Vanishing cycles of matrix singularities

18^th November 2019, 14:00 MATH-103
Victor Goryunov
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Abstract

The talk is about holomorphic map germs M : (C^s , 0) ? M atn, where the target is the space of either square, or symmetric, or skew-symmetric n × n matrices. The target contains the set ? of all degenerate matrices, and our main object will be the vanishing topology of M?1(?). Our attention is on the singular Milnor fibre of M, that is, the local inverse image V of ? under a generic small perturbation of M. The variety V is highly singular, but, according to Le Dung Trang’s theorem, it is homotopic to a wedge of (s ? 1)-dimensional spheres.

The talk will start with introduction of local models for the spheres vanishing in the matrix context.
We will then prove the µ = ? conjecture formulated by Damon for corank 1 map-germs M with a generic linear part, and a generalisation of this conjecture
to the matrix version of boundary function singularities.

Bifurcation diagrams of matrix singularities will also be discussed, and a rather unexpected appearance of the discriminants of certain Shephard-Todd
groups as such diagrams will be highlighted.

If time permits, possible approaches to the study of the monodromy will be mentioned