We study the geometry of the cuspidal edge $M$ in $\mathbb{R}^3$ derived from its contact with planes and lines (referred to as flat geometry). The contact of $M$ with planes is measured by the singularities of the height functions on $M$. We classify submersions on a model of $M$ by diffeomorphisms and recover the contact of $M$ with planes from that classification. The contact of $M$ with lines is measured by the singularities of orthogonal projections of $M$. We list the generic singularities of the projections and obtain the generic deformations of the apparent contour (profile) when the direction of projection varies locally in $S^2$. We also relate the singularities of the height functions and of the projections to some geometric invariants of the cuspidal edge.

Enriques varieties have been defined as higher-dimensional generalizations of Enriques surfaces. Bloch's conjecture implies that Enriques varieties should have trivial Chow group of zero-cycles. We prove this is the case for all known examples of irreducible Enriques varieties of index larger than $2$. The proof is based on results concerning the Chow motive of generalized Kummer varieties.

A special rank-one perturbation $S_{t,n}$ of a weighted shift on a directed tree is constructed. Partial normality and weak hyponormality (including quasinormality, $p$-hyponormality, $p$-paranormality, absolute-$p$-paranormality and $A(p)$-class) of $S_{t,n}$ are characterized.

This paper studies strongly local symmetric Dirichlet forms on general measure spaces. The underlying space is equipped with the intrinsic metric induced by the Dirichlet form, with respect to which the metric measure space does not necessarily satisfy volume-doubling property. Assuming Nash-type inequality, it is proved in this paper that outside a properly exceptional set, if a pointwise on-diagonal heat kernel upper bound in terms of the volume function is known a priori, then the comparable heat kernel lower bound also holds. The only assumption made on the volume growth rate is that it can be bounded by a continuous function satisfying doubling property, in other words, is not exponential.

In this paper we show that there exist infinitely many Mazur type manifolds and corks with shadow complexity one among the 4-manifolds constructed from contractible special polyhedra having one true vertex by using the notion of Turaev's shadow. We also find such manifolds among 4-manifolds constructed from Bing's house. Our manifolds with shadow complexity one contain the Mazur manifolds $W^{\pm }(l,k)$ which were studied by Akbulut and Kirby.

We introduce a plus-space of Jacobi forms, which is a certain subspace of Jacobi forms of half-integral weight of matrix index. This is an analogue to the Kohnen plus-space in the framework of Jacobi forms. We shall show a linear isomorphism between the plus-space of Jacobi forms and the space of Jacobi forms of integral weight of certain matrix index. Moreover, we shall show that this linear isomorphism is compatible with the action of Hecke operators of both spaces. This result is a kind of generalization of Eichler-Zagier-Ibukiyama correspondence, which is an isomorphism between the generalized plus-space of Siegel modular forms of general degree and Jacobi forms of index $1$ of general degree.

In this paper we give a positive answer to a question raised by Nakamura, Nakanishi, and Satoh concerning an inequality involving crossing numbers of knots. We show it is an equality only for the trefoil and for the figure-eight knots.

It is well known that the Burau representation of the braid group can be used to recover the Alexander polynomial of the closure of a braid. We define $L^2$-Burau maps and use them to compute some $L^2$-Alexander torsions of links. As an application, we prove that the $L^2$-Burau maps distinguish more braids than the Burau representation.

We consider an initial-boundary value problem for the degenerate linear hyperbolic equation as a model of the motion of an inextensible string fixed at one end in the gravity field. We shall show the existence and the uniqueness of the solution and study the regularity of the solution.

It is known that the antipodal set of a Riemannian symmetric space of compact type $G/K$ consists of a union of $K$-orbits. We determine the dimensions of these $K$-orbits of most irreducible symmetric spaces of compact type. The symmetric spaces we are not going to deal with are those with restricted root system $\mathfrak{a}_r$ and a non-trivial fundamental group, which is not isomorphic to $\mathbb{Z}_2$ or $\mathbb{Z}_{r+1}$. For example, we show that the antipodal sets of the Lie groups $Spin(2r+1)\:\: r\geq 5$, $E_8$ and $G_2$ consist only of one orbit which is of dimension $2r$, 128 and 6, respectively; $SO(2r+1)$ has also an antipodal set of dimension $2r$; and the Grassmannian $Gr_{r,r+q}(\mathbb{R})$ has a $rq$-dimensional orbit as antipodal set if $r\geq 5$ and $r\neq q>0$.