How to Raise Harmonic Families?

Nicolas Templier, Princeton University
Fine Hall 314

The talk will present a step towards answering the delicate question in the title! In the colloqium lunch I describe the work of Montgomery on the pair correlation of zeros of the Riemann zeta function. The result is beautifully connected with the eigenvalue statistics of random matrices studied by Gaudin, Mehta and Dyson. This was the beginning of fruitful connections between L-functions and Random Matrix Theory. Katz and Sarnak have emphasized the importance of families and introduced a new perspective in which the existence of a Symmetry Type plays a central role. In joint work with Sug-Woo Shin we establish a general Plancherel equidistribution theorem that is strong enough for such applications. This will be presented along with attached results in harmonic analysis on semisimple groups.