The lecture series focuses on the interesting and complex topic of holomorphic dynamics and its application to root finding. The course is taught by Dierk Schleicher (Aix-Marseille University, LMS Distinguished Visiting Fellow at ICMU). Lecturers are devoted to the fundamental problems of finding roots of analytic equations, exploring both classical and modern methods.

The goals of this series of lectures are to:

• Explore the close relationship between root-finding methods and the iteration of holomorphic functions.

• Discuss research progress on the Newton method, including questions about starting points, complexity, and implementation for high-degree polynomials.

• Present recent developments in root-finding implementations, including those that have factored polynomials of extremely large degrees.

• Highlight open questions and current challenges in the field.

These lectures are suitable for graduate students and researchers with an interest in theoretical mathematics, numerical analysis, (statistical) physics, and various areas of engineering. The material can be adjusted depending on the audience's interests, and connections to other fields such as geometry, algebra, and number theory can be discussed.

The Ukrainian host is Jakub Konieczny, a Simons Professor in Mathematics at Kyiv School of Economics

Prerequisites for the course participants: A solid foundation in undergraduate-level mathematics is necessary, knowledge of complex analysis and holomorphic functions would be beneficial. The course material can be adjusted to the audience's interests, so a willingness to engage in discussions about connections to other fields like geometry, algebra, and number theory is a plus.

Image: One picture shows the actual Mandelbrot set. One shows a detail of the space of those cubic polynomials where Newton’s method fails to work well. And another one shows a detail of the space of those cubic polynomials where the Weierstrass-(Durand-Kerner-) method fails to work well. One is a definition, one is a theorem, and one is a wide open research question. — Dierk Schleicher

Dierk Schleicher is a professor of mathematics at Aix-Marseille University. He obtained his PhD at Cornell University, NY, and held visiting positions in Berkeley, Stony Brook, Paris, Toronto, and München. In 2001-2019 Dierk was a professor of the Jakobs University Bremen and devoted a lot of his efforts to the creation of the Department of Mathematics there.  His research interests are  in dynamical systems and chaos theory, particularly focusing on holomorphic dynamics and Newton's method for finding roots of complex polynomials.

Photo: © BMBF / Yves Sucksdorff