Mathematics Colloquium
15 March 2023, 4 pm
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Venue: Ramanujan Hall
Host: Mayukh Mukherjee
Speaker: Arunima Ray
Affiliation: MPI, Bonn
Title: Embedding surfaces in 4-manifolds
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.
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.