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Focusing on two central conjectures of Asymptotic Geometric Analysis, the Kannan-Lovász-Simonovits spectral gap conjecture and the variance conjecture, these Lecture Notes present the theory in an accessible way, so that interested readers, even those who are not experts in the field, will be able to appreciate the treated topics. Offering a presentation suitable for professionals with little background in analysis, geometry or probability, the work goes directly to the connection between isoperimetric-type inequalities and functional inequalities, giving the interested reader rapid access to the core of these conjectures.

In addition, four recent and important results in this theory are presented in a compelling way. The first two are theorems due to Eldan-Klartag and Ball-Nguyen, relating the variance and the KLS conjectures, respectively, to the hyperplane conjecture. Next, the main ideas needed prove the best known estimate for the thin-shell width given by Guédon-Milman and an approach to Eldan´s work on the connection between the thin-shell width and the KLS conjecture are detailed.

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The Conjectures.- Main Examples.- Relating the Conjectures.- Appendix.- Index.

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"This book serves as an excellent and well-written introduction to a fascinating and active research subject. It is a must have for specialists as well as students interested in diving into this subject, but it is also suitable for mathematicians with a different focus who are interested in a taste of this theory. It can also easily be used as a basis for an advanced course." (Ronen Eldan, Mathematical Reviews, October, 2015)

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