Program

Linear Invariants for poset representations

Let \(P\) be a poset. An \( \mathbb Z \)-invariant on \( \operatorname{mod} P \) is a \( \mathbb Z \)-linear map \( \operatorname{K}_0^{\rm sp}(\operatorname{mod} P) \to \mathbb Z^N \). In this talk, I will dicuss several examples of invariants that appear in persistence theory coming either from relative exact structures or from order embeddings. This is a joint work with Thomas Brüstle and Eric Hanson.

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Indecomposables in multiparameter persistence

I will discuss various aspects of multi-parameter persistence related to representation theory and decompositions into indecomposable summands, based on joint work with Magnus Botnan, Steffen Oppermann, Johan Steen, Luis Scoccola, and Benedikt Fluhr.

A classification of indecomposables is infeasible; the category of two-parameter persistence modules has wild representation type. We show that this is still the case if the structure maps in one parameter direction are epimorphisms, a property that is commonly satisfied by degree 0 persistent homology and related to filtered hierarchical clustering. Furthermore, we show that indecomposable persistence modules are dense in the interleaving distance, and that being nearly-indecomposable is a generic property of persistence modules. On the other hand, the two-parameter persistence modules arising from interleaved sets (relative interleaved set cohomology) have a very well-behaved structure that is encoded as a complete invariant in the extended persistence diagram. This perspective reveals some important but largely overlooked insights about persistent homology; in particular, it highlights a strong reason for working at the level of chain complexes, in a derived category.

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Tiled surfaces, string algebras and laminations

We consider tilings of surfaces. We show how they can be used as a geometric model for string algebras, a large class of algebras satisfying monomial relations. We can also use the tiled surfaces to consider a new type of laminations, called T-laminations. These give us certain infinite dimensional modules for locally gentle algebras.

This is joint work with R. Coelho Simoes and with B. Marsh.

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The homotopy theory of multiparameter interlevel sets persistence

It has been noticed for a long time that 2-persitence modules arising from interlevel sets of real valued function carry extra homotopical structure ('middle exactness', 'Mayer-Vietoris systems', etc..), that allow to obtain interval decomposition theorems for these. In this ongoing work joint with Grégory Ginot, I propose to generalise these results to the multiparameter setting. For identifying the appropriate homotopical structure carried by \( (n+1) \)-parameters persistence modules coming from interlevel sets filtration of \( \mathbb{R}^n \) valued functions, it will be necessary to take an infinity categorical point of view, and take inspiration from Goodwillie calculus of functors.

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How to prove hardness for Lp distances

In recent joint work with Håvard Bjerkevik initiated at CAS, we show that computing the p-presentation distance is NP-hard for merge trees and multiparameter persistence modules, two central objects in topological data analysis. The former is a representation of \( A_n \) in finite sets, and the latter is a representation of a commutative grid in finite-dimensional vector spaces. While the technical details differ, the two proofs follow the same general strategy. In this talk, I will detail this strategy and outline the proof for the case of merge trees.

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Decomposing rooted tree modules with the elder rule

In many applications of topological data analysis only zero-dimensional homology is relevant. The zero-dimensional homology of a space is conceptually and computationally simpler than higher dimensional homology since it can be characterized as the linearization of the set of path-connected components of the space. In particular, the additive hull of all zero-dimensional homology modules of a filtration indexed by a poset \( P \) will in general form a proper subcategory of \( \operatorname{mod} P \) (to the contrary, \( \operatorname{add} \operatorname{Im} {\rm H}_i \) is dense for \( i \) greater than zero).

We show that when the indexing poset \( P \) is a rooted tree, only finitely many indecomposables can be obtained as zero-dimensional homology of a filtration. In fact, all these indecomposables are reduced rooted tree modules in the sense of Kinser. Moreover, we give an algorithm for the efficient decomposition of rooted tree modules, generalizing the so-called elder rule used in TDA for linear quivers.

