In this paper, we develop an approach to the problem of closing lemma based on KAM normal form. The new approach differs from existing $C^1$ perturbation approach and spectral approach, and can handle the high regularity, high dimensional cases and even Riemannian metric perturbations. Moreover, the proof is constructive and effective. We apply the method to the original nearly integrable setting of Poincaré and confirm several old and new conjectures with weak formulations. First, for Poincaré's original setting of nearly integrable systems, we prove that for typical perturbations, periodic orbits are asymptotically dense as the size of perturbation tends to zero. Second, we prove that typical smooth perturbation of the geodesic flow on the flat torus has asymptotically dense periodic orbits, which partially solves an open problem since Pugh-Robinson's $C^1$-closing lemma. Third, we prove that for typical Hamiltonian or contact perturbation of the geodesic flows of the ellipsoid has asymptotically dense orbit on the energy level, which enhances the recent researches on strong closing lemma, and also confirms partially a conjecture of Fish-Hofer in this setting and a problem of Arnold. We also discuss the relation of our models to the recent researches on many-body localization in physics.

It is well-known that there are global small data smooth solutions for the 3-D semilinear Klein-Gordon equations $\square u+u=F(u, \partial u)$ with cubic nonlinearities. However, for the short pulse initial data $\left(u, \partial_t u\right)(0, x)=\left(\delta^{\nu+1} u_0\left(\frac{x}{\delta}\right), \delta^\nu u_1\left(\frac{x}{\delta}\right)\right)$ with $\nu \in \mathbb{R}$ and $\left(u_0, u_1\right) \in C_0^{\infty}(\mathbb{R})$, which are a class of large initial data, we establish that when $\nu \leq-\frac{1}{2}$, the solution $u$ can blow up in finite time for some suitable choices of ($u_0, u_1$) and cubic nonlinearity $F(u, \partial u)$; when $\nu>-\frac{1}{2}$, the smooth solution $u$ exists globally. Therefore, $\nu=-\frac{1}{2}$ is just the critical power corresponding to the global existence or blowup of smooth short pulse solutions for the cubic semilinear Klein-Gordon equations.

This paper is a review of $K$-theory methods in solid state physics. Our goal is to demonstrate that $K$-theory is a natural language for the mathematical description of solid bodies. Our main tool is the $K$-theory of $C^*$-algebras. We follow Kitaev's idea that the symmetry algebras of solid bodies belong to the class of Clifford algebras which reduces the quantization problem for solid states to the representation theory of Clifford algebras.