We study the capacity of a uniform G_{\delta} set. Changing the speed at which the lengths of intervals generating it decrease, we observe a sharp phase transition from full to zero capacity. Such a G_{\delta} set is also interesting because it can be considered as a model case for the set of exceptional energies in the parametric version of the Furstenberg theorem. In the talk, we will demonstrate the techniques used and as well as some interesting capacity-related examples.

Based on a weak Wegner estimate and Peierls argument   I will present the complete proof of Anderson localization.

The materials are  from Bourgain–Kenig [Section 7] and Germinet–Klein [Sections 6 and 7].

In the beginning of this talk, I will sketch the proof  a Liouville theorem of Buhovsky–Logunov–Malinnikova–Sodin .

https://arxiv.org/pdf/1809.09041.pdf

Based on   BLMS, I will present a unique continuation result of  the discrete  Laplacian on $\Z^2$.

https://arxiv.org/abs/1712.07902

In this talk, I will talk about the quasi-periodic solutions of Hamiltonian perturbations of higher dimensional NLS and NLW equations.

One of the approaches to constructing quasi-periodic solutions is by KAM in infinitely dimensional phases.  I listed some authors here   Bambusi, Eliasson, Grébert, Kuksin,  Wayne, Poschel, Procesi, .......

Another approach was initiated by Craig and Wayne based on Lyapunov-Schmidt decomposition, which splits the problem into a Q-equation and P-equation. This approach was highly developed by Bourgain.   This approach essentially is the control of inverse of certain linear operators, which arise from consecutive linearizations when applying the Newton iteration scheme. It turns out that the control of inverse of the matrices is closely related to the proof of Anderson localization of quasi-periodic operators.    With the additional help of separate properties of eigenvalues (arithmetic lemma), Bourgain constructed the quasi-periodic of higher dimensional NLS and NLW equations.

I am going to use two lectures to talk about Bourgain's proof.  I will talk about the small divisor problems,  the large deviation theorem, semi-algebraic sets and the arithmetic lemma.