Approaching the Kannan Lovász Simonovits and variance conjectures Focusing on two central conjectures from the field 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 topics treated. Employing a style 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 allowing readers to quickly access the core of these conjectures. In addition four recent and important results concerning this theory are presented. The first two are theorems attributed to Eldan Klartag and Ball Nguyen which relate the variance and the KLS conjectures respectively to the hyperplane conjecture. The remaining two present in detail the main ideas needed to 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