Symplectic Embeddings and results in TDA
Tid: Må 2022-05-16 kl 10.00
Respondent: Alvin Jin , Matematik (Avd.)
Opponent: Associate Professor Henry Adams, Colorado State University, USA
Handledare: Professor Wojciech Chacholski, Matematik (Inst.); Professor Gregory Arone,
This thesis is a collection of work under the theme of “applied topology." The linking idea behind seemingly disjoint fields is the existence of a filtration that one uses to study a space. In turn, given the ubiquitous nature of filtrations, applications range from theoretical fields (e.g. symplectic geometry) to applied fields (machine learning).
In paper A, we study when homological information of a simplicial complex can be determined from its components in the following manner: given a data cloud, partition the points in the cloud into two (or more) sets. Form separate simplicial complexes from these sets, and compare the homologies of these simplicial complexes from that of the simplicial complex formed from the point cloud itself. In applied topology, very rarely does a decomposition of a space yield information about the space itself - meaning that it is rare for a Mayer-Vietoris sequence to hold. We study “obstruction complexes" and show that in nice enough cases, there is a relationship between homological information of the space and its decomposition.
In paper B, we study a construction called “realisation" that we apply to posets. This enables the generation of a wealth of examples of posets that might not necessarily be the nonnegative reals in topological data analysis. We define various properties of these realizations, and in the end we link these properties to homological properties of the functors that are being studied.
In paper C, we study the classic evasion-path problem. This problem is well-known in robotics and machine learning, and more recently became of interest in the applied topology community through works of Krishnan and Ghrist in addition to work of Adams and Carlsson. The key point is that just studying homology and barcodes could not determine if an evasion path exists. We study a higher invariant, using tools of Goodwillie calculus to yield an obstruction to the existence of an evasion path.
In symplectic geometry, work has been done to try to use the filtration to study symplectic embeddings. The work in this thesis does not get to the direct relationship between the filtration and symplectic embeddings, but it does study the relationship between symplectic embeddings of ellipsoids and polydiscs in dimension four, yielding a rigid-flexible result similar to the one given by the famous nonsqueezing theorem. This is the topic of paper D. There is still much work to be done linking applied topology and symplectic geometry.