Markus Klemetti’s doctoral thesis is motivated by topological data analysis, which is a recent field of mathematics studying the shape of data. One of the main methods of topological data analysis is persistent homology. In persistent homology the data is studied by associating a filtered topological space to it. By taking homology with coefficients in a field, a diagram of vector spaces and linear maps is obtained.

“This diagram is called a persistence module. In the standard case, this diagram is indexed by Z or R, but in general the indexing set can be any poset. Persistence modules need not be finitely presented. For computational reasons, several notions of ‘tameness’ have been introduced,” Klemetti explains.

The correspondence theorem of Carlsson and Zomorodian, which states that persistence modules can be viewed as modules over a polynomial ring of one variable, opened the graded perspective in topological data analysis.

In his thesis, Klemetti investigates generalized persistence modules from this perspective by considering monoid actions on preordered sets. If the indexing set is a partially ordered set (poset), he introduces a new tameness condition for a generalized persistence module by defining the notion of S-determinacy, where S is a subposet containing all the ‘births’ and the ‘deaths’.

“My first focus is on the correspondence between generalized persistence modules and graded modules in the case the indexing set has a monoid action. I introduce the notion of an action category over a monoid graded ring. Then I show that the category of additive functors from this category to the category of Abelian groups is isomorphic to the category of modules graded over the set with a monoid action, and to the category of unital modules over a certain smash product,” Klemetti says.

In the case S is finite, the notion of S-determinacy leads to a new characterization for a generalized persistence module being finitely presented. Moreover, it’s shown that after adding ‘infinitary points’ to Z, ‘S-determined’ is equivalent to ‘finitely determined’ as defined by Miller.

The doctoral dissertation of MSc Markus Klemetti in the field of mathematics titled *Generalized Persistence and Graded Structures* will be publicly examined in the Faculty of Information Technology and Communication Sciences at Tampere University **at 12 o’clock on 13 May** on City Centre Campus in the auditorium Pinni B 1096 (Kalevantie 4, Tampere). The Opponent will be Professor **Ran Levi **from the University of Aberdeen. The Custos will be Professor **Eero Hyry** from Tampere University.

The dissertation is available online at https://urn.fi/URN:ISBN:978-952-03-2372-1