Embedding surfaces in 4-manifolds

Speaker: Arunima Ray

Affiliation: University of Melbourne

Abstract

Manifolds are fundamental objects in topology since they locally model Euclidean space. Within a given ambient manifold, we are often interested in finding embedded submanifolds, which would then enable cutting and pasting operations, such as surgery. The study of surfaces in 4-dimensional manifolds has led to breakthroughs such as Freedman's proof of the 4-dimensional Poincare conjecture. Important open questions on 4-manifolds can also be reduced to the question of finding certain embedded surfaces. In this talk, I will consider the following question: When is a given map of a surface to a 4-manifold homotopic to an embedding? I will give a survey of related results, including the celebrated work of Freedman and Quinn, and culminating in a general surface embedding theorem, arising in joint work with Daniel Kasprowski, Mark Powell, and Peter Teichner.

Bio: Dr. Arunima Ray is a Senior Lecturer at the University of Melbourne. Her research lies in low-dimensional topology, with a particular focus on knot theory, knot concordance, and the topology of 3- and 4-manifolds. She is a co-editor and contributor of The Disc Embedding Theorem (OUP, 2021), a book that modernised the (Fields medal winning) proof of Freedman's disc embedding theorem. Prior to joining Melbourne, she led a Lise Meitner research group at the Max-Planck-Institut für Mathematik in Bonn. Dr. Ray has published extensively in leading mathematics journals and has been an active organiser of international workshops and conferences in topology.