Qi-Man Shao - Gaussian correlation conjecture and small ball probabilities

The Gaussian correlation conjecture states that for any two symmetric convex sets in n-dimensional space and for any centered Gaussian measure on that space, the measure of the intersection is greater than or equal to the product of the measures. It is known that the conjecture is true for some special cases, which has been found a very useful tool in the study of the so called small ball probabilities for Gaussian processes. This talk will review recent progress on the conjecture. A new Gaussian correlation inequality as well as its application to the existence of small ball constant for fractional Brownian motion will be discussed.