Vol 76, No 2 (2021)

This paper is a survey of contemporary results and applications of the theory of homotopes. The notion of a well-tempered element of an associative algebra is introduced, and it is proved that the category of representations of the homotope constructed by a well-tempered element is the heart of a suitably glued t-structure. The Hochschild and global dimensions of homotopes are calculated in the case of well-tempered elements. The homotopes constructed from generalized Laplace operators in Poincare groupoids of graphs are studied. It is shown that they are quotients of Temperley–Lieb algebras of general graphs. The perverse sheaves on a punctured disc and on a 2-dimensional sphere with a double point are identified with representations of suitable homotopes. Relations of the theory to orthogonal decompositions of the Lie algebras $\operatorname{sl}(n,\mathbb{C})$ into a sum of Cartan subalgebras, to classifications of configurations of lines, to mutually unbiased bases, to quantum protocols, and to generalized Hadamard matrices are discussed.Bibliography: 56 titles.

This paper is devoted to the resolution of singularities of holomorphic vector fields and one-dimensional holomorphic foliations in dimension three, and it has two main objectives. First, within the general framework of one-dimensional foliations, we build upon and essentially complete the work of Cano, Roche, and Spivakovsky (2014). As a consequence, we obtain a general resolution theorem comparable to the resolution theorem of McQuillan–Panazzolo (2013) but proved by means of rather different methods.The other objective of this paper is to consider a special class of singularities of foliations containing, in particular, all the singularities ofcomplete holomorphic vector fields on complex manifolds of dimension three. We then prove that a much sharper resolution theorem holds for this class of holomorphic foliations. This second result was the initial motivation for this paper. It relies on combining earlier resolution theorems for (general) foliations with some classical material on asymptotic expansions for solutions of differential equations.Bibliography: 34 titles.

A local normal form for Roussarie vector fields with degenerate quadratic part is presented.