BIREP – Representations of finite dimensional algebras at Bielefeld

Hideto Asashiba (Shizuoka)

Computations of persistence diagrams by almost split sequences

This is a joint work with K. Nakashima and M. Yoshiwaki.

Let $A$ be a finite-dimensional algebra over a field $k$ with a quiver presentation $A = kQ/I$ ($Q$ a finite quiver, $I$ an admissible ideal of $kQ$). We assume that all $A$-modules considered here are finite-dimensional. We set $\text{(Lst)}$ to be a complete set of representables of isoclasses of indecomposable $A$-modules. Thus $\text{(Lst)}$ is identified with the vertex set of the AR-quiver of $A$. Denote by $\text{(Nat)}$ the set of non-negative integers. Then the Krull-Schmidt theorem states the following:

For each $A$-module $M$, there exists a unique map $d_M : \text{(Lst)} \to \text{(Nat)}$ such that

(*) $M$ is isomorphic to the direct sum of $L^{(d_M(L))}$, where $L$ runs through $\text{(Lst)}$.

(*) is called the indecomposable decomposition or the persistence diagram of $M$. Note that $M$ is isomorphic to $N$ if and only if $d_M = d_N$ for all $A$-modules $M$ and $N$, i.e., the map $d_M$ is a complete invariant of $M$ under isomorphisms. Note further that since $M$ is finite-dimensional, the support $\operatorname{supp}(d_M):= \{L \in \text{(Lst)} \,|\, d_M(L) \text{ is nonzero}\}$ of $d_M$ is a finite set. We give a way to compute the persistence diagram of a module $M$ by giving a formula of $d_M$ with a very simple proof using AR-theory, and also a finite subset $S_M$ of $\text{(Lst)}$ that contains $\operatorname{supp}(d_M)$.

This formula uses dimensions of Hom-spaces from finitely many indecomposable modules to $M$, which can be calculated by ranks of some matrices over $k$. We give explicit formula of $d_M$ for some examples of $A$ such as Kronecker algebra and commutative ladder algebra of finite type. The same formula of $d_M$ was given by Dowbor—Mroz in 2007 with a different proof.

Ulrich Bauer (München)

Persistent homology: from theory to computation

In this talk, I will survey some recent results on theoretical and computational aspects of applied topology. I will focus on three aspects of persistent homology as a topological descriptor: its use for the inference and simplification of topological features, its stability with respect to perturbations of the data, and efficient methods for its computation on a large scale.

These questions will be motivated and illustrated by concrete examples and problems, such as

- reconstruction of a shape and its topological properties from a point cloud,

- denoising of isosurfaces for the visualization of medical images,

- faithful simplification of contours lines of a real-valued function, and

- the existence of unstable minimal surfaces.

Magnus Bakke Botnan (Amsterdam)

The Computational Complexity of Comparing Quiver Representations

It has been shown that one can determine in polynomial time if two representations of a quiver (with relations) are isomorphic. But what about checking if a representation injects as a sub-representation of another representation, or determining if two modules are approximately isomorphic? Here the notion of an approximate isomorphism is made precise through the language of interleavings.

In this talk I will report on recent work with H. B. Bjerkevik and M. Kerber where we show that a wide range of decision problems (including the ones mentioned above) are NP-Hard when the quiver is a commutative grid.

Basic concepts from computational complexity theory will be introduced.

Mickaël Buchet (Graz)

On Interval Decomposability of 2D Persistence Modules

In persistent homology of filtrations, the indecomposable decompositions provide the persistence diagrams. In multidimensional persistence it is known to be impossible to classify all indecomposable modules. One direction is to consider the subclass of interval-decomposable persistence modules, which are direct sums of interval representations. We introduce the definition of pre-interval representations, a more algebraic definition, and study the relationships between pre-interval, interval, and indecomposable thin representations. We show that over the "equioriented" commutative 2D grid, these concepts are equivalent. Moreover, we provide an algorithm for determining whether or not an nD persistence module is interval/pre-interval/thin-decomposable, under certain finiteness conditions and without explicitly computing decompositions.

Emerson G. Escolar (Kyoto)

Every 1D Persistence Module is a Restriction of Some Indecomposable 2D Persistence Module

A recent work by Lesnick and Wright proposed a visualisation of 2D persistence modules by using their restrictions onto lines, giving a family of 1D persistence modules. We explore what 1D persistence modules can be obtained as a restriction of indecomposable 2D persistence modules to a single line. To this end, we give a constructive proof that any 1D persistence module can in fact be found as a restriction of some indecomposable 2D persistence module. More generally, we show that any finite-rectangle-decomposable nD persistence module can be found as a restriction of some indecomposable (n+1)D persistence module. As a consequence of our construction, we are able to exhibit indecomposable persistence modules whose support has holes.

