Geometric Structures - Miruna-Stefana Sorea - March 30 - 4:15 pm (Rome Time)

[Antonio Lerario]

Dear All,

This is to announce the next seminar from the series "Geometric Structures".

Speaker: Miruna-Stefana Sorea (SISSA)
Title: Poincaré-Reeb trees of real Milnor fibres

Time: March 30, 2021, 4:15 pm (Rome Time)
Venue: from remote only, on zoom at this link
https://sissa-it.zoom.us/j/85675591787?pwd=TUo2VXpmcEhOU1paRzBXUWp2MU1odz09
Passcode: geometry

Abstract:
We study the real Milnor fibre of real bivariate polynomial functions
vanishing at the origin, with an isolated local minimum at this point. We
work in a neighbourhood of the origin in which its non-zero level sets are
smooth Jordan curves. Whenever the origin is a Morse critical point, the
sufficiently small levels become boundaries of convex disks. Otherwise,
they may fail to be convex, as was shown by Coste.

In order to measure the non-convexity of the level curves, we introduce a
new combinatorial object, called the Poincaré-Reeb tree, and show that
locally the shape stabilises and that no spiralling phenomena occur near
the origin. Our main objective is to characterise all topological types of
asymptotic Poincaré-Reeb trees. To this end, we construct a family of
polynomials with non-Morse strict local minimum at the origin, realising a
large class of such trees.

As a preliminary step, we reduce the problem to the univariate case, via
the interplay between the polar curve and its discriminant. Here we give a
new and constructive proof of the existence of Morse polynomials whose
associated permutation (the so-called “Arnold snake”) is separable, using
tools inspired from Ghys’s work.


More information can be found here:
https://sites.google.com/view/geometric-structures/

Everyone is welcome!

Best,
Antonio


-- 
*http://people.sissa.it/~lerario <http://people.sissa.it/~lerario>*