There is a classical theorem of Iwasawa which concerns certain modules X for the formal power series ring Λ = Zp[[T]] in one variable.  Here p is a prime and Zp is the ring of p-adic integers.  Iwasawa's theorem asserts that X has no nonzero, finite Λ-submodules. We will begin by describing the modules X which occur in Iwasawa's theorem and explaining  how the theorem is connected with the title of my talk. Then we will describe generalizations of this theorem for  certain Λ-modules (the so-called "Selmer groups")  which arise naturally in Iwasawa theory.  The ring Λ can be a formal power series ring over Zp in any number of variables, or even a non-commutative analogue of such a ring. 

One of the classical questions about the evolution of a one
dimensional quantum system is the asymptotic rate of propagation of
the wave packet. It is usually captured through the notion of
transport exponents. Several methods were developed to estimate these
quantities in various models. However many authors only treated the
case of a state initially localized at a single site (in the discrete
setting). We show that some of these results can be extended to a
broad class of initial states with compact or even infinite support,
and explain what are the methods and obstacles to further
generalizations.

We complete the proof of the main theorem by showing that if X^3 is isomorphic to X, then X^{\omega} has a parity-reversing automorphism. By our previous results this implies X^2 is isomorphic to X as well. The proof generalizes to show that for any n > 1, if X^n is isomorphic to X, then X^2 is isomorphic to X. Time permitting we will discuss related results, including the existence of an A and X such that A^2X is isomorphic to X, while AX is not.

We compute the limit of the free energy of the mean field generated by the independent Brownian particles through pairwise interaction. Our main theorem is relevant to the high moment asymptotics for the parabolic Anderson models with Gaussian noise that is white in time, white or colored in space. Our approach makes a novel connection to the celebrated Donsker-Varadhan’s large deviation principle for the i.i.d. random variables in infinite dimensional spaces. As an application of our main theorem, we provide a probabilistic treatment to the Hartree’s theory  on the asymptotics for the ground state energy of bosonic quantum system.

This is a collaborative work with Tuoc V. Phan.