Topological data analysis is a field of mathematics that is mainly concerned with studying the shape of data. One of the core tools of the field is persistent homology. Single parameter persistent homology is well understood and has found many applications in practice. On the other hand, multiparameter persistent homology is not practical yet due to a lack of a discrete complete invariant.
“A key problem in topological data analysis is to find discrete invariants that are not complete but provide enough interpretable information about multiparameter persistence modules to make multiparameter persistent homology useful in practice,” Ville Puuska states.
One promising approach to tackle this problem was offered by Miller in his work on finitely determined and finitely encoded persistence modules. He proposed using flat covers and injective hulls to encode multiparameter persistence modules instead of the more typical approach of using free presentations.
In his thesis, Puuska translates the theory of flat covers and flat cotorsion modules originally developed by Enochs and Xu in the setting of commutative algebra to the setting of persistence modules. With this theory, Puuska generalizes the idea of using flat covers and injective hulls to encode persistence modules to work for all persistence modules to move beyond the previously required finiteness assumptions. With this general theory, Puuska also studies different connections between flat covers and injective hulls which lead to connections between the graded Enochs-Xu and Bass numbers as well.
“By decomposing the flat cover and injective hull of a multiparameter persistence module we get information about the generators and cogenerators of the multiparameter persistence module. These generators and cogenerators give us interesting discrete invariants – the graded Enochs-Xu and Bass numbers – that describe the shape of the multiparameter persistence module,” Puuska says.
Public defence on Friday 20 October
The doctoral dissertation of MSc Ville Puuska in the field of mathematics titled Flat Covers and Cotorsion in Persistence will be publicly examined at the Faculty of Information Technology and Communication Sciences at Tampere University, at 12 o’clock on Friday 20 October 2023. The venue is auditorium 1096 of the Pinni B building, (address: Kanslerinrinne 1, Tampere). The Opponent will be Professor Wojciech Chachólski from KTH Royal Institute of Technology, Sweden. The Custos will be Professor Eero Hyry from the Faculty of Information Technology and Communication Sciences.
The doctoral dissertation is available online.
The public defence can be followed via a remote connection.