The formal weight enumerators were first introduced by M. Ozeki, and it was shown in the author's previous paper that there are various families of similar divisible polynomials. Among them, three families are dealt with in this paper and their properties are investigated: they are analogues of the Mallows--Sloane bound, the extremal property, the Riemann hypothesis analogue, etc. In the course of the investigation, some generalizations of the theory of invariant differential operators developed by I. Duursma and T. Okuda are deduced.

In this paper we compare the combinatorial Ricci curvature on cell complexes and Lin-Lu-Yau's Ricci curvature defined on graphs. On a cell complex, the combinatorial Ricci curvature is introduced by the Bochner-Weitzenb\"{o}ck formula. A cell complex corresponds to a graph such that the vertices are cells and the edges are vectors on the cell complex. We compare these two kinds of Ricci curvatures by the coupling and the Kantorovich duality.

The aim of this article is to generalize the notion of the cut locus and to get the structure theorem for it. For this purpose, we first introduce a class of 1-Lipschitz functions on a Finsler manifold, each member of which is called an {\em almost distance function}. Typical examples of an almost distance function are the distance function from a point and the Busemann functions. The generalized notion of the cut locus in this paper is called the {\em singular locus} of an almost distance function. The singular locus consists of the {\em upper singular locus} and the {\em lower singular locus}. The upper singular locus coincides with the cut locus of a point $p$ for the distance function from the point $p$, and the lower singular locus coincides with the set of all copoints of a ray $\gamma$ when the almost distance function is the Busemann function of the ray $\gamma$. Therefore, it is possible to treat the cut locus of a closed subset and the set of copoints of a ray in a unified way by introducing the singular locus for the almost distance function. In this article, some theorems on the distance function from a closed set and the Busemann function are generalized by making use of the almost distance function.

We present a way of viewing Lucas sequences in the framework of group scheme theory. This enables us to treat the Lucas sequences from a geometric and functorial viewpoint, which was suggested by Laxton $\langle$On groups of linear recurrences, I$\rangle$ and by Aoki-Sakai $\langle$Mod $p$ equivalence classes of linear recurrence sequences of degree 2$\rangle$.

There is the close relation between smooth $SU(p,q)$-actions on ${\mathbf P}^{p+q-1}_{\mathbb C}$ and triples of smooth functions satisfying four conditions. To construct smooth $SU(p,q)$-actions on ${\mathbf P}^{p+q-1}_{\mathbb C}$, we construct triples of smooth functions satisfying four conditions. As a result, we can show that for given positive integer $m$ there exist uncountably infinite equivalence classes of smooth $SU(p,q)$-actions on ${\mathbf P}^{p+q-1}_{\mathbb C}$ with $m$ closed and $m+1$ open orbits, and furthermore we have new smooth $SU(p,q)$-actions on $S^{2p+2q-1}$ with $m$ closed and $m+1$ open orbits.

We introduce a direct method to analyze the blow-up of solutions to systems of ordinary differential inequalities, and apply it to study the blow-up of solutions to a weakly coupled system of semilinear heat equations. In particular, we give upper and lower estimates of the lifespan of the solution in the subcritical case.

Let $k_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of an algebraic number field $k$. We denote by $S$ a finite set of prime numbers which does not contain $p$, and $S(k_\infty)$ the set of primes of $k_\infty$ lying above $S$. In the present paper, we will study the structure of the Galois group $\mathcal{X}_S (k_\infty)$ of the maximal pro-$p$ extension unramified outside $S (k_\infty)$ over $k_\infty$. We mainly consider the question whether $\mathcal{X}_S (k_\infty)$ is a non-abelian free pro-$p$ group or not. In the former part, we treat the case when $k$ is an imaginary quadratic field and $S = \emptyset$ (here $p$ is an odd prime number which does not split in $k$). In the latter part, we treat the case when $k$ is a totally real field and $S \neq \emptyset$.

Let $k$ be a field, $n \geqslant 5$ be an integer, $x_1, \dots, x_n$ be independent variables and $L_n = k(x_1, \dots, x_n)$. The symmetric group $\Sigma_n$ acts on $L_n$ by permuting the variables, and the projective linear group $\text{PGL}_2$ acts by applying (the same) fractional linear transformation to each variable. The fixed field $K_n = L_n^{\text{PGL}_2}$ is called ``the field of cross-ratios''. Let $S \subset \Sigma_n$ be a subgroup. The Noether Problem asks whether the field extension $L_n^S/k$ is rational, and the Noether Problem for cross-ratios asks whether $K_n^S/k$ is rational. In an effort to relate these two problems, H.~Tsunogai posed the following question: Is $L_n^S$ rational over $K_n^S$? He answered this question in several interesting situations, in particular in the case where $S = \Sigma_n$. In this paper we extend his results by recasting the problem in terms of Galois cohomology. Our main theorem asserts that the following conditions on a subgroup $S \subset \Sigma_n$ are equivalent: (a) $L_n^S$ is rational over $K_n^S$, (b) $L_n^S$ is unirational over $K_n^S$, (c) $S$ has an orbit of odd order in $\{ 1, \dots, n \}$.

We show a method to construct a special Lagrangian submanifold $L^\prime$ from a given special Lagrangian submanifold $L$ in a Calabi-Yau manifold with the use of generalized perpendicular symmetries. We use moment maps of the actions of Lie groups, which are not necessarily abelian. By our method, we construct some non-trivial examples in non-flat Calabi-Yau manifolds $\mathrm{T}^\ast S^n$ which equipped with the Stenzel metrics.

We consider a product of fundamental crystals of type A in monomial realization. Then we shall show that the product holds crystal structure and describe how it is decomposed into irreducible crystals, which is, in general, different from the decomposition for tensor product of the fundamental crystals.

In this paper, an algorithm to enumerate non-hyperelliptic superspecial curves of genus $4$ over finite fields of characteristic $p \geq 5$ is constructed. As an application, the algorithm is used to enumerate non-hyperelliptic superspecial curves of genus $4$ over prime fields of characteristic $p \leq 11$. Thanks to the fact that the number of $\mathbb{F}_{p^a}$-isomorphism classes of superspecial curves over $\mathbb{F}_{p^a}$ of a fixed genus depends only on the parity of $a$, this paper contributes to the odd-degree case for genus $4$, whereas our previous work (Kudo-Harashita: Superspecial curves of genus $4$ in small characteristic) contributes to the even-degree case. This paper is the full-version of our conference paper (Kudo-Harashita: Enumerating Superspecial Curves of Genus $4$ over Prime Fields) presented at WCC2017.