In this paper we investigate what kind of manifolds arise as the total spaces of iterated $S^{1}$-bundles. A real Bott tower studied in [2], [13] and [14] is an example of an iterated $S^{1}$-bundle. We show that the total space of an iterated $S^{1}$-bundle is homeomorphic to an infra-nilmanifold. A real Bott manifold, which is the total space of a real Bott tower, provides an example of a closed flat Riemannian manifold. We also show that real Bott manifolds are the only closed flat Riemannian manifolds obtained from iterated $\mathbb{R}P^{1}$-bundles. Finally we classify the homeomorphism types of the total spaces of iterated $S^{1}$-bundles in dimension 3.

We introduce an algebraic version of the translation algebra of a group. We prove that a quasi-isometry of two finitely generated groups induces Morita equivalence of their algebraic translation algebras.

We give a simple characterization of the highest weight vertices in the crystal graph of the level $l$ Fock spaces. This characterization is based on the notion of totally periodic symbols viewed as affine analogues of reverse lattice words classically used in the decomposition of tensor products of fundamental $\mathfrak{sl}_{n}$-modules. This yields a combinatorial decomposition of the Fock spaces in their irreducible components and the branching law for the restriction of the irreducible highest weight $\mathfrak{sl}_{\infty}$-modules to $\widehat{\mathfrak{sl}}_{e}$.

The localized induction hierarchy in three-dimensional space forms is studied. In particular, we determine all the generating curves of congruence solutions to each evolution equation belonging to the localized induction hierarchy. Here, a congruence solution means a solution moving without changing shape. Also, we give the characterization of some low-order soliton curves.

In this paper, we give the representation theorem for harmonic Bergman functions and harmonic Bloch functions on smooth bounded domains. As an application, we discuss Toeplitz operators.

In this paper, we study the nonlinear stability of wavefronts in a delayed stage-structured population model on a 2-D spatial lattice. For all wavefronts with the speed \begin{equation*} c>\max\{c(\eta_{0}),c_{*}(\theta)\}, \end{equation*} where $\eta_{0}$ is some positive constant, $c_{*}(\theta)>0$ is the critical wave speed and $\theta$ is the direction of propagation, we prove that these wavefronts are asymptotically stable, when the initial perturbation around the wavefronts decays exponentially as $i\cos\theta+j\sin\theta \to -\infty$, but it can be arbitrary large in other locations. This essentially improves the previous work with more strongly restricted wave speed and the small initial perturbation. Our approach adopted in this paper is the weighted energy method and the squeezing technique.

We prove the existence of invariant almost complex structure on any positively omnioriented quasitoric orbifold. We construct blowdowns. We define Chen--Ruan cohomology ring for any omnioriented quasitoric orbifold. We prove that the Euler characteristic of this cohomology is preserved by a crepant blowdown. We prove that the Betti numbers are also preserved if dimension is less or equal to six. In particular, our work reveals a new form of McKay correspondence for orbifold toric varieties that are not Gorenstein. We illustrate with an example.

For a block $b$ of a normal subgroup of a finite group $G$, E.C. Dade has defined a subgroup $G[b]$ of $G$. We give a character-theoretical interpretation of his result on $G[b]$. In the course of proofs we determine a defect group of a block of $G[b]$ covering $b$. We also consider character-theoretical characterizations of isomorphic blocks with respect to normal subgroups.

It is well-known that the zeta-function $\zeta(s,F)$ attached to a normalized Hecke-eigen cusp form $F$ of weight $\kappa$ is universal in the sense that their shifts $\zeta(s + i\tau,F)$ with appropriate $\tau \in \mathbb{R}$ approximate any analytic function uniformly on compact subsets of the strip $\{s = \sigma+it \in \mathbb{C}\colon \kappa/2 < \sigma < (\kappa+1)/2\}$ with any prescribed accuracy. In this paper we consider some classes of operators $\Phi$ such that the function $\Phi(\zeta(s,F))$ is universal in the above sense. In particular, this implies the universality of the functions, for example, $\zeta(s,F)^{N}$ ($N$-th power) and $\zeta^{(N)}(s,F)$ ($N$-th derivative) with $N \in \mathbb{N}$, $e^{\zeta(s,F)}$, $\sin\zeta(s,F)$, and $\cos\zeta(s,F)$.

The class of invariant affine reflection algebras is the most general known extension of the class of affine Kac--Moody Lie algebras, introduced in 2008. We develop a method known as ``affinization'' for the class of invariant affine reflection algebras, and show that starting from an algebra belonging to this class together with a certain finite order automorphism, and applying the so called ``affinization method'', we obtain again an invariant affine reflection algebra. This can be considered as an important step towards the realization of invariant affine reflection algebras.

We introduce a fake HOMFLY polynomial of a knot and show existence of such polynomials of a given knot.