A polynomial invariant for multi-knotoids in $S^2$ is given by an elementary and combinatorial method. It is shown that the invariant is an extension of ``a'' HOMFLY polynomial for multi-knotoids and that there exist infinitely many non-trivial knotoids with trivial HOMFLY polynomial. Furthermore, formulas between the polynomials for a given multi-knotoid, its mirror and reverse images are given. By using the formulas, it is revealed that each knotoid with less than four crossings is invertible.

The contraction of the image of the Johnson homomorphism is called the Chillingworth class. In this paper, we derive a combinatorial description of the Chillingworth class for Putman's subsurface Torelli groups. We also prove the naturality and uniqueness properties of the map whose image is the dual of the Chillingworth classes of the subsurface Torelli groups. Moreover, we relate the Chillingworth class of the subsurface Torelli group to the partitioned Johnson homomorphism.

In this paper we define the Reidemeister torsion as a rational function on the geometric components of the character variety of a one-cusped hyperbolic manifold $M$. We study its poles and zeros, and we deduce sufficient conditions on the manifold $M$ for this function being non-constant.

Let $D$ be a closed unit disk in dimension two and $G_{\rm rel}$ the group of symplectomorphisms on $D$ preserving the origin and the boundary $\partial D$ pointwise. We consider the flux homomorphism on $G_{\rm rel}$ and construct a central $\mathbb{R}$-extension called the flux extension. We determine the Euler class of this extension and investigate the relation among the extension, the group $2$-cocycle defined by Ismagilov, Losik, and Michor, and the Calabi invariant of $D$.

We consider iterated function systems on the unit interval generated by two contractive similarity transformations with the same similarity ratio. When the ratio is greater than or equal to $1/2$, the limit set is the interval itself and the code map is not one-to-one. We study the set of points of the limit set having unique addresses. We obtain a formula for the Hausdorff dimension of the set when the similarity ratio belongs to certain intervals by applying the concept of graph directed Markov system.

We determine the center of a meta-nilpotent quotient of a mapping-torus group. As a corollary, we introduce two invariants, which are quadratic forms, of knots and of mapping classes.

In this paper, we re-compute the cohomology of the Morava stabilizer algebra $S(3)$ [12, 16]. As an application, we show that for $p \geq 7$, if $s\not \equiv 0, \pm 1 \,\, mod \,(p) $, $n\not \equiv 1 \,\, mod\, 3$, $n>1$, then $\zeta_n\gamma_s$ is a nontrivial product in $\pi_*(S)$ by Adams-Novikov spectral sequence, where $\zeta_n$ is created by R. Cohen [1], $\gamma_s$ is a third periodic homotopy elements.

We develop a method to compute the genera and slopes of essential surfaces in 2-bridge link exteriors with one longitudinal boundary component. The tools we use are those developed by Floyd and Hatcher in [4]. Such computations allow us to compute the genera of satellite tunnel number one knots and torti-rational knots.

We classify four-dimensional homogeneous conformal $C$-spaces and show that they are conformally Cotton-flat.

Beta Laguerre ensembles, generalizations of Wishart and Laguerre ensembles, can be realized as eigenvalues of certain random tridiagonal matrices. Analogous to the Wishart ($\beta=1$) case and the Laguerre $(\beta = 2)$ case, for fixed $\beta$, it is known that the empirical distribution of the eigenvalues of the ensembles converges weakly to Marchenko-Pastur distributions, almost surely. The paper restudies the limiting behavior of the empirical distribution but in regimes where the parameter $\beta$ is allowed to vary as a function of the matrix size $N$. We show that the above Marchenko-Pastur law holds as long as $\beta N \to \infty$. When $\beta N \to 2c \in (0, \infty)$, the limiting measure is related to associated Laguerre orthogonal polynomials. Gaussian fluctuations around the limit are also studied.

Let $f:S\to B$ be a finite cyclic covering fibration of a fibered surface. We study the lower bound of the slope $\lambda_{f}$ when the relative irregularity $q_{f}$ is positive.

In this article we prove that the Kawamata-Viehweg vanishing theorem holds for regular del Pezzo surfaces over imperfect ground fields of characteristic $p>3$.