Jeffrey Giansiracusa (Swansea)

Multiparameter persistence vs parametrized persistence

One of the key properties of 1-parameter persistent homology is that its output can entirely encoded in a purely combinatorial way via persistence diagrams or barcodes. However, many applications of topological data analysis naturally present themselves with more than 1 parameter. Multiparameter persistence suggests itself as the natural invariant to use, but the problem here is that the moduli space of multiparameter persistence diagrams has a much more complicated structure and no combinatorial diagrammatic description is possible. An alternative approach was suggested by work of Giansiracusa-Moon-Lazar, where they investigated calculating a series of 1-parameter persistence diagrams as a second parameter is varied. In this talk I will discuss work in progress to produce a refinement of their perspective, making use the Algebraic Stability Theorem for persistent homology and work of Bauer-Lesnick on induced matchings.

Yasuaki Hiraoka (Kyoto)

Persistent homology and its applications in materials science

In this talk, I will give a survey about our recent activity on generalization of persistent homology from a viewpoint of representation theory and its applications to materials science.

Michael Lesnick (Albany)

Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent Homology

This is joint work with Matthew Wright.

Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm takes as input a short chain complex of free modules $\displaystyle F_2\xrightarrow{\partial_2} F_1 \xrightarrow{\partial_1} F_0$ such that $M\cong \ker{\partial_1}/\operatorname{im}{\partial_2}$. It runs in time $O(\sum_i |F_i|^3)$ and requires $O(\sum_i |F_i|^2)$ memory, where $|F_i|$ denotes the size of a basis of $F_i$. We observe that, given the presentation computed by our algorithm, the bigraded Betti numbers of $M$ are readily computed. These algorithms have been implemented in RIVET, a software tool for the visualization and analysis of two-parameter persistent homology. In experiments on topological data analysis problems, our approach outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin.

Rene Marczinzik (Stuttgart)

Representation theory of algebras - discovering new results with the help of QPA

We present several results and problems that were suggested by experiments with QPA for important classes of finite dimensional algebras such as hereditary algebras, incidence algebras of posets and Frobenius algebras.

Killian Meehan (Kyoto)

Zigzag Metrics from Quiver Theory

This talk will present enough background from TDA and Quiver Theory to discuss a new zigzag bottleneck metric that entwines the two fields. Common metrics on persistence modules (zigzag and otherwise) will be discussed, as well a brief survey of Auslander-Reiten quivers, from which the new metric will be derived. The various features of this and other zigzag metrics from the literature will be compared, including stability theorems.

Steffen Oppermann (Trondheim)

Cotorsion torsion triples and the representation theory of filtered hierarchical clustering

This talk is based on joint work with U. Bauer, M. B. Botnan, and J. Steen.

Filtered hierarchical clustering leads to representations of grids, where all structure maps in one direction are epimorphisms. In my talk we will observe that we can use the torsion and cotorsion pairs associated with a tilting module in order to describe the subcategories described by this epimorphism property. In particular we will be able to conclude when they are finite, tame, or wild.

Steve Oudot (Palaiseau)

Two decomposition results for bipersistence modules

Thanks to recent work by Botnan and Crawley-Boevey, we know that all pointwise finite-dimensional representations of the plane R^2 decompose into indecomposables. A question the arises immediately is to determine the shape of the indecomposables. As it is a difficult one in full generality, we want to tackle it from different perspectives. One of them, inspired by applications in topological data analysis, is to try to work out "local" or easily-verifiable conditions under which certain types of summands will (or will not) appear in the decomposition. In this talk I will present two results in this vein:

- first, a condition called "middle exactness", which guarantees that the indecomposables are indicator modules supported on blocks (i.e. upper-right or lower-left quadrants, horizontal or vertical bands);

- second, a somewhat weakened condition which guarantees that the indecomposables are indicator modules supported on axis-aligned rectangles. I will try to illustrate the interest of these conditions through recent contributions on the stability of bipersistence modules and on the connection of these modules to sheaves of vector spaces over the real line.

This talk is based on joint work with J. Cochoy on the one hand, with M. Botnan on the other hand.

Øyvind Solberg (Trondheim)

Examples of using QPA in research

We will review some examples of how QPA has been used in research with a particular focus on representation theory. This will involve finding counterexamples, verifying counterexamples, exploring numerous examples and finding the "right thing" to prove.

