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Group Meeting on Algebraic Topology and Its Applications
An explicit construction of path homology chains in dimension 3
An explicit construction of path homology chains in dimension 3
演讲者
时间
2025年06月04日 13:00 至 14:00
地点
A3-1a-205
线上
Zoom 435 529 7909
(BIMSA)
摘要
The path homology introduced by Grigor'yan, Lin, Muranov and Yau plays a central role in digraph topology and the emerging field of digraph homotopy theory more generally. Unfortunately, the computation of the path homology of a digraph is a two-step process, and until now no complete description of even the underlying chain complex has appeared in the literature. In particular, our understanding of the path chains is the primary obstruction to the development of fast path homology algorithms, which in turn would enable the practicality of a wide range of applications to directed networks.
In recent work join with Tyron Cutler, we set out the construction of inductive elements that can be directly applied to understand path homology, under no restriction on the digraph. Inductive elements form a generating set with coefficients of prime characteristic that coincides with naturally occurring generating sets in low dimensions and agrees with the basis of Fu and Ivanov when the digraph contains no multisquares.
During this talk I intend to demonstrate the construction of inductive elements by applying them to provided a precise description of the dimension 3 path chains. In particular, generalising a result due to Grigor'yan in the case when the digraph contains not multisquares or double edge.
In recent work join with Tyron Cutler, we set out the construction of inductive elements that can be directly applied to understand path homology, under no restriction on the digraph. Inductive elements form a generating set with coefficients of prime characteristic that coincides with naturally occurring generating sets in low dimensions and agrees with the basis of Fu and Ivanov when the digraph contains no multisquares.
During this talk I intend to demonstrate the construction of inductive elements by applying them to provided a precise description of the dimension 3 path chains. In particular, generalising a result due to Grigor'yan in the case when the digraph contains not multisquares or double edge.