Calculus of Variations in Probability and Geometry

Recently, the techniques from calculus of variations have been extensively used to tackle isoperimetric-type inequalities in Euclidean space. In particular, progress was made on a number of newly emerged questions in geometric probability theory. Understanding these questions will shed light on how symmetry and structure influence various families of isoperimetric-type inequalities. This circle of ideas has been used in Riemannian geometry for decades in the fields of geometry and probability such as hypercontractive inequalities and their interactions with curvature. Recently, these ideas have found new applications. Conversely, questions motivated purely by differential geometry, such as mean curvature flow, are ameliorated by studying isoperimetric-type problems with respect to the Gaussian measure. This connection was evidenced by Huisken's monotonicity formula and studies of singularities in mean curvature flow. The above isoperimetric-type questions have tight connections in