PITA

Problems I Think About – 02. November, 656 Simastuen

Marius Nielsen : A machinery for deforming exact categories

Abstract: In joint work in progress with Erlend Børve we produce to any exact \(\infty\)-category \(\mathcal{E}\) a stable \(\infty\)-category \(\mathrm{Syn}_{\mathcal{E}}\) satisfying the universal property that \(\mathrm{Syn}_{\mathcal{E}}\) is the initial presentable stable \(\infty\)-category, receiving an exact functor from \(\mathcal{E}\). This provides a strong relationship between homological algebra and stable homotopy theory with the help of deformation theory. In this talk I will give an gentle introduction to these ideas and I present some of our results and speculations. If time permit I will also discuss how to study derived functors from this perspective and generalized Grothendieck spectral sequence.

Problems I Think About – 12. October, 656 Simastuen

Trygve Poppe Oldervoll : The operadic principle in symplectic topology

Abstract: Invariants defined by counting J-holomorphic curves have become essential tools in symplectic topology. These invariants assemble into rich algebraic structures, such as \(A_\infty\)-algebras and \(A_\infty\)-categories. In this talk, I will give an “analysis free” introduction to some of these invariants, and motivate why certain algebraic structures show up. In particular, I will introduce the operadic principle, which can be stated as follows: The algebraic nature of an invariant defined by counting J-holomorphic curves is determined by the operadic structure of the moduli space of domains. Motivating examples throughout the talk will be Floer homology and Fukaya categories.

Problems I Think About – Wednesday 26 April, 11:15-12:15, S1

Torgeir Aambø : 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.

Problems I Think About – Wednesday 26 April, 11:15-12:15, Simastuen

Marius Verner Bach Nielsen : 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.

Problems I Think About – Wednesday 12 April, 11:15-12:15, S1

Jacob Fjeld Grevstad : 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.

Problems I Think About – Wednesday 08 March, 11:15-12:15, Simastuen

William Hornslien : 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.

Problems I Think About – Wednesday 22 February, 11:15-12:15, Simastuen

Johanne Haugland : 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.

Problems I Think About – Wednesday 8 February, 11:15-12:15, Simastuen

Clover May : 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.

Problems I Think About – 12. March, 734

Peder Thompson : Trace modules and enveloping classes

Abstract: The idea of trace modules appears in a number of settings, often related to endomorphism invariance. For example, the classic notion of a quasi-injective module is simply a module that is trace in its injective envelope. I will introduce some general properties of trace modules and sketch some ongoing joint work with Haydee Lindo on relating trace modules and enveloping classes of modules over any ring. In particular, we will give some characterizations of rings (such as semi-simple, regular, and Gorenstein rings) in terms of their ideals being trace in certain envelopes. I will also discuss a number of open questions related to the theory of trace modules, such as their relationship to the Auslander-Reiten conjecture.

Problems I Think About – 27. February, 734

Didrik Fosse : A combinatorial rule for tilting mutation

Abstract: Tilting objects are important in the study of the representation theory of algebras. For example, Rickard’s derived Morita theorem tells us that two rings are derived equivalent precisely when there exists a certain tilting object over one of them. Tilting mutation is a way to modify a known tilting object in such a way that you get a new tilting object, which thus allows us to create an algebra that is derived equivalent to a given algebra. In this talk we will develop a set of combinatorial rules for tilting mutation of algebras which are given as path algebras of a certain class of quivers with relations. We will also see an explicit example of how we can use these rules to calculate a sequence of derived equivalent algebras, which in turn can be used as a tool for identifying when two given algebras are derived equivalent. If we manage to find such a sequence which starts with one of the algebras and ends with the other, then they must be derived equivalent.

Problems I Think About – 13. February, 734

Mads Hustad Sandøy : Generalized T-Koszul algebras

Abstract: Koszul algebras are in some sense the graded algebras that are easiest to understand while not being semi-simple. Moreover, they abound in nature. We will review key properties of Koszul algebras before introducing a generalization of them, namely generalized T-Koszul algebras, and sketch connections to \(n\)-hereditary algebras. The novel parts of this talk are based on joint work with Johanne Haugland.

Problems I Think About – 30. Jan 2020, 734

Louis-Philippe Thibault : The quiver of \(n\)-hereditary algebras

Abstract: Auslander-Reiten theory is fundamental in the study of modules over Artin algebras. In this setting, the number ‘2’ appears often. For example, the Auslander correspondence gives a bijection between Morita-equivalence classes of Artin algebras of finite representation type and algebras satisfying \(gl.dim A \leq 2 \leq dom.dim A\). It is thus natural to generalize some of the central ideas to a “higher dimensional” Auslander-Reiten theory. This was introduced by Iyama in 2004 and has generated many interesting ideas over the years.

One important concept is that of \(n\)-hereditary algebras, which enjoy some of the key properties of hereditary algebras in the context of higher AR-theory. They are divided into two classes: the \(n\)-representation-finite and the \(n\)-representation-infinite algebras. A lot of research has been done on these objects but, in contrast to classical hereditary algebras, very little is known as to what their quivers actually look like. In this talk, I will introduce (higher) AR-theory, give some examples of \(n\)-hereditary algebras and ask whether we can deduce some of their quiver properties. These questions are part of a project with Mads.

Problems I Think About – 16. Jan 2020, 734

Johanne Haugland : Extriangulated categories – functors and subcategories

Abstract: In this talk we give a basic introduction to extriangulated categories. Extriangulated categories were first presented in a paper by Hiroyuki Nakaoka and Yann Palu in 2016, and has turned out to be a useful framework to unify and extend known concepts and results. We discuss the relevant definitions and explain how extriangulated categories is a simultaneous generalization of exact and triangulated categories. After this, we move on to the question of what the correct definitions of extriangulated functors and extriangulated subcategories should be.