Applied algebraic topology
We are interested in the algebraic foundations of topological data-analysis. Persistent homology is the main tool of topological data analysis.
While the theory of single parameter persistent homology is well understood, the general theory of persistent homology over arbitrary posets poses at the moment the biggest challenge of topological data analysis. Graded algebra and representation theory of posets have revealed to be powerful tools for both the theoretical understanding of persistent homology as well as for its practical use. However, the mathematical questions that arise in the general case are very different than the ones encountered classically, and call for new algebraic foundations.
We are therefore developing completely new fundamental algebra for modules over posets.
Members:
Eero Hyry, Professor
Marja Kankaanrinta, University Lecturer
Markus Klemetti, PhD, currently a postdoc at the University of Aberdeen, UK
Manu Harsu, Doctoral student
Former member: Ville Puuska, PhD
Discrete geometry
The research focuses on discrete Ricci curvatures on graphs, examining how global properties of networks can be derived from local data. This work intersects graph theory, analysis, combinatorics, linear algebra, and stochastics, with the aim of uncovering fundamental theoretical results alongside practical applications and software development. The project is part of an international collaboration involving researchers from several countries, contributing to advancements in the study of graph curvature.
Members: Riikka Kangaslampi, University Lecturer
Number theory
- We investigate mainly arithmetic functions (i.e., sequences of numbers) and matrices of integers
- Sequences and matrices appear almost everywhere in science and technology
- Our purpose is to explain the behaviour of these objects with theaid of algebraic, linear algebraic, number theoretic and discretemathematics tools
- We also apply computer-aided experimental methods jointly withJori Mäntysalo and others
- As an example let us mention that we could disprove latticetheoretically the famous Bourque-Ligh conjecture which claimsthat LCM matrices on GCD closed sets are always nonsingular.