This paper is an announcement of the minimal model theory for log surfaces in all characteristics and contains some related results including a simplified proof of the Artin–Keel contraction theorem in the surface case.

We obtain a determinant expression of the zeta function of a generalized permutation over a finite set. As a corollary we prove the functional equation for the zeta function. In view of absolute mathematics, this is an extension from $GL(n,\mathbf{F}_{1})$ to $GL(n,\mathbf{F}_{1^{m}})$, where $\mathbf{F}_{1}$ and $\mathbf{F}_{1^{m}}$ denote the imaginary objects “the field of one element” and “its extension of degree $m$”, respectively. As application we obtain a certain product formula for the zeta function, which is analogous to the factorization of the Dedekind zeta function into a product of Dirichlet $L$-functions for an abelian extention.

We introduce the notion of a quasi DG category, and give a procedure to construct a triangulated category associated to it. Then we apply it to the construction of the triangulated category of mixed motivic sheaves over a base variety.

We introduce several definitions of energy density of Brody curves and show that they give the same value in an appropriate situation.

Small time existence of a strong solution to the free surface problem of primitive equations for the ocean with the variable turbulent viscosity terms is shown in this paper. The turbulent viscosity coefficients, which include the Richardson number depending on the unknown variables, are formulated explicitly. We consider the problem in the 3-dimensional stripe-like region, and construct the strong local-in-time solution in Sobolev-Slobodetskiĭ spaces. The details of the proofs will be provided in another full paper.