In this article, we generalize the classification of genus one Lefschetz fibrations to genus one simplified broken Lefschetz fibrations, which have fibers of genera one and zero. We classify genus one Lefschetz fibrations over the 2-disk with certain non-trivial global monodromies using chart descriptions, and identify the 4-manifolds admitting genus one simplified broken Lefschetz fibrations up to blow-ups.

It was conjectured, twenty years ago, the following result that would generalize the so-called rank rigidity theorem for homogeneous Euclidean submanifolds: let $M^n$, $n\geq2$, be a full and irreducible homogeneous submanifold of the sphere $S^{N-1}\subset \mathbb{R}^N$ such that the normal holonomy group is not transitive (on the unit sphere of the normal space to the sphere). Then $M^n$ must be an orbit of an irreducible $s$-representation (i.e. the isotropy representation of a semisimple Riemannian symmetric space).

If $n=2$, then the normal holonomy is always transitive, unless $M$ is a homogeneous isoparametric hypersurface of the sphere (and so the conjecture is true in this case). We prove the conjecture when $n=3$. In this case $M^3$ must be either isoparametric or a Veronese submanifold. The proof combines geometric arguments with (delicate) topological arguments that use information from two different fibrations with the same total space (the holonomy tube and the caustic fibrations).

We also prove the conjecture for $n\geq 3$ when the normal holonomy acts irreducibly and the codimension is the maximal possible $n(n+1)/2$. This gives a characterization of Veronese submanifolds in terms of normal holonomy. We also extend this last result by replacing the homogeneity assumption by the assumption of minimality (in the sphere).

Another result of the paper, used for the case $n=3$, is that the number of irreducible factors of the local normal holonomy group, for any Euclidean submanifold $M^n$, is less or equal than $[n/2]$ (which is the rank of the orthogonal group $SO(n)$). This bound is sharp and improves the known bound $n(n-1)/2$.

A construction of general formal solutions for members $(P_{\mathrm{J}})_{m}$ $({\rm J = II,\,IV})$ of the second and the fourth Painlevé hierarchies with a large parameter is discussed. We also investigate a relation between formal solutions of $(P_{\mathrm{J}})_{m}$ $({\rm J = II,\,IV})$.

Seiberg–Witten theory is used to obtain new obstructions to the existence of Einstein metrics on 4-manifolds with conical singularities along an embedded surface. In the present article, the cone angle is required to be of the form $2\pi /p$, $p$ a positive integer, but we conjecture that similar results will also hold in greater generality.

Beirão da Veiga [ 5] proves that for a straight channel in $\mathbb{R}^n$ ($n\ge2$) and for a given time periodic flux there exists a unique time periodic Poiseuille flow. As a by product, existence of the time periodic Poiseuille flow in perturbed channels (Leray's problem) is shown for the Stokes problem ($n\ge2$) and for the Navier–Stokes problem ($n\le4$). Concerning the Navier–Stokes case, in [ 5] a quantitative condition required to show the existence of the solutions depends not just on the flux of the time periodic Poiseuille flow but also on the domain itself.

Kobayashi [ 16], [ 18] proves that for a perturbed channel in $\mathbb{R}^n$ $(n=2,3)$ there exists a time periodic solution of the Navier–Stokes equations with the Poiseuille flow applying the theory of the steady problem to the time periodic problem.

In this paper, applying Fujita [ 8] and Kobayashi [ 18], we succeed in proving the existence of a time periodic solution for a symmetric perturbed channel in $\mathbb{R}^2$.

In this paper, we give the Fourier coefficients of Siegel Eisenstein series of degree 2, level $p$, in order to calculate the dimensions of the space of Eisenstein series for low weights. The main methods of the calculation is to compute the Siegel series of level $p$ directly, following the similar way to that of Kaufhold.

The sum formula is one of the most well-known relations among multiple zeta values. This paper proves a conjecture of Kaneko predicting that an analogous formula holds for finite multiple zeta values.