This is a report on joint work with Riju Bindua and Luis Scoccola, see arXiv:2411.19319

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The g-representation type of concealed algebras and incidence algebras of posets

The representation type of an algebra is a measure of complexity when classifying modules up to isomorphism. A rougher notion of g-representation type is currently being envisioned. One may define the notions of g-finite representation type and g-tame representation type in terms of the g-vector fan. In this talk, we highlight certain cases where traditional representation type and g-representation type coincide. We outline a proof that no wild concealed algebra is g-tame.

The behaviour for concealed algebras has direct consequences for incidence algebras of posets. A finite poset is of g-finite representation type if and only if it is of finite representation type. If the poset is also simply connected, then the incidence algebra is of g-tame representation type if and only if it is of tame representation type.

Our original results are joint with Jacob Grevstad and Endre Rundsveen arXiv:2407.17965.

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Realizations of posets and their homological algebra

I will explain the role that upper semilattices play in Topological Data Analysis (TDA) and introduce a construction called realisation, which shares many of the desirable properties of upper semilattices in the context of TDA.

For example, I will demonstrate that the homological dimension (the length of the minimal free resolution) and the Betti numbers of tame vector space-valued functors indexed by both realisations and upper semilattices can be computed using Koszul complexes. As a result, calculating these invariants for functors indexed by these two types of posets can be done directly, without the need to construct explicit resolutions.

Additionally, I will highlight why the homological algebra of realisations associated with dimension-1 posets (such as trees) is particularly manageable.

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Metric completions of discrete cluster categories

Methods to construct new triangulated categories from old are few and far between. Recently, Neeman defined such a method emulating the completion of a metric space. However, computing these completions often requires using the properties of an already completed ambient triangulated category, like the derived category. In this talk, based on joint work with Sira Gratz, we present an example from cluster theory that avoids this requirement by focusing on categories with combinatorial models. In particular, we show that categorical completions of discrete cluster categories are intimately connected to topological completions of their associated disc model.

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Coxeter diagrams give one presentation for each finite (real) reflection group and its associated braid group. For some types, new families of presentations of these groups have been found using cluster algebras and their surface models. This talk will give an overview of a class of such families and introduce its complex analogue for complex braid groups. This is based on joint work with Bethany Marsh.

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Harder-Narasimhan filtrations of persistence modules

The Harder-Narasimhan types are a family of discrete isomorphism invariants for representations of finite quivers. We evaluate their discriminating power in the context of persistence modules over a finite poset, including multiparameter persistence modules (over a finite grid). In particular, we introduce the skyscraper invariant and proved amongst other that it is strictly finer than the rank invariant. In order to study the stability of the skyscraper invariant, we extend its definition from the finite to the infinite setting and consider multiparameter persistence modules over Zn and Rn. We then establish an erosion-type stability result for the skyscraper invariant in this setting. This talk is based on the content of the preprints arXiv:2303.16075 (with E. Jacquard, V. Nanda and U. Tillmann) and arXiv:2406.05069.

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Cotorsion pairs, thick subcategories and g-finiteness in the category of projective presentations

An algebra is said to be g-finite if it admits finitely many isomorphism classes of basic \( \tau \)-tilting pairs. This notion was introduced and thoroughly studied by L. Demonet O. Iyama and G. Jasso, who showed that this property is equivalent to the module category admitting finitely many isomorphism classes of bricks, finitely many functorially finite torsion classes, and equivalent to all torsion classes being functorially finite. Many of these concepts and their relationships have been shown to have counterparts in the extriangulated category of 2-term complexes of projective modules. In this talk, we introduce new equivalent conditions to an algebra being g-finite in the context of the category of 2-term complexes. Namely, we establish that being g-finite is equivalent to the category of 2-term complexes admitting finitely many thick subcategories, finitely many complete cotorsion pairs and equivalent to all cotorsion pairs being complete.

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Mutation of \( \tau \)-exceptional sequences

By the work of Crawley-Boevey and Ringel, the set of complete exceptional sequences over a finite-dimensional hereditary algebra admits a transitive braid group action. This can also be viewed as a "mutation theory" for exceptional sequences. In this talk, we discuss recent joint work with Aslak Buan and Bethany Marsh which extends this into a mutation theory for (complete) tau-exceptional sequences over an arbitrary finite-dimensional algebra. In addition to giving the formulas for this mutation, we discuss the existence of non-mutable sequences, the problem of transitivity, and the (lack of) braid relations.

