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.