We prove that for a given continuous function $H(s)$, $(-\infty$ < $s$ < $\infty)$, there exists a globally defined generating curve of a rotational hypersurface in a Euclidean space such that the mean curvature is $H(s)$. We also prove a similar theorem for generalized rotational hypersurfaces of $O(l+1)\times O(m+1)$-type. The key lemmas in this paper show the existence of solutions for singular initial value problems of ordinary differential equations satisfied using generating curves of those hypersurfaces.

For an arbitrary positive integer $T$ we introduce the notion of a $(V,T)$-module over a vertex algebra $V$, which is a generalization of a twisted $V$-module. Under some conditions on $V$, we construct an associative algebra $A^{T}_{m}(V)$ for $m\in(1/T)\mathbb N$ and an $A^{T}_{m}(V)$-$A^{T}_{n}(V)$-bimodule $A^{T}_{n,m}(V)$ for $n,m\in(1/T)\mathbb N$ and we establish a one-to-one correspondence between the set of isomorphism classes of simple left $A^{T}_{0}(V)$-modules and that of simple $(1/T)\mathbb N$-graded $(V,T)$-modules.

The main result of this note gives an efficient presentation of the $S^1$-equivariant cohomology ring of Peterson varieties (in type $A$) as a quotient of a polynomial ring by an ideal $J$, in the spirit of the well-known Borel presentation of the cohomology of the flag variety. Our result simplifies previous presentations given by Harada-Tymoczko and Bayegan-Harada. In particular, our result gives an affirmative answer to a conjecture of Bayegan and Harada that the defining ideal $J$ is generated by quadratics.

We correct the proof of Theorem 1.1 in our previous paper specially in the non-irreducible case.

In a recent paper, Y. Hu has given a sufficient condition for the fundamental group of the $r$-th cyclic branched covering of $S^3$ along a prime knot to be left-orderable in terms of representations of the knot group. Applying her criterion to a large class of two-bridge knots, we determine a range of integers $r > 1$ for which the $r$-th cyclic branched covering of $S^3$ along the knot is left-orderable.

This paper introduces the notion of $k$-isoparametric hypersurface in an $(n+1)$-dimensional Riemannian manifold for $k=0,1,\dots,n$. Many fundamental and interesting results (towards the classification of homogeneous hypersurfaces among other things) are given in complex projective spaces, complex hyperbolic spaces, and even in locally rank one symmetric spaces.

We confirm a conjecture of Kyoji Saito on the growth functions of graphs, which was originally posed for hyperbolic groups.

We show that every connected graph can be realized as the cut locus of some point on some Riemannian surface $S$ which, in some cases, has constant curvature. We study the stability of such realizations, and their generic behavior.

An operator $C^*$-algebra $\mathfrak E$ associated with a dynamical system on a metric graph is introduced. The system is governed by the wave equation and controlled from boundary vertices. Algebra $\mathfrak E$ is generated by eikonals, which are self-adjoint operators related with reachable sets of the system. Its structure is studied. Algebra $\mathfrak E$ is determined by the boundary inverse data. This shows promise of its possible applications to inverse problems.

It is well-known that an orthonormal scaling function generates an orthonormal wavelet function in the theory of multiresolution analysis. We consider two families of unitary operators. One is a family of extensions of the Hilbert transform called fractional Hilbert transforms. The other is a new family of operators which are a kind of modified translation operators. A fractional Hilbert transform of a given orthonormal wavelet (resp. scaling) function is also an orthonormal wavelet (resp. scaling) function, although a fractional Hilbert transform of a scaling function has bad localization in many cases. We show that a modified translation of a scaling function is also a scaling function, and it generates a fractional Hilbert transform of the corresponding wavelet function. We also show a good localization property of the modified translation operators. The modified translation operators act on the Meyer scaling functions as the ordinary translation operators. We give a class of scaling functions, on which the modified translation operators act as the ordinary translation operators.