Georgia Institute of Technology College of Sciences

This formula of Haagerup gives an expression of the log|x-y| in terms of Chebyshev polynomials of the first kind. This is very useful for problems involving the logarithmic potentials which plays a prominent role in random matrices, free probability, orthogonal polynomials and other areas. We will show how one can go from this to several things, for example the counting problems of planar diagrams and functional inequalities in free probability in particular an intriguing Poincare inequality and some related other inequalities. If time allows I will also talk about a conjecture related to the Hilbert transform, semicircular and arcsine distribution. Parts of this was with Stavros Garoufalidis and some other parts with Michel Ledoux.