Torgeir Aambø, 10. May 2023
Title: Torsion in topology and algebra
Abstract: Torsion is a concept used in algebra, topology and their topology-focused intersection: algebraic topology. One can compare topological and algebraic structures using abstract homology theories, and one can ask how torsion interacts under these comparisons. During the talk we will set up a general framework for these questions and put forward several ideas about how this should behave — particularily focusing on ideas trying to relate TTF triples, chromatic homotopy theory and (co)monads.
Marius Verner Bach Nielsen, 26. Apr 2023
Title: Line bundles, tensor-invertible objects and how homotopy theory helps us compute them
Abstract: In this talk I will introduce the picard group functor. This functor associates to a symmetric monoidal category $\mathcal{C}$ the group of isomorphism classes of “linebundles” in $\mathcal{C}$ which is a strong invariant of the category. However this group is generally not that easy to compute on its own.
Fear not, this is where homotopy theory comes into play. Using homotopy theory I will introduce the “Picard space” of $\mathcal{C}$. This is an infinite loopspace with strong formal properties where the path components form a group, which canonically identifies with the Picard group of $\mathcal{C}$. In particular, this picard space functor takes limits to homotopy limits which allows for computations using spectral sequences.
Finally, I will relate this to my research where I try to compute picard groups of categories of mackey functors. Which has strong relations to quiver representations.
Jacob Grevstad, 26. Apr 2023
Title: Lower homological algbera
Abstract: An important result in higher homological algebra is the Auslander—Iyama correspondence, relating n-cluster tilting modules to algebras with specific homological dimensions. This has been generalized in various ways, relating higher homological algebra data to homological dimensions. For n=1 this recovers classical results in homological algebra, but for n=0 there is no sensible notian of 0-cluster tilting module. This means that there is no sensible setting to do “lower homological algebra”, but we need not give up. By constraining homological dimensions approriately and study the resulting algebras one gets a proxy for studying this nonexistent theory.
In this talk I will talk about the problem of classifying 0-minimal Auslander—Gorenstein algebras, and give a spurious connections to knot theory.
William Hornslien, 08. Mar 2023
Title: Homotopies of toric varieties and a linear algebra problem I don’t know how to solve
Abstract: Motivic homotopy theory is the homotopy theory of smooth varieties. My favorite lemma in motivic homotopy theory is called “Jouanolou’s trick”. It states that any smooth variety is “homotopic” to a smooth affine variety, also called their Jouanolou device. Toric varieties are rich class of algebraic varieties that are combinatorial in nature, this often makes computations easier, and they serve as a nice testing ground for theorems. In this talk I’ll provide an algorithm for computing the Jouanolou device of any smooth toric variety. A simple (?) linear algebra problem will also be presented.
Johanne Haugland, 22. Feb 2023
Title: Structure-preserving functors in higher homological algebra
Abstract: Tools from homological algebra play a fundamental role both in algebraic topology and representation theory. In this talk, I give an introduction to a higher-dimensional generalization of classical homological algebra. I furthermore discuss the problem of describing what it means for higher homological structures to relate to each other in a compatible way.
The talk is based on ongoing joint work with Raphael Bennett-Tennenhaus, Mads H. Sandøy and Amit Shah.
Clover May, 08. Feb 2023
Title: Quivers and equivariant homotopy theory
Abstract: One of the problems I’m thinking about these days is describing the bounded derived category of representations over a certain quiver with relations. My interest in this quiver stems from equivariant homotopy theory. I will talk about the problem at hand, give some background about equivariant homotopy theory, and describe the winding path that has taken me from equivariant homotopy theory to quivers.