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Fatou-Bieberbach domains and attracting basins.

Mathematics Colloquium | Sayani Bera | Sep 27th, 2023

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Fatou-Biberbach domains in $\\mathbb{C}^k, k \\ge 2$ are proper subdomains of $\\mathbb{C}^k, k \\ge 2$ that are biholomorphic to $\\mathbb{C}^k, k \\ge 2$. Such domains clearly do not exist in $\\mathbb{C}$ and, further arise naturally from dynamics of automorphisms of $\\mathbb{C}^k, k \\ge 2$, admitting an attracting fixed point. The goal of this talk is to discuss a few pathological examples of such domains, and also their relevance in the context of geometry, complex function theory and dynamics.

Further, we will introduce non-autonomous (dynamical) families and their basins attractions, and discuss an affirmative solution to a long standing open problem, called the Bedford's conjecture, related to the (non-autonomous) dynamical origins of a Fatou-Bieberbach domain. This is a joint work with Dr. Kaushal Verma.

About the speaker : 

Currently Sayani is working as an assistant professor in Indian Association for the Cultivation of Science. Prior to that she had held post doctoral fellowships in HRI, Prayagraj (Harish Chandra Research Institute, Prayagraj) and RKM-VERI, Belur (Ramakrishna Vivekananda Educational Research Institute, Belur). She completed her Ph.D. from Indian Institute of Science, Bangalore in 2015, under the supervision of Dr. Kaushal Verma. Her area of specialisation is holomorphic dynamics in several complex variables.


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