Let $G=GL(V)$ for an $N$-dimensional vector space $V$ over an algebraically closed field $k$, and $G^{\theta }$ the fixed point subgroup of $G$ under an involution $\theta$ on $G$. In the case where $G^{\theta }=O(V)$, the generalized Springer correspondence for the unipotent variety of the symmetric space $G/G^{\theta }$ was described in [SY], assuming that $ch k\ne 2$. The definition of $\theta$ given there, and of the symmetric space arising from $\theta$, make sense even if $ch k=2$. In this paper, we discuss the Springer correspondence for those symmetric spaces with even characteristic. We show, if $N$ is even, that the Springer correspondence is reduced to that of symplectic Lie algebras in $ch k=2$, which was determined by Xue. While if $N$ is odd, the number of $G^{\theta }$-orbits in the unipotent variety is infinite, and a very similar phenomenon occurs as in the case of exotic symmetric space of higher level, namely of level $r=3$.

For three dimensional cyclic quotient singularities of type $\frac{1}{r}\left(1,a,r-a-1\right)$ (resp. type $\frac{1}{r}\left(1,a,r-a\right)$), the Fujiki-Oka resolution coincides with one of crepant resolutions (resp. an economic resolution). In this paper, we will characterize binary trees which gives the Fujiki-Oka resolution for the above two series of cyclic quotient singularities.

We consider Fano sevenfolds $Y$ obtained by intersecting the Grassmannian $\text{Gr}(3,6)$ with a codimension 2 linear subspace (with respect to the Plücker embedding). We prove that the motive of $Y$ is Kimura finite-dimensional. We also prove the generalized Hodge conjecture for all powers of $Y$.

In this paper we consider the Cauchy problem for the gravity water-wave equations, in a domain with flat bottom and in arbitrary space dimension. We prove that if the data are of size $\epsilon$ in a space of analytic functions which have a holomorphic extension in a strip of size $\sigma$, then the solution exists up to a time of size $C/\epsilon$ in a space of analytic functions having at time $t$ a holomorphic extension in a strip of size $\sigma -C^{\prime }\epsilon t$.

We construct ${\mathbb{Z}}_2$-perfect Morse functions of $G_2/\text{SO}\left(4\right)$ whose set of all critical points is a great antipodal set of $G_2/\text{SO}(4)$. In particular, we give the reason why the $2$-number ${\#}_2(G_2/{\text{SO}}_4))$ matches the Betti number of the ${\mathbb{Z}}_2$-coefficient homology group of $G_2/\text{SO}\left(4\right)$.

Given a homomorphism from a knot group to a fixed group, we introduce an element of a $K_1$-group, which is a generalization of (twisted) Alexander polynomials. We compare the $K_1$-class with other Alexander polynomials. In terms of semi-local rings, we compute the $K_1$-classes of some knots and show their non-triviality. We also introduce metabelian Alexander polynomials.

We study equivariant Iwasawa theory for two-variable abelian extensions of an imaginary quadratic field. One of the main goals of this paper is to describe the Fitting ideals of Iwasawa modules using $p$-adic $L$-functions. We also provide an application to Selmer groups of elliptic curves with complex multiplication.