I will discuss the use of persistent cohomology to discover meaningful circle-valued coordinates on a data set, in the spirit of nonlinear dimensionality reduction (NLDR) algorithms such as Isomap, LLE, and Laplacian Eigenmaps. The construction is closely related to the Abel-Jacobi map in algebraic geometry. There are possible applications to the study and simulation of dynamical systems. My collaborators in this work (some of it completed, some of it in progress) include Dmitriy Morozov, Mikael Vejdemo-Johansson, Primoz Skraba, Konstantin Mischaikow.
Vin de Silva studied mathematics at Cambridge and Oxford, completing a doctorate in symplectic geometry under the supervision of Simon Donaldson. Since 2000, he has worked in applied topology, spending five years working in Gunnar Carlsson’s research group at Stanford. His work with Josh Tenenbaum and John Langford on the Isomap algorithm is widely cited to this day, and his collaboration with Robert Ghrist on sensor network topology was honored by a SciAm50 award in 2007. He is currently an associate professor of mathematics at Pomona College, California, and holds a Digiteo Chair at INRIA Saclay Ile-de-France.`