We study nonorientable maximal surfaces in Lorentz-Minkowski 3-space. We prove some existence results for surfaces of this kind with high genus and one end.

We construct the bounded solution of the second problem of Cousin in the analytic polyedra and in the domain of holomorphy of ${\mathbb C}^2$.

We study the existence of transfers on a generalization of Milnor K-theory called Milnor-Witt K-theory. We give a new proof of the fact that Milnor-Witt K-theory has geometric transfers. Moreover, we explain how our proof yields a simplification of Morel's conjecture about Bass-Tate-Kato transfers on contracted homotopy sheaves in the context of motivic homotopy theory.

In this paper various irreducibilities for Markov processes related to topologies and excessive measures are discussed and their relations are presented. We shall mainly prove that, while the fine irreducibility is sufficient for the symmetric measure (and stationary distribution) to be unique if exists, it is almost necessary, namely $X$ is $m$-irreducible if $m$ is the unique symmetric measure for $X$.

We study the bilinear estimates in the Sobolev spaces with the Dirichlet and the Neumann boundary condition. The optimal regularity is revealed to get such estimates in the half space case, which is related to not only smoothness of functions and but also boundary behavior. The crucial point for the proof is how to handle boundary values of functions and their derivatives.

It has been known that there exist exactly three left-invariant Lorentzian metrics up to scaling and automorphisms on the three dimensional Heisenberg group. In this paper, we classify left-invariant Lorentzian metrics on the direct product of three dimensional Heisenberg group and the Euclidean space of dimension $n-3$ with $n \geq 4$, and prove that there exist exactly six such metrics on this Lie group up to scaling and automorphisms. Moreover we show that only one of them is flat, and the other five metrics are Ricci solitons but not Einstein. We also characterize this flat metric as the unique closed orbit, where the equivalence class of each left-invariant metric can be identified with an orbit of a certain group action on some symmetric space.

We prove that Cheng's maximal diameter theorem for hypergraphs with positive coarse Ricci curvature. Coarse Ricci curvature on hypergraphs is defined by using the resolvent of the nonlinear Laplacian. As a byproduct of the main theorem, the first non-zero eigenvalue on a hypergraph coincides with the lower bound of the curvature if the hypergraph has the maximal diameter.

It is well-known that cross caps on surfaces in the Euclidean 3-space can be expressed in Bruce-West's normal form, which is a special local coordinate system centered at the singular point. In this paper, we show a certain kind of uniqueness of such a coordinate system. In particular, the functions associated with this coordinate system produce new invariants on cross cap singular points. Using them, we characterize the possible symmetries on cross caps.