalities as well as the original proofs that Talagrand gave and published.
• After introducing the necessary tools, we present simpler proofs of the inequalities, that were given later, emphasizing a common principle behind all of them, which is based on the notion of entropy, the logarithmic Sobolev inequality and hypercontractivity. This unified presentation was presented in a recent survey article of Ledoux.
• Finally, we present a sample of applications of the four inequalities to different areas of Mathematics.
Main subject category:
Science
Keywords:
Entropy, hypercontractivity, logarithmic Sobolev inequality, Talagrand , concentration of measure
Number of references:
103
