October 20, 2020: Hitesh Gakhar
Title: A topological analysis of quasiperiodicity using persistent Künneth formulae
Abstract: Classically, sliding window embeddings were used in the study of dynamical systems to reconstruct the topology of underlying attractors from generic observation functions. In 2015, Perea and Harer developed a technique for recurrence detection in time series data using sliding window embeddings of periodic functions and persistent homology---which is an algebraic and computational tool used to quantify multiscale features of shapes. We study a closely related class of functions, namely quasiperiodic functions, which are defined as a superposition of periodic functions with non-commensurate harmonics. The sliding window embeddings of such functions are dense in high dimensional tori, where the dimension depends on the number of incommensurate harmonics. The study of persistent homology of Rips filtration on these embeddings motivated our work on persistent Künneth theorems---that is, results relating persistent homology of two filtered spaces to persistent homology of their products (we define two such products). In this talk, I will present some theoretical results and demonstrate that in certain cases a Künneth-type theorem helps recover quasiperiodicity better. I will also showcase an application of sliding windows persistence to the identification of dissonant music samples.
November 10, 2020: Reza Niazi
Title: Computing Homology of Submanifolds from Samples via Cubical Complexes
Abstract: Manifold reconstruction is a process of determining the properties of an unknown manifold M from a point cloud sampled from this manifold. The particular properties of interest depend on the application at hand. In computer graphics, it is often desirable to create a triangulation of the manifold. Unfortunately, building triangulations of high-dimensional manifolds is a computationally intractable problem. However, if one is only interested in the 'shape' of the manifold then understanding its homology groups is often enough. If the point cloud sampled from M is sufficiently dense, then the union of balls based at the sample points deformation retracts to M. So, the homology groups of M can be computed from a Cech complex associated with the point cloud but this approach is computationally expensive. In the first part of this talk, we will summarize the above mentioned results of Niygoi, Smale, and Weinberger. While in the second part we will present our ideas for building a cubical approximation of the manifold M. The main advantage of this approximation is that it can be used for efficient computations of the homology groups of higher dimensional manifolds.