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BIMSA Topology Seminar
BIMSA Topology Seminar
Nested homotopy models of finite metric spaces and their spectral homology
Nested homotopy models of finite metric spaces and their spectral homology
演讲者
时间
2024年01月22日 14:30 至 16:00
地点
A3-4-101
线上
Zoom 230 432 7880
(BIMSA)
摘要
For over a decade, two theories have been actively developed: theory of magnitude and magnitude homology of metric spaces, and GLMY-theory of path homology of directed graphs. Recently Asao showed that for the case of directed graphs there is a unified approach to these theories via a spectral sequence which is now known as the magnitude-path spectral sequence. He also introduced a notion of r-homotopy for directed graphs and proved that the r+1-st page of the spectral sequence is r-homotopy invariant. We extend this theory to the general case of quasimetric spaces that include metric spaces and directed graphs. We show that for a real number r and a finite quasimetric space X there is a unique (up to isometry) r-homotopy equivalent quasimetric space of the minimal possible cardinality. It is called the r-minimal model of X. We use this to construct a decomposition of the magnitude-path spectral sequence of a digraph into a direct sum of spectral sequences with certain properties. We also construct an r-homotopy invariant of a quasimetric space X called spectral homology, that generalizes many other invariants: the pages of the magnitude-path spectral sequence, including path homology, magnitude homology, blurred magnitude homology and reachability homology.
演讲者介绍
Sergei Ivanov教授是来自俄罗斯圣彼得堡的数学家。 他的研究兴趣包括同调代数、代数拓扑、群论、单纯同伦论和单纯群。