CAM Colloquium: Yulij Ilyashenko (National Research University Higher School of Economics, Moscow) - Global bifurcation theory in the plane

Description

Abstract:
This talk is devoted to a new chapter in the bifurcation theoty: global bifurcations in the plane. The planar bifurcation theory splits in three parts: local, semilocal and global bifurcations. It became clear five years ago that the third part is yet to be created. Local bifurcation theory (we will discuss planar bifurcations only) studies the perestroykas of the phase portraits of vector fields near singular points. This theory is almost completed, though recently some new open questions occurred. Semilocal theory studies bifurcations of polycycles (separatrix polygons).

Recently some unexpected results were found in this theory; they will be included to the talk. The goal of the talk is to outline the development of the new theory and to state open problems. The main new results are:
-Existence of an open set of structurally unstable families of vector fields in the plane, as well as of families having a functional invariant (joint with Yu. Kudryashov and I. Schurov)
-A complete classification of global bifurcations in generic one parameter families (joint with N. Goncharuk, N. Solodovnikov and V. Starichkova)
-Description of large bifurcation supports (the sets near which the bifurcation really occurs) for the families with an arbitrary large number of parameters (joint with N. Goncharuk) -Discovery of structurally unstable families in the semilocal bifurcation theory (joint with A. Dukov)
-New invariants of topological classification of multiparameter families of vector fields on the sphere (N. Goncharuk and Yu. Kudryshov) Thirty five years ago Arnold stated six conjectures aimed to predict the future development of the global planar bifurcation theory. Now all the six are disproved, but they motivated the modern development of the theory.

Bio:
Yulij Ilyashenko is currently a professor at the National Research University Higher School of Economics in Moscow. He received his PhD from the Mechanics and Mathematics Department of Moscow State University, where he has been a faculty member since 1968. In 1994, he became the Dean of the Higher College of Mathematics at the Independent University of Moscow, and he was a Professor in the Department of Mathematics at Cornell University from 1997 to 2016. In 2017, he was elected a Fellow of the American Mathematical Society. From 2000 to the present time, he has been the President of the Independent University of Moscow.

Zoom Link Access:
This talk will be given via Zoom, and the link is emailed to the CAM Seminar listserv the week of the talk. If you are not on the listserv, please contact Erika Fowler-Decatur to request the link.