In the late 1980s Andreas Floer revolutionized low-dimensional and symplectic topology by discovering the existence an extension of Morse theory to an infinite-dimensional setting where the standard methods of variational calculus fail. While he foresaw that his theory should be able to encompass generalized homology theory (bordism, K-theory, …), severe foundational difficulties prevented any significant progress on this question until two years ago. I will explain the advances that have been made on two fronts: (I) defining concrete models, in terms of equivariant vector bundles, for the moduli spaces that appear in Floer theory, and (II) understanding the geometric consequences of lifting Floer homology to generalized homology theories. I will end by formulating how the notion of derived orbifold bordism provides a universal receptacle for Floer’s invariants, and its descendants.
This video was produced by the Hausdorff Center for Mathematics, and was part of the conference Panorama of Mathematics II.
