On the basis of fractional calculus, we introduce an integral of weakly controlled paths, which is a generalization of integrals in the context of rough path analysis. As an application, we provide an alternative proof of Lyons' extension theorem for geometric Hölder rough paths together with an explicit expression of the extension map.

Recently, David Asperó and Miguel Angel Mota discovered a new method of iterated forcing using models as side conditions. The side condition method with models was introduced by Stevo Todorčević in the 1980s. The Asperó–Mota iteration enables us to force some $\Pi_2$-statements over $H(\aleph_2)$ with the continuum greater than $\aleph_2$. In this article, by using the Asperó–Mota iteration, we prove that it is consistent that $\mho$ fails, there are no weak club guessing ladder systems, $\mathfrak{p}= {\mathrm{add}}(\mathcal{N}) = 2^{\aleph_0}>\aleph_2$ and ${MA}_{\aleph_1}$ fails.

This paper has three main goals. First, we set up a general framework to address the problem of constructing module bases for the equivariant cohomology of certain subspaces of GKM spaces. To this end we introduce the notion of a GKM-compatible subspace of an ambient GKM space. We also discuss poset-upper-triangularity, a key combinatorial notion in both GKM theory and more generally in localization theory in equivariant cohomology. With a view toward other applications, we present parts of our setup in a general algebraic and combinatorial framework. Second, motivated by our central problem of building module bases, we introduce a combinatorial game which we dub poset pinball and illustrate with several examples. Finally, as first applications, we apply the perspective of GKM-compatible subspaces and poset pinball to construct explicit and computationally convenient module bases for the $S^1$-equivariant cohomology of all Peterson varieties of classical Lie type, and subregular Springer varieties of Lie type $A$. In addition, in the Springer case we use our module basis to lift the classical Springer representation on the ordinary cohomology of subregular Springer varieties to $S^1$-equivariant cohomology in Lie type $A$.

This paper is devoted to studying transformations on metric spaces. It is done in an effort to produce qualitative version of quasi-isometries which takes into account the asymptotic behavior of the Gromov product in hyperbolic spaces. We characterize a quotient semigroup of such transformations on Teichmüller space by use of simplicial automorphisms of the complex of curves, and we will see that such transformation is recognized as a “coarsification” of isometries on Teichmüller space which is rigid at infinity. We also show a hyperbolic characteristic that any finite dimensional Teichmüller space does not admit (quasi)-invertible rough-homothety.

We prove that for $n \gt 2$ there exists a quandle of cyclic type of size $n$ if and only if $n$ is a power of a prime number. This establishes a conjecture of S. Kamada, H. Tamaru and K. Wada. As a corollary, every finite quandle of cyclic type is an Alexander quandle. We also prove that finite doubly transitive quandles are of cyclic type. This establishes a conjecture of H. Tamaru.

We establish the mapping properties of the fractional integral operators with homogeneous kernels on Morrey spaces with variable exponents.

We consider several classes of knotted objects, namely usual, virtual and welded pure braids and string links, and two equivalence relations on those objects, induced by either self-crossing changes or self-virtualizations. We provide a number of results which point out the differences between these various notions. The proofs are mainly based on the techniques of Gauss diagram formulae.

We prove a non-negativity result for shifted products of two Fourier coefficients of a Siegel Hecke eigenform of degree two not in the Maass space.

Around 1990 Soibelman constructed certain irreducible modules over the quantized coordinate algebra. A. Kuniba, M. Okado, Y. Yamada [8] recently found that the relation among natural bases of Soibelman's irreducible module can be described using the relation among the PBW-type bases of the positive part of the quantized enveloping algebra, and proved this fact using case-by-case analysis in rank two cases. In this paper we will give a realization of Soibelman's module as an induced module, and give a unified proof of the above result of [8]. We also verify Conjecture 1 of [8] about certain operators on Soibelman's module.

In the study of the holomorphic equivalence problem for tube domains, it is fundamental to investigate tube domains with polynomial infinitesimal automorphisms. To apply Lie group theory to the holomorphic equivalence problem for such tube domains $T_\Omega$, investigating certain solvable subalgebras of ${\frak {g}}(T_{\Omega})$ plays an important role, where ${\frak {g}}(T_{\Omega})$ is the Lie algebra of all complete polynomial vector fields on $T_\Omega$. Related to this theme, we discuss in this paper the structure and equivalence of a class of tube domains with solvable groups of automorphisms. Besides, we give a concrete example of a tube domain whose automorphism group is solvable and contains nonaffine automorphisms.

