Borel's stability and vanishing theorem gives the stable cohomology of $\operatorname{GL}(n,\mathbb{Z})$ with coefficients in algebraic $\operatorname{GL}(n,\mathbb{Z})$-representations. We compute the improved stable range that Borel suggested. In order to further improve Borel's stable range, we adapt the method of Kupers-Miller-Patzt to any algebraic $\operatorname{GL}(n,\mathbb{Z})$-representations.

We determine the first homology group with coefficients in $H_1(N;\mathbb{Z})$ for various mapping class groups of a non-orientable surface $N$ with punctures and/or boundary.

We determine the representation type for block algebras of the quiver Hecke algebras $R^{\Lambda_{k}}(\beta)$ of type $C^{(1)}_\ell$ for all $k$, generalising results of Ariki-Park for $\Lambda = \Lambda_0$.

We investigate some geometric properties of orbits of the isotropy group action on quaternionic irreducible symmetric spaces of compact type. We show that such orbits, except for a one-point set, satisfy one of some four properties and classify which orbits satisfy which properties in each such symmetric space. In a symmetric space, a connected component of the fixed point set of a geodesic symmetry, except for a one-point set, is called a polar. A polar is a totally geodesic submanifold and an orbit of the isotropy group action. By the classification, we show that an orbit which is a quaternionic submanifold or the image of a totally complex immersion is a polar, and a polar becomes a quaternionic submanifold or the image of a totally complex immersion.

We prove that the transcendental motive of any quasi-elliptic surface is trivial. To prove this, we focus on the uniruledness of quasi-elliptic surfaces.

Twisted knot theory introduced by M. Bourgoin is a generalization of knot theory. It leads us to the notion of twisted virtual braids. In this paper we show theorems for twisted links corresponding to the Alexander theorem and the Markov theorem in knot theory. We also provide a group presentation and a reduced group presentation of the twisted virtual braid group.

Mangum and Shanahan construct a complex curve of irreducible $\mathrm{SL}_3(\mathbb{C})$-representations of the fundamental group of a once-punctured torus bundle over the circle. In this paper, we provide an explicit formula for the twisted Alexander polynomial of a once-punctured torus bundle with tunnel number one associated with the Mangum-Shanahan representations. As a corollary, we exhibit the interesting phenomenon that the Reidemeister torsion of the complement of the figure-eight knot is constant on the curve.

A $4$-move is a local move of links replacing two parallel arcs with $4$ half twists. The notion of $4$-moves can be extended to handlebody-links naturally. In this paper, we detect $4$-move inequivalent handlebody-links by using Alexander type invariants obtained from an $f$-twisted Alexander matrix, which is defined by way of a derivative for multiple conjugation quandles. We give a link-homotopically trivial handlebody-link which cannot be reduced to a trivial handlebody-link by $4$-moves.

In the article we establish the existence of solutions of a system of integro-differential equations in the case of the double scale anomalous diffusion. Each equation of the system contains the sum of the two negative Laplace operators raised to two distinct fractional powers in ${\mathbb R}^{d}, d=4,5$. The proof of the existence of solutions relies on a fixed point technique. We use the solvability conditions for the non-Fredholm elliptic operators in unbounded domains.