Classification of Monoidal Involutions Having a Fixed Tangent Cone

Around the tangent cone theorem

A cornerstone of the theory of cohomology jump loci is the Tangent Cone theorem, which relates the behavior around the origin of the characteristic and resonance varieties of a space. We revisit this theorem, in both the algebraic setting provided by cdga models, and in the topological setting provided by fundamental groups and cohomology rings. The general theory is illustrated with several cl...

Fixed Point Shifts of Inert Involutions

Given a mixing shift of finite type X , we consider which subshifts of finite type Y ⊂ X can be realized as the fixed point shift of an inert involution of X . We establish a condition on the periodic points of X and Y that is necessary for Y to be the fixed point shift of an inert involution of X . We show that this condition is sufficient to realize Y as the fixed point shift of an involution...

Fixed Point Sets of Involutions on Spheres

Fixed point free involutions on Riemann surfaces

Involutions without fixed points on hyperbolic closed Riemann surface are discussed. Notably, the following question is answered: for τ such an involution on a surface (S, d), can a point p be chosen so that d(p, τ(p)) is bounded by a constant dependent solely on the genus of S? This is proved to be true for even genus and false otherwise. Furthermore, the sharp constant is given in genus 2.

Tangent cone algorithm for homogenized differential operators

We extend Mora’s tangent cone or the écart division algorithm to a homogenized ring of differential operators. This allows us to compute standard bases of modules over the ring of analytic differential operators with respect to sufficiently general orderings which are needed in the D-module theory.