In the present paper hypergroup structures are investigated on distinguished dual objects related to a given hypergroup $K$, especially to a semi-direct product hypergroup $K = H \rtimes_\alpha G$ defined by an action $\alpha$ of a locally compact group $G$ on a commutative hypergroup $H$. Typical dual objects are the sets of equivalence classes of irreducible representations of $K$, of infinite-dimensional irreducible representations of type I hypergroups $K$, and of quasi-equivalence classes of type $\text{II}_1$ factor representations of non-type I hypergroups $K$. The method of proof relies on the notion of a character of a representation of $K = H \rtimes_\alpha G$.

We introduce a class of minimal submanifolds $M^n$, $n\geq 3$, in spheres $\mathbb{S}^{n+2}$ that are ruled by totally geodesic spheres of dimension $n-2$. If simply-connected, such a submanifold admits a one-parameter associated family of equally ruled minimal isometric deformations that are genuine. As for compact examples, there are plenty of them but only for dimensions $n=3$ and $n=4$. In the first case, we have that $M^3$ must be a $\mathbb{S}^1$-bundle over a minimal torus $T^2$ in $\mathbb{S}^5$ and in the second case $M^4$ has to be a $\mathbb{S}^2$-bundle over a minimal sphere $\mathbb{S}^2$ in $\mathbb{S}^6$. In addition, we provide new examples in relation to the well-known Chern-do Carmo–Kobayashi problem since taking the torus $T^2$ to be flat yields minimal submanifolds $M^3$ in $\mathbb{S}^5$ with constant scalar curvature.

To each of regular polyhedra and a soccer ball, we associate degenerating families (degenerations) of Riemann surfaces. More specifically: To each orientation-preserving automorphism of a regular polyhedron (and also of a soccer ball), we associate a degenerating family of Riemann surfaces whose topological monodromy is the automorphism. The complete classification of such degenerating families is given. Besides, we determine the Euler numbers of their total spaces. Furthermore, we affirmatively solve the compactification problem raised by Mutsuo Oka — we explicitly construct compact fibrations of Riemann surfaces that compactify the above degenerating families. Their singular fibers and Euler numbers are also determined.

We consider a holomorphic foliation $\mathcal{F}$ of codimension $k\geq 1$ on a homogeneous compact Kähler manifold $X$ of dimension $n \gt k$. Assuming that the singular set ${\rm Sing}(\mathcal{F})$ of $\mathcal{F}$ is contained in an absolutely $k$-convex domain $U\subset X$, we prove that the determinant of normal bundle $\det(N_{\mathcal{F}})$ of $\mathcal{F}$ cannot be an ample line bundle, provided $[n/k]\geq 2k+3$. Here $[n/k]$ denotes the largest integer $\leq n/k.$

We investigate the partial order $(\mathrm{FIN})^\omega$ of infinite block sequences, ordered by almost condensation, from the forcing-theoretic point of view. This order bears the same relationship to Hindman's Theorem as $\mathcal{P}(\omega) /\mathrm{fin}$ does to Ramsey's Theorem. While $(\mathcal{P}(\omega) / \mathrm{fin})^2$ completely embeds into $(\mathrm{FIN})^\omega$, we show this is consistently false for higher powers of $\mathcal{P} (\omega) / \mathrm{fin}$, by proving that the distributivity number $\mathfrak{h}_3$ of $(\mathcal{P} (\omega) /\mathrm{fin})^3$ may be strictly smaller than the distributivity number $\mathfrak{h}_{\mathrm{FIN}}$ of $(\mathrm{FIN})^\omega$. We also investigate infinite maximal antichains in $(\mathrm{FIN})^\omega$ and show that the least cardinality $\mathfrak{a}_{\mathrm{FIN}}$ of such a maximal antichain is at least the smallest size of a nonmeager set of reals. As a consequence, we obtain that $\mathfrak{a}_{\mathrm{FIN}}$ is consistently larger than $\mathfrak{a}$, the least cardinality of an infinite maximal antichain in $\mathcal{P} (\omega) / \mathrm{fin}$.

This paper is devoted to a functional analytic approach to the subelliptic oblique derivative problem for second-order, elliptic differential operators with a complex parameter $\lambda$. We prove an existence and uniqueness theorem of the homogeneous oblique derivative problem in the framework of $L^{p}$ Sobolev spaces when $\vert\lambda\vert$ tends to $\infty$. As an application of the main theorem, we prove generation theorems of analytic semigroups for this subelliptic oblique derivative problem in the $L^{p}$ topology and in the topology of uniform convergence. Moreover, we solve the long-standing open problem of the asymptotic eigenvalue distribution for the subelliptic oblique derivative problem. In this paper we make use of Agmon's technique of treating a spectral parameter $\lambda$ as a second-order elliptic differential operator of an extra variable on the unit circle and relating the old problem to a new one with the additional variable.