This is a survey article on the cobordism theory of non-spherical knots studied in [BM, B2, BS1, BMS, BS2, BS3]. Special emphasis is put on fibered knots.

We first recall the classical results concerning cobordisms of spherical knots. Then we give recent results on cobordisms of simple fibered $(2n-1)$-knots for $n \ge 2$ together with relevant examples. We discuss the Fox-Milnor type relation and show that the usual spherical knot cobordism group modulo the subgroup generated by the cobordism classes of fibered knots is infinitely generated for odd dimensions. The pull back relation on the set of knots is also discussed, which is closely related to the cobordism theory of knots via the codimension two surgery theory. We also present recent results on cobordisms of surface knots in $S^4$ and 4-dimensional knots in $S^6$. Finally we give some open problems related to the subject.

This paper surveys some results surrounding Lehmer's problem in the context of fibered links and Hopf plumbing. Topics addressed here are Mahler measures of fibered links, the relation between iterated Hopf plumbings and Salem-Boyd polynomials, and the question of when monotone growth occurs under iterated plumbing. Explicit calculations for certain "perturbations" of links associated to the ADE singularities are computed.

A real polynomial in one real variable is hyperbolic if all its roots are real. Denote the set of monic hyperbolic polynomials of degree $n$ by $\prod_n$. Suppose that for a real polynomial $P(x)$ of degree $n$ there exists $k \in \mathbf{N}$ and a polynomial $Q(x)$ of $\mathrm{degree} \le k-1$ such that $x^k P+Q \in \prod_{n+k}$. Denote the set of such polynomials $P$ by $\prod_n (k)$. Call the set $\prod_n (\infty) = \overline{\cup_{k=0}^{\infty} \prod_n (k)}$ the domain of stably hyperbolic polynomials of degree $n$. In the present paper we explore the geometric properties of the set $\prod_4 (\infty)$.

In this paper, a notion of a weighted-degree is introduced to algebraic local cohomology classes associated with a semiquasihomogeneous function. Utilizing weighted-degrees, computations of a dual basis of Milnor algebra and membership problems are considered as applications.

We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence of a canonical plumbing structure on the abstract boundaries (also called links) of normal surface singularities. The duality between supplementary cones gives in particular a geometric interpretation of a duality discovered by Hirzebruch between the continued fraction expansions of two numbers $\lambda \gt 1$ and $\lambda / (\lambda -1)$.

We define here the Artin functions of a germ of analytic space. Artin functions are analytic invariants of the germ and a measure of its singularity. In general these functions are very difficult to compute. We give a few properties of these functions and we present some examples.

We review some recent applications of perverse sheaves (intersection cohomologies) in singularity theory. Milnor fibers over general complete intersection varieties will be treated. We also give a proof of a result announced in [31].

In this note we give a rational uniformisation equation of the discriminant loci associated to a non-degenerate affine complete intersection variety. To show this formula we establish a relation of the fibre-integral with the hypergeometric function of Horn and that of Gel'fand-Kapranov-Zelevinski.

We compare the Euler-Poincaré characteristic to the global Euler obstruction, in case of singular affine varieties, and point out a certain duality among their expressions in terms of strata of a Whitney stratification.

We extend some results on augmented valuations and key-polynomials to limit augmented valuations and limit key-polynomials. As any valuation $\mu$ of $K[x]$ is obtained as a limit of an admissible family of valuations, we deduce a description of the graded algebra associated to $\mu$.

Bayer and Hefez described irreducible plane curve singularities whose Milnor and Tjurina numbers differ by one or two, modulo analytic equivalence. After their work, we classify the case in which their difference is three.