We introduce a concept of viscosity solutions of Hamilton-Jacobi equations in metric spaces and in some cases relate it to viscosity solutions in the sense of differentials in the Wasserstein space. Our study is motivated physical systems which consist of infinitely many particles in motion (This is a joint work with Andzrej Swiech).

We first briefly review Dwork's trace formula and Wan's decomposition theorems. As an application, we consider a family of Laurent polynomials which is a generalization of the Laurent polynomials appeared in Iwaniec's work, and determine $p$-adic valuations for all the roots of the $L$-functions associated to an Zariski open dense subset of the space of Laurent polynomials. For lower dimension cases, we represent the Zariski open subset explicitly by computing an explicit Hasse polynomial.

In the first talk we discussed some models that can be attacked via the trace map as well as some model-independent result. In this talk we shall apply our model-independent results to some specific models (Jacobi operators, CMV matrices, quantum and classical Ising models) and derive answers to questions that until quite recently were open. We will also present a connection between CMV matrices and Ising models. We shall state also some open problems and propose some routes for further development.

We give a Chabauty-like method for finding p-adic approximations to
integral points on hyperelliptic curves when the Mordell-Weil rank of
the Jacobian equals the genus. The method uses an interpretation of
the component at p of the p-adic height pairing in terms of iterated
Coleman integrals.  This is joint work with Amnon Besser and Steffen
Mueller.