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Quiver Heisenberg algebras and radical power approximations

In my talk I will introduce the quiver Heisenberg algebra associated to an acyclic quiver Q. It is a certain kind of central extension of the preprojective algebra of Q. Just as the preprojective algebra, the quiver Heisenberg algebra encodes important information about the path algebra of Q. By considering dimensions they have a combinatorial interpretation. I will explain how the quiver Heisenberg algebra gives rise to a universal Auslander-Reiten sequence from which all others can be obtained. I will also explain how it can be used to construct minimal left and right approximations with respect to the powers of the radical ideal. Several homological properties of the quiver Heisenberg algebra will also be discussed. The talk is based on joint work with Hiroyuki Minamoto.

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What are triangulations in the surface model?

To any gentle algebra we can associate an oriented marked surface with a dissection, giving us a geometric model of the bounded derived category of that algebra, where curves taken up to homotopy correspond to indecomposable objects.

In joint work with Haugland, Schiffler and Schroll, we investigate the natural question: What do the triangulations of such a surface correspond to?

In answering the question, we arrive at a definition of maximal almost rigid subcategories in the bounded derived category. Furthermore we study interplay between algebraic and geometric mutation.

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Estimating the persistent homology of \( \mathbb{R}^n \)-valued functions using functional-geometric multifiltrations

This talk is about the following applied problem in topological data analysis, where the structure and stability of persistence modules plays a key role: given an unknown \( \mathbb{R}^n \)-valued function \( f \) on a metric space \(X\), how can we approximate the persistent homology of \( f \) from a finite sampling of \( X \) with known pairwise distances and function values? This question was answered more than a decade ago in the case \( n=1 \), assuming \( f \) is \( c \)-Lipschitz and \( X \) is a sufficiently regular geodesic metric space, and using filtered geometric complexes with fixed scale parameter for the approximation. Here we consider the question for arbitrary \( n \), which has remained open to this day, and we answer it using a class of multiparameter filtrations called functional-geometric multifiltrations, which have gained attraction recently and have been studied for their stability properties.

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Convex geometry for fans of abelian categories

Arising in cluster theory, the g-vector fan is a convex-geometric invariant encoding the mutation behaviour of clusters. In representation theory, the g-vector fan encodes the mutation theory of support tau-tilting objects or, equivalently, two-term silting objects.

In this talk, we will introduce cofans as the convex-geometric object associated to c-vectors and describe the heart fan as the dual of the c-vector cofan. The heart fan generalises and completes the g-vector fan. This convex geometric invariant encodes many important homological properties: e.g. one can detect from the convex geometry whether an abelian category is length, whether it has finitely many torsion pairs, and whether a given Happel-Reiten-Smalø tilt is length. This talk will be a report on joint work with Nathan Broomhead, David Ploog and Jon Woolf.

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Skew-gentle algebras through skew-group A-infinity categories

Derived categories of gentle algebras have a nice geometric model using A-infinity categories defined from surfaces. Viewing skew-gentle algebras as gentle algebras with the action of an involution, we can study them through surfaces with the action of an involution. Several approaches have been put forward recently; I will present on which uses skew-group A-infinity categories.

In this talk, I will emphasize some of the technical aspects of this strategy, in particular in working in the split closure of an A-infinity category. This will allow us to describe objects and morphisms in the derived category of a skew-gentle algebra using (tagged) curves on a surface.

This is a joint work with Claire Amiot, which was done in large part during the program at the CAS in Oslo.

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Hochschild cohomology, a generalization of a method of Etingof and Eu, and higher representation finite algebras

While computing Hochschild cohomology can be hard work, Etingof and Eu (2006) showed that it could be done easily for preprojective algebras associated to ADE Dynkin diagrams, at least if you only wanted to know the graded vector space structure of each Hochschild cohomology group. This method has since been used by Evans and Pugh (2012) on higher preprojective algebras that arise from “higher” ADE Dynkin diagrams.

In this talk, we present a generalization of the method used by Etingof and Eu (jt. with Jon W. Anundsen) obtained in part through a theory of projective resolutions for almost T-Koszul algebras (jt. with Johanne Haugland). In many cases, this generalization is easier to use, and we present applications to computations of Hochschild cohomology for higher preprojective algebras. Moreover, time permitting, we relate our results to questions and conjectures concerning higher representation finite algebras.

This is based on joint work with Jon Wallem Anundsen and joint work with Johanne Haugland.

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Algebras derived equivalent to skew-gentle algebras

Gentle algebras and skew-gentle algebras are two closely related classes of finite dimensional algebras which have received much attention in recent years. One of the reasons for this is that they are closely related to many other areas of mathematics such as cluster algebras and homological mirror symmetry. What is remarkable about both classes is that not only their module categories but also their derived categories are tame. However, one fundamental difference between the two classes is that while gentle algebras are closed under derived equivalence, skew-gentle algebras are not. In this talk we will use dissections of orbifold surfaces to define a new class of finite dimensional algebras derived equivalent to skew-gentle algebras.

This is based on joint work with Severin Barmeier and Zhengfang Wang, as well as joint work in progress with Severin Barmeier, Cheol-Hyun Cho, Kyoungmo Kim, Kyungmin Rho and Zhengfang Wang.

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The index: from rigid to non-rigid

The index with respect to cluster tilting subcategories in triangulated categories plays a key role in the categorification of cluster algebras. In previous work with Peter Jørgensen, we defined the index with respect to a contravariantly finite and rigid subcategory of a triangulated category using extriangulated categories. This talk is about current work, joint with Francesca Fedele and Peter Jørgensen, in which we remove the rigid restriction. In particular, this index theory works in completed discrete cluster categories of type A for fan triangulations, which are often not rigid. I’ll give some theory and some examples.

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Birth and death curves for representations of a grid poset

In the setting of representations of a two-dimensional grid poset, I will explain how to make a formal definition of "birth curves" and "death curves" which match the obvious intuition for direct sums of spread modules. I will go on to discuss a kind of pairing between birth curves and death curves, and important properties of the invariant of representations which counts the number of birth curves. This is joint work with Thomas Brüstle, Steve Oudot, and Luis Scoccola.

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We consider an additive generator X of the fundamental domain of cluster category C associated to a Dynkin quiver. Let Γ be the endomorphism algebra of X when considering cluster category morphisms, and let Σ be the endomorphism algebra of the same X when considering only derived category morphisms. We show that the higher preprojective algebra of Σ is isomorphic to Γ. We extend this result to both k-cluster categories and higher cluster categories as well.

Joint with Osamu Iyama and Emre Sen.

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Higher \( \tau \)-tilting theory for linear Nakayama algebras

\( \tau \)-tilting theory and torsion theory are well-established areas of study in representation theory. In recent years, efforts have been made to extend these concepts into the framework of higher homological algebra. In particular, Jørgensen introduced d-torsion classes, while various notions of higher analogues of \( \tau \)-tilting modules have been introduced by several authors (Jacobsen - Jørgensen, Martinez - Mendoza, Zhou - Zu and others). This talk aims to present an explicit classification of d-torsion classes and certain higher \( \tau \)-tilting modules for truncated linear Nakayama algebras. Additionally, the classification will serve to highlight the distinct properties of the different proposed definitions of higher \( \tau \)-tilting modules. This is joint work with Endre S. Rundsveen arXiv:2410.19505).

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Tilting theory for extended module categories

Let \( A \) be a finite-dimensional algebra and \( m \) be a positive integer. The \( m \)-extended module category of \( A \) is defined as the full subcategory of the bounded derived category \( {\rm D^b}(\operatorname{mod} A) \) of finitely generated \( A \)-modules consisting of complexes whose cohomologies are concentrated in degrees between \( -(m-1) \) and \( 0 \). We show the existence of Auslander-Reiten (ex)triangles in \( m \)–\(\operatorname{mod} A \) and bijections between positive tau-tilting pairs in \( m \)–\( \operatorname{mod} A \), \( (m+1) \)-term silting complexes, and functorially finite s-torsion pairs in \( m \)–\( \operatorname{mod} A \).

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