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Structure rings of singularities and differential equations. (English) Zbl 0907.58067
Demuth, Michael (ed.) et al., Differential equations, asymptotic analysis, and mathematical physics. Papers associated with the international conference on partial differential equations, Potsdam, Germany, June 29–July 2, 1996. Berlin: Akademie Verlag. Math. Res. 100, 325-347 (1997).
In this note the authors describe singular points of manifolds with singularities, beginning with the examples of circular cone and circular cusp. The goal of this paper is to study the behaviour of local function rings near singular points of a manifold which is smooth everywhere except for some discrete set of singular points.
The authors consider a ring of germs of the structure ring at some arbitrary singular point. This ring will refer to a structure ring of the singular point that gives all the analysis in a neighbourhood of the given point as, for example, the ring of differential equations, the class of Riemann matrices. The choice of structure rings leads to different classes of different equations. More precisely, for different structure rings we will arrive at equations with conical degeneracy, cuspidal degeneracy or other types of degeneracy; the interest of the authors is to consider some of these kinds of degeneracy.
For the entire collection see [Zbl 0864.00061].

58J20 Index theory and related fixed-point theorems on manifolds
35S05 Pseudodifferential operators as generalizations of partial differential operators
19K56 Index theory
46E40 Spaces of vector- and operator-valued functions