Abstract:
I will report on recent and ongoing works joint with Charles Bordenave, where we evaluate the spectrum of non-commutative polynomials in various random tensor objects, in particular iid random permutations. Concisely stated, our results state that non-trivial outliers do not exist in large dimension for the models we investigate. In particular, this settles a conjecture of Friedman about random lifts of graphs. Time allowing, we will describe some of the techniques involved in the proof of our results, as they are of independent interest. They include moment methods, the linearization trick, and a non-commutative version of non-backtracking theory.
Biography:
Benoit Collins is a leading probabilist, working primarily in Random Matrix Theory (RMT). He developed systematically the Weingarten Calculus, a technique for computing moments of Haar measures of compact groups, which is widely used today. Professor Collins is interested in applications of RMT, both to fundamental and to applied fields. On the pure side, he made substantial contributions to Operator Algebras and in particular Free Probability Theory, but also to Quantum groups and asymptotic Representation Theory. On the applied side, he pioneered the systematic use of RMT and Free Probability Theory to Quantum Information Theory.
Benoit Collins is a professor at Kyoto University. He earned his PhD from Paris University in 2003 after his studies at Ecole Normale Superieure in Paris. He held permanent positions at CNRS (Lyon, France) and at the University of Ottawa before moving to Kyoto University in 2014.
He is an associate editor for journals covering a broad scope of mathematics (Annales IHP, Journal of Mathematical Physics, Annales Blaise Pascal, Letters in Mathematical Physics). He has mentored more than 30 graduate students and postdoctoral researchers.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai