Abstract: The centralizer Z(f) of a diffeomorphism f: M--> M of a closed manifold M is the group of all diffeomorphisms commuting with f; it is the collection of dynamical symmetries of f. The centralizer of f always contains the group generated by f as a normal subgroup, and conjecturally the two typically coincide (that is, ``the generic diffeomorphism has only trivial symmetries'').  In this talk, I will describe some results and conjectures in a project with Danijela Damjanovic, Chengyang Wu and Disheng Xu that addresses the question: what happens when Z(f) is bigger than ?

Bio-Sketch: 

Amie Wilkinson is a Professor of Mathematics at the University of Chicago and a world-leading expert in ergodic theory and smooth dynamical systems. Her research lies in the area of smooth dynamical systems and is concerned with the interplay between dynamics and other structures in pure mathematics. She has earned numerous awards, including the Levi L. Conant Prize and the Ruth Lyttle Satter Prize, and is a member of the American Academy of Arts and Sciences, a Foreign Member of the Academia Europaea, and a Fellow of the American Mathematical Society. Professor Wilkinson is widely recognized for solving deep problems in dynamics, including proving the ergodicity of the Weil-Petersson geodesic flow, establishing that the $C^1$ generic diffeomorphism has a trivial centralizer (a partial solution of Smale's 12th problem), making major breakthroughs on the Pugh-Shub stable ergodicity conjecture, and systematically investigated pathological foliations. Her work has been published in the most prestigious mathematical journals, including the Annals of Mathematics, Inventiones Mathematicae, and Publications Mathématiques de l'IHES. She has recently authored the textbook Geometry, Dynamics, and Rigidity (Princeton University Press), a generously illustrated volume that integrates two major areas of mathematics for graduate students and researchers. Reflecting the fundamental impact of her research, she was selected as an invited speaker at the 2010 International Congress of Mathematicians.