Title: Limiting dynamics of large quantum systems and Egorov-type theorems

 Abstract:
 In certain limiting regimes, the microscopic Hamiltonian dynamics of a 
 quantum system may be replaced by a simpler, "emergent" description. 
 Well-known examples are the semiclassical limit of a quantum system of 
 finitely many degrees of freedom and the mean-field limit of a Bose 
 gas. Traditionally, the study of these limits is confined to a 
 particular class of initial states, such as coherent states. In the 
 spirit of Egorov's classical result, I derive results in terms of 
 Heisenberg-picture observables, thus removing any restrictions on the 
 initial state. In the first part of the talk, I review Egorov's result 
 and discuss the semiclassical and mean-field limits of quantum spin 
 systems. I also discuss the mean-field limits of Bose and Fermi gases. 
 In the second part of the talk, I give an outline of the proofs, and 
 in particular describe a new and simple method to deal with singular 
 interaction potentials (such as the Coulomb potential) in the 
 mean-field limit of a quantum gas.