<!-- *** Custom HTML *** --> For an odd prime number $p$ and a number field $k$ which is an elementary abelian $p$-extension of the rationals, we prove the equivalence between the vanishing of all Iwasawa invariants of the cyclotomic $\mathbb{Z}_p$-extension of $k$ and an arithmetical condition described by the linking numbers of primes from a viewpoint of analogies between pro-$p$ Galois groups and link groups. A criterion of Greenberg's conjecture for $k$ of degree $p$ is also described in terms of linking matrices.

<!-- *** Custom HTML *** --> The main conjectures in Iwasawa theory predict the relationship between the Iwasawa modules and the $p$-adic $L$-functions. Using a certain proved formulation of the main conjecture, Greither and Kurihara described explicitly the (initial) Fitting ideals of the Iwasawa modules for the cyclotomic $\mathbb{Z}_p$-extensions of finite abelian extensions of totally real fields. In this paper, we generalize the algebraic theory behind their work by developing the theory of “shifts of Fitting invariants.” As applications to Iwasawa theory, we obtain a noncommutative version and a two-variable version of the work of Greither and Kurihara.

<!-- *** Custom HTML *** --> In this paper we give an overview of some aspects of Iwasawa theory for modular forms. We start with the classical formulation in terms of $p$-adic $L$-functions in the ordinary case and the $\pm$-formulation for supersingular elliptic curves. Then we discuss some recent progresses in the proof of the corresponding Iwasawa main conjectures formulated by Kato (Conjecture 4.1), which relates the index of his zeta element to the characteristic ideal of the <i>strict</i> Selmer groups.

<!-- *** Custom HTML *** --> We give an explicit description of the image of a quantum LS path, regarded as a rational path, under the action of root operators, and show that the set of quantum LS paths is stable under the action of the root operators. As a by-product, we obtain a new proof of the fact that a projected level-zero LS path is just a quantum LS path.

We consider a blow-up curve for the one dimensional wave equation. Merle–Zaag [5] showed that there is a possibility that the blow-up curve for $\partial_t^2 u - \partial_x^2 u = |u|^{p-1}u$ is not differentiable if the sign of the solution changes. To show the result, they use the variational structure of the equation. In this paper, we study the blow-up curve $\partial_t^2 u - \partial_x^2 u = |\partial_t u|^{p-1}\partial_tu $ which does not have the variational structure. We state that the blow-up curve is not differentiable if the initial data is an odd function which satisfy some conditions. Next, we show the key of the proof. This paper is an announcement of our forthcoming paper.

<!-- *** Custom HTML *** --> Simons cone is a minimal cone in $\mathbb{R}^8$, which provides a counterexample to the Bernstein problem. Corresponding objects for the Allen-Cahn equation are saddle solutions. They are conjectured to be minimal in dimension 8. We discuss some progress on this problem, in particular, the existence of minimizing solutions to Allen-Cahn equation in $\mathbb{R}^8$.

In this paper, we briefly summarize recent results on a generalization of the Lagrangian mean curvature flow in Kahler-Einstein manifolds. In particular, we focus on the geometry of generalized LMCF in a Kahler manifold with positive weighted Ricci-form and exhibit the extended results according to [6].

This paper gives a general framework for the computation of several classes of partial differential equations (PDEs) on hypersurfaces. The approach is to work in a sufficiently thin narrow band, surrounding the surface, in the embedding Euclidean space and appropriately extend the differential operators so that the solutions on the narrow band and surface are equivalent. The solutions in the narrow band are equivalent to the solutions on the surface as in the former are constant along surface normal extensions of the latter. Consequently, it is possible to use existing (high order) numerical methods developed on grids in Euclidean space to solve PDEs on surfaces, with narrow bands whose widths are a small constant multiple of uniform Cartesian grid spacing. We apply the formulation to PDEs that originate from variational principles defined on surfaces as well as Hamilton-Jacobi-Bellman equations on surfaces that are derived from optimal control problems. This framework also provides a systematic way for solving PDE's on the unstructured point clouds that are sampled from the surface.