Mr. **Alexander Brudnyi** completed his Ph.D. Thesis in 1996 at the Technion, Israel. Afterwards he held a NATO postdoctoral fellowship at the University of Toronto and the Fields Institute. After spending some time at the Ben Gurion and Sundsval Universities, he joined the staff at the University of Calgary in 2000. Working in complex analysis and geometry, he has, in more than 25 articles, made significant contributions to four different areas: fundamental groups of compact Kähler manifolds, local inequalities for holomorphic and plurisubharmonic functions, limit cycles and the distribution of zeros of families of analytic functions, maximal ideals of the space of bounded analytic functions and matrix-valued corona theorem. Professor Brudnyi delivered a lecture on February 21, 2003, for which he wrote the following summary:

*Center Problem for Ordinary Differential Equations*

**Abstract**: The classical H. Poincaré Center-Focus problem is to describe planar polynomial vector fields such that all their integral trajectories are closed curves around some point. This situation is called a center. In some cases this problem can be reduced to a similar one for ordinary differential equations. In the talk I present a new general approach to the Center problem for ODEs. I will also explain how in this approach the Center problem is related to the Hilbert 16th problem (on the number of limit cycles for planar polynomial vector fields), and the Composition problem for Lipschitz functions defined on the unit circle.

Centre de recherches mathematiques

Université de Montréal

Pavillon Andre-Aisenstadt

2920, chemin de la Tour

Salle / Room 6254

*Le vendredi 21 février 2003*

Friday, February 21, 2003

16 h / 4:00 p.m.