Homological algebra (seminar course)
Scientific Disciplinary Sector (SSD)
MAT/02 - ALGEBRA
Secondo semestre dal Mar 7, 2022 al Jun 10, 2022.
This reading course is devoted to some topics in homological algebra and representation theory. Prerequisites: Representation Theory.
The first part will consist in an introductory lecture series "Quivers and representations" by Raquel Coelho Simoes
from Lancaster University in the period March 15-25, see
With this background, we will then focus on some algebraic and homological methods that are applied in topological data analysis. Given a set of data, which is viewed as a set of points in euclidean space, the idea is to detect the topological features which are relevant and thus persist when varying the scale of observation. This leads to the notion of persistent homology, which is investigated with methods from homological algebra and representation theory of quivers.
We will study some aspects of this algebraic approach to persistence by reading the following material:
• Chapters 1.1. and 1.2 in
Oudot, S.Y., Persistence theory: from quiver representations to data analysis. Mathematical Surveys and Monographs, 209. American Mathematical Society, Providence, RI, (2015).
• B. Blanchette, T. Brüstle, E.J. Hanson: Homological approximations in persistence theory, preprint 2021, https://arxiv.org/abs/2112.07632
Students interested in a short introduction to persistence homology may watch https://www.youtube.com/watch?v=DJSaTaEQWDA
Please note: the prerequisite Representation Theory is helpful, but not strictly necessary for attending this course.
Students actively participate in the course and deliver a seminar talk.