## Select Seminar Series

All seminars- Home
- ›
- Studying at the Faculty
- ›
- Seminars ›
- Seminar in Geometry and Topology

# Seminar in Geometry and Topology

I will present a joint work with A. Alvarez, J.L. Bravo and C. Christopher.

We study the analogue of the classical infinitesimal center problem in the plane, but for zero cycles. We define the displacement function in this context and prove that it is identically zero if and only if the deformation has a composition factor. That is, we prove that here the composition conjecture is true, in contrast with the tangential center problem on zero cycles. Finally, we give examples of applications of our results.

Zoom link: https://us02web.zoom.us/j/88255165329

The second part of Hilbert's sixteenth problem asks for a study of the number and relative positions of the limit cycles of an ODE system associated to a planar vector field with polynomial component functions. Seeking a probabilistic perspective on Hilbert's problem, we present recent results on the distribution of limit cycles when the vector field component functions are random polynomials. We present a lower bound for the average number of limit cycles for the Kostlan-Shub-Smale model, and we present asymptotic results for a special class of limit cycle enumeration problems concerning a randomly perturbed center focus. We will discuss some ideas from the proofs which combine techniques from dynamical systems (such as transverse annuli, Poincare maps, and Melnikov functions) with those coming from the theory of random analytic functions (such as real zeros of random series, the Kac-Rice formula, and the barrier construction from the study of nodal sets of random eigenfunctions). We conclude by discussing future directions and open problems.

Relevant text: preprint on arXiv

**Zoom link**: https://weizmann.zoom.us/j/94594972747?pwd=cWl0bjB1T2ZYSUp0dzdaK3NNa3Mzdz09