Algebraic topology is about the study of quantitative features of spaces that are independent of continuous deformations. It often studies the intricate relationships between continuous operations like pulling or stretching and properties pertaining to corresponding algebraic structures. It relies on central concepts of homotopy and homology. The abstraction and discretization of algebraic topology can be applied to the study of higher-order structures and dynamics of complex systems. The research area focuses on the following directions: 1) Fundamental research on all subjects in algebraic topology, 2) application-guided research developing topological foundations of higher-order interaction networks focusing on digraphs, mixture graphs, hyper(di-)graphs and hyper-networks related to GLMY homology and homotopy of digraphs introduced by Shing-Tung Yau, and introducing new topology theories in data, 3) practical applications of algebraic topology in data science and artificial intelligence focusing on testing the effectiveness of topological approaches developed at BIMSA.
Focus topics: 1) GLMY-theory and magnitude homology theory, 2) Non-commutative patterns in unstable homotopy theory, 3) Exponents of homotopy groups, 4) Decomposition of loop spaces, 5) Homotopy aspects of knot theory, 6) Topological foundations of complex networks, 7) AI methods in algebraic topology, 8) Combinatorial topology based data analysis, 9) Knot theory based data analysis, 10) Representations and calculus of functors.
The members in the group will work along one or more above-mentioned directions, with aiming to one or more of the focus topics via collaborative interactions among group members.