Topology Seminar

Computational topology is an emerging discipline developed since around 2000. It studies the determination and modeling of topological problems in computer applications, as well as the design of algorithms for topological problems. As a subject to apply computational topology to the field of data processing, topological data analysis has been developed. It has been widely used in biomedicine, drug design, financial analysis, machine learning, and other fields. The main tools in computational topology and topological data analysis are persistent homology and Mapper. Persistent homology constructs a sequence of gradually "growing" simplicial complexes in a metric space, computes the persisting homological features (generators in the persistent homology groups), and infers the importance of the features based on the life span of these homological features, thereby enabling the inference and extraction of global topological features of the discrete data set. On the other hand, Mapper extracts the main topological structure of the data set by defining a reference mapping on the data set and using data segmentation and clustering. Since almost 10 years ago, the CAGD research group of Zhejiang University has applied computational topology methods to geometric design and geometric processing, and developed a series of computer-aided topological design methods. This talk will introduce a series of work in this area, including curve and surface reconstruction technology based on topological understanding, and topological control methods in implicit curve and surface reconstruction. Furthermore, persistent homology has been applied to the field of porous structure processing, and a variety of topological descriptors have been designed for porous structure retrieval and classification; porous structure generation technology that ensures connectivity has been developed; porous thickness computation technology that preserves topological structure and porous model slicing method have been proposed in 3D printing of porous structures.

Up to continuous deformations, all based continuous maps between two spheres form an abelian group, and is called a homotopy group of the target sphere. It turns out that determination of these groups is a very hard problem in topology. The structures of the homotopy groups of spheres are closely related to many topics in topology, such as the Hopf invariant problem, the Kervaire invariant problem, and the number of smooth structures on a given sphere.
In this talk, I will review some classical methods of computing these groups, and discuss some recent progress.