The depth of a cell in an arrangement of $n$ (non-vertical) great-spheres in $S^d$ is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn--Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements.
This is work with Ranita Biswas, Sebastiano Cultrera, and Morteza Saghafian.
Herbert Edelsbrunner is one of pioneers for topological and geometric approaches to data science, by his excellent research on the topic during the decades, from his PhD thesis on computational geometry in 1982 and assistant professorship in Information Processing at the Graz University of Technology during 1981-1985, to his professorship positions at UIUC, Duke and ISTA. Professor Edelsbrunner received Alan T. Waterman Award, he is members of the Academia Europaea, the American Academy of Arts and Sciences and the German Academy of Sciences Leopoldina. Professor Edelsbrunner is widely recognized as one of founding fathers of the new-born area of topological data analysis (TDA).
In the area of topology, there has been a lot of exciting progress in recent years. Topological tools and ideas have been used in arithmetic geometry, for instance, topological cyclic homology, and in low dimensional topology, like constructions of new knot and link invariants using homotopy theory and so on. On the other hand, topology has found exciting applications in science and technology. In mathematical physics, topics like topological quantum field theory, quantum anomaly cancellation, topological T-duality, topological insulators, topological order, and others, have made tremendous progress. In modern technology, topology has also played prominent roles. In particular, TDA (topological data analysis) has demonstrated great potential for big data and has been widely used in the study of robotics, materials, chemistry, biology, drug design and discovery. More and more topological tools and ideas have been used in machine learning and deep learning models. The aim of the seminar is to invite top experts and young researchers to introduce and share their progress in the study of topology and its applications. We expect that the seminar will bring potential collaborations and broaden the horizon of young students.
Please find out collective information of this seminar here: http://www.bimsa.cn/newsinfo/624195.html (copy and open in browser)