L. Buhovsky, A. Logunov and S. Tanny proved the (strong) Poisson bracket conjecture by Leonid Polterovich in dimension $2$. In this note, instead of open covers consisting of displaceable sets in their work, we consider open covers constituting of topological disks and give a necessary and sufficient condition that Poisson bracket invariants of these covers are positive.

The duality relation of one-variable multiple polylogarithms was proved by Hirose, Iwaki, Sato and Tasaka by means of iterated integrals. In this paper, we give a new proof using the method of connected sums, which was recently invented by Seki and the author. Interestingly, the connected sum involves the hypergeometric function in its connector. Moreover, we introduce two kinds of $q$-analogues of the one-variable multiple polylogarithms and generalize the duality to them.

We study anisotropic weighted Sobolev spaces defined on a half-space. These spaces are important in the study of singular elliptic operators which degenerate at the boundary. We prove trace properties and density of smooth functions. Hardy inequalities are studied in detail.

Let $q$ be a prime with $q \equiv 7 \mod 8$, and let $K=\mathbb{Q}(\sqrt{-q})$. Then $2$ splits in $K$, and we write $\mathfrak{p}$ for either of the primes $K$ above $2$. Let $K_\infty$ be the unique $\mathbb{Z}_2$-extension of $K$ unramified outside $\mathfrak{p}$. For certain quadratic and biquadratic extensions $\mathfrak{F}/K$, we prove a simple exact formula for the $\lambda$-invariant of the Galois group of the maximal abelian 2-extension unramified outside $\mathfrak{p}$ of the field $\mathfrak{F}_\infty = \mathfrak{F} K_\infty$. Equivalently, our result determines the exact $\mathbb{Z}_2$-corank of certain Selmer groups over $\mathfrak{F}_\infty$ of a large family of quadratic twists of the higher dimensional abelian variety with complex multiplication, which is the restriction of scalars to $K$ of the Gross curve with complex multiplication defined over the Hilbert class field of $K$. We also exhibit computations of the associated Selmer groups over $K_n$ in the case when the $\lambda$-invariant is equal to $1$; here $K_n$ denotes the $n$-th layer of $K_\infty/K$.

Welded knotted objects are a combinatorial extension of knot theory, which can be used as a tool for studying ribbon surfaces in $4$-space. A finite type invariant theory for ribbon knotted surfaces was developed by Kanenobu, Habiro and Shima, and this paper proposes a study of these invariants, using welded objects. Specifically, we study welded string links up to $w_k$-equivalence, which is an equivalence relation introduced by Yasuhara and the second author in connection with finite type theory. In low degrees, we show that this relation characterizes the information contained by finite type invariants. We also study the algebraic structure of welded string links up to $w_k$-equivalence. All results have direct corollaries for ribbon knotted surfaces.

We prove that oriented and standard shadowing properties are equivalent for topological flows with finite singularities that are Lyapunov stable or backward Lyapunov stable. Moreover, we prove that the direct product $\phi_1 \times \phi_2$ of two topological flows has the oriented shadowing property if $\phi_1$ with finite singularities has the oriented shadowing property, while $\phi_2$ has the limit set consisting of finite singularities that are Lyapunov stable or backward Lyapunov stable.

We determine all possible configurations for the unit signature rank for real biquadratic fields relative to the signature ranks of their quadratic subfields, providing infinite families for each possibility, and make some additional remarks for higher rank real multiquadratic fields.

The automorphism group of a Riemann surface naturally acts on the vector space of holomorphic quadratic differentials on it, which determines the canonical representation of the automorphism group. In light of the theory of linear quotient families developed by S. Takamura, the universal family over the moduli space around the Riemann surface is determined by this representation together with the automorphism group action on the Riemann surface — it is essential to determine the stabilizer poset of the former. Unfortunately this theory does not tell us how to determine the canonical representation and its stabilizer poset. The aim of this paper is to explicitly determine the canonical representations of Fermat curves of all genera together with the stabilizer poset for genus 3 case. We then describe, with a group-theoretic flavor, the local moduli of the Fermat curve of genus 3.

Ng constructed differential graded algebras for knots in ${\mathbb{R}}^3$, and extended the definitions to other knotted objects. Modifying his definitions, we construct differential graded algebras for handlebody-knots in ${\mathbb{R}}^3$. We calculate 0-dimensional homology for handlebody-knots up to 6 crossings.

In a previous paper, we introduced functions $\zeta_r(s,a,\Delta;q)$, which are $q$-analogues of zeta functions associated with root systems $\Delta$, and their $p$-deformations $\zeta_r(s,a,\beta,\Delta;p,q)$. It is known that zeta functions of root systems satisfy certain functional relations including Witten's volume formula. In this paper, we consider $q$-extensions of these functional relations for $\Delta=\Delta(A_2), \ \Delta(A_3), \ \Delta(B_2), \ \Delta(G_2)$. We also generalize expressions of ``Weyl group symmetric'' linear combination of $\zeta_r(s,a,\beta,\Delta;p,q)$ for $\Delta=\Delta(A_1), \ \Delta(A_2), \ \Delta(A_3)$, which were obtained in the previous paper, to arbitrary root systems $\Delta$ by using an identity due to Macdonald.

First we give a classification of Hopf real hypersurfaces with $\eta$-parallel shape operator in the complex quadric $Q^{m}$, $m \geq 3$. As an application of this result, we give another classification of cyclic parallel structure Jacobi operator on Hopf real hypersurfaces in $Q^{m}$.