Title: Nakayama functors are wannabe Serre functors
Abstract: Classic Auslander-Reiten theory is a neat tool used to paint a portrait of the category of modules over an Artinian ring. Nakayama functors play an important role in this painting. In suitable settings, the theory generalises to abelian categories, triangulated categories and their subcategories. In this talk, we will construct Nakayama functors on proper abelian subcategories. These categories, defined by Jørgensen in 2022, are generalisations of hearts of t-structures.
Talk is based on the following arXiv preprint: arXiv:2312.07323.
Given at: SSS conference (Lake District, UK), CHARMS Summer School (Versailles, France), University of Leeds Algebra Seminar, Köln Algebra and Representation Theory Seminar and Aarhus Homological Algebra Seminar.
Title: Rank functions in the framework of higher homological algebra
Abstract: Chuang and Lazarev introduced the concept of rank functions on triangulated categories as a generalisation of classical work by Cohn and Schofield on Sylvester rank functions. In this talk, we propose a generalisation of this notion to the broader framework of higher homological algebra.
Talk is based on the following arXiv preprint: arXiv:2405.19042.
Given at: BIREP seminar (Bielefeld University, Germany), Algebra and Representation Theory Oberseminar (Bonn, Germany), Köln Algebra and Representation Theory Seminar and Flash Talks in Representation Theory at NTNU.
Title: Homological algebra without grading
Abstract: Complexes are a fundamental object in homological algebra, and as such, grading is deeply ingrained in the subject. But what happens if we remove the grading from a complex? This leads us to the notion of a differential module. In this talk, we will explore how the finiteness of the injective dimension of a finitely generated module over a local commutative noetherian ring can be detected using this ungraded framework. As an immediate application, we will see that this perspective also detects when such a ring is Cohen–Macaulay. This approach offers a new perspective on a classical question posed by H. Bass.
Talk is based on the following arXiv preprint: arXiv:2504.15981
Given at: IDEAL conference (Leeds, UK), ARTA X (Poster) (Köln, Germany), Geometric Models in Representation Theory and Beyond (NTNU, Norway), Queer and Trans Mathematicians in Algebra and Representation Theory (Bonn, Germany).
Lectures at masterclasses
A geometric model for the derived category of a gentle algebra - Lecture 1. Course by Pierre-Guy Plamondon, given at Geometric Models in Representation Theory and Beyond (NTNU, Norway)
Cluster categories and thick subcategories - Lecture 1. Course by Sira Gratz, given at Interactions between homotopy theory and representation theory (Copenhagen, Denmark).