The Killing operator on a Riemmanian manifold is a linear differetial operator on vector fields whose kernel provides the infinitesimal Riemannian symmetries. The Killing operator is best understood in terms of its prolongation, which entails some simple tensor identities. These simple identities can be viewed as arising from the vanishing of certain Lie algebra cohomologies. The point is that this case provides a model for other more complicated operators similiarly concerned with symmetry.

Let $V_1, V_2$ be hypersurface germs in $\mathbb C^m$, with $m \geq 2$, each having a quasi-homogenous isolated singularity at the origin. In our recent joint article with G. Fels, W. Kaup and N. Kruzhilin we reduced the biholomorphic equivalence problem for $V_1, V_2$ to verifying whether certain polynomials arising from the moduli algebras of $V_1, V_2$ are equivalent up to scale by means of a linear transformation. In the present we illustrate this result by the examples of simple elliptic singularities of $types \tilde{E}_6, \tilde{E}_7, \tilde{E}_8$ and compare our method with that due to M. G. Eastwood who has also introduced certain polynomials that distinguish non-equivalent singularities within each of these three types.

We compute the first order CR deformation of cyclic quotient surface singularities

Let $f(\mathbf{z, \bar{z}})$ be a mixed strongly polar homogeneous polynomial of 3 variables $z = (z1, z2, z3)$. It defines a Riemann surface $V := \{[\mathbf{z}] \in \mathbb{P}^2 | f(\mathbf{z, \bar{z}) =0\}$ in the complex projective space $\mathbb{P}^2$. We will show that for an arbitrary given $g \geq 0$, there exists a mixed polar homogenous polynomial with polar degree $1$ which defines a projective surface of genus $g$. For the construction, we introduce a new type of weighted homogeneous polynomials which we call polar weighted homogenous polynomials of the twisted join type.

In this short note, we first show (1) if $(n, p)$ lies inside Mather’s nice region then any $A-$stable multigerm $f : (\mathbb{R}^n, S) \rightarrow (\mathbb{R}^p, 0)$ and any $C^\infty$ unfolding of $F$ are $A-$simple, and (2) for any $(n, p)$ there exists a non-negative integer $i$ such that for any integer $j ((I \leq j))$ there exists an $A$ stable multigerm $f : (\mathbb{R}^n x \mathbb{R}^j, S x \{0\}) \rightarrow (\mathbb{R}^p x \mathbb{R}^j, (0,0))$ which is not $A-$simple. Next, we obtain a characterization of curves among multigerms of corank at most one from the view point of $A-$stable multigerms and $A-$simple multigerms. It turns out that for any $(n, p)$ such that $n \lt p$ an asymmetric Cantor set is naturally constructed by using upper bounds for multiplicities of $A-$stable multigerms and upper bounds for multiplicities of $A-$simple multigerms, and the desired characterization of curves can be obtained by cardinalities of constructed asymmetric Cantor sets.

This is a survey on recent progress on Saito free divisors and systems of uniformization equations with respect to such divisors. We first develop general theory on torsion free integrble connections along Saito free divisors based on $[Sa0]$. Then we show many examples of uniformization equations with respect to Saito free divisors in $C^3$ including discriminant sets of real and complex irreducible finite reflection groups of rank three.

In [S3] I solved the Thom’s conjecture that a proper Thom map is triangulable. In this paper I drop the properness condition in the semialgebraic case and, moreover, in the definable case in an o-minimal structure.

We review our recent results on monodromies at infinity of polynomial maps and $A$-hypergeometric functions. By using the theory of mixed Hodge modules, we introduce motivic global Milnor fibers of polynomial maps which encode the information of their monodromies at infinity into mixed Hodge structures with finite group actions. The numbers of the Jordan blocks in the monodromy at infinity of the polynomial will be described by its Newton polyhedron at infinity