Seminar/ Colloquium

Speaker: Dr. Manjunath Krishnapur, Department of Mathematics, IISc Bangalore

Abstract:  Log-concavity of a sequence (a_n^2\ge a_{n-1}a_{n+1}) or function (f(x)^2\ge f(x-h)f(x+h)) or measure (\mu(C)^2\ge \mu(A)\mu(B) if C=(A+B)/2)  gives many strong conclusions, such as its decay or `concentration’ properties. For example, the log-concavity of Lebesgue measure (known as Brunn-Minskowski inequality) implies the isoperimetric inequality.

In this lecture we show that many interesting distributions that arise in probability theory (particularly random matrix theory) are log-concave. We also give a partial solution to a problem of Chen that states that the distribution of the longest increasing sequence of a random permutation is log-concave. This is joint work with Jnaneshwar Baslingker and Mokshay Madiman. The talk is intended to be accessible to students with some knowledge of analysis and very basic notions of probability.