The Brunn-Minkowski inequality is a fundamental inequality in Geometry, and has found many applications in Analysis and Probability. This inequality states that for every compact set A,B in R^n, one has Vol(A+B)^{1/n} >= Vol(A)^{1/n} + Vol(B)^{1/n}, where Vol denotes Lebesgue measure, and A+B = {a+b : a in A, b in B} is the Minkowski sum of A and B. The class of log-concave measures contains fundamental probability measures, such as the Gaussian measure, the exponential measure, and the uniform measure. In this talk, we show that the Brunn-Minkowski inequality is valid for log-concave measures, under symmetry assumptions. We then discuss some applications to new isoperimetric-type inequalities. Speaker(s): Arnaud Marsiglietti (University of MInnesota)