In this paper we give a formula for counting the number of isolated stable singularities of a stable perturbation of corank 1 germs f : C, 0 → C , 0 with n < p that appear in the image f(C). We also define a set of A-invariants and show that their finiteness is a necessary and sufficient condition for the A-finiteness of the germ f.

We describe a general construction of finiteness spaces which subsumes the interpretations of all positive connectors of linear logic. We then show how to apply this construction to prove the existence of least fixpoints for particular functors in the category of finiteness spaces: these include the functors involved in a relational interpretation of lazy recursive algebraic datatypes along the...

We prove the finiteness of class numbers and Tate-Shafarevich sets for all affine group schemes of finite type over global function fields, as well as the finiteness of Tamagawa numbers and Godement’s compactness criterion (and a local analogue) for all such groups that are smooth and connected. This builds on the known cases of solvable and semisimple groups via systematic use of the recently ...

The purpose of this paper is to present finiteness theorems and several properties of cohomologies of holomorphic solution sheaves of maximally overdetermined systems of linear differential equations. The proof relies on the finiteness theorem for elliptic systems due to T. Kawai [4], as an analytic tool, and on the theory of stratifications of analytic sets introduced by H. Whitney [8] and [9]...

This paper is a survey of finiteness results in hyperkähler geometry. We review some classical theorems by Sullivan, Kollár-Matsusaka, Huybrechts, as well as theorems in the recent literature by Charles, Sawon, and joint results of the author with Verbitsky. We also strengthen a finiteness theorem of the author. These are extended notes of the author’s talk during the closing conference of the ...

A set of matrices is said to have the finiteness property if the maximal rate of exponential growth of long products of matrices drawn from that set is realised by a periodic product. The extent to which the finiteness property is prevalent among finite sets of matrices is the subject of ongoing research. In this article we give a condition on a finite irreducible set of matrices which guarante...

Craig Squier proved that, if a monoid can be presented by a finite convergent string rewriting system, then it satisfies the homological finiteness condition left-FP3. Using this result, he constructed finitely presentable monoids with a decidable word problem, but that cannot be presented by finite convergent rewriting systems. Later, he introduced the condition of finite derivation type, whic...

An arbitrary renormalizable quantum field theory is considered as finite if its dimensionless couplings conspire to yield at every order of its perturbative expansion no ultraviolet-divergent renormalizations of the physical parameters of the theory. The “finiteness conditions” resulting from these requirements form highly complicated, non-linear systems of relations. A promising type of soluti...

In 1987, Craig Squier proved that, if a monoid can be presented by a finite convergent string rewriting system, then it satisfies the homological finiteness condition left-FP3. Using this result, he has constructed finitely presented decidable monoids that cannot be presented by finite convergent rewriting systems. In 1994, Squier introduced the condition of finite derivation type, which is a h...

Finiteness spaces constitute a categorical model of Linear Logic whose objects can be seen as linearly topologised spaces, (a class of topological vector spaces introduced by Lefschetz in 1942) and morphisms as continuous linear maps. First, we recall definitions of finiteness spaces and describe their basic properties deduced from the general theory of linearly topologised spaces. Then we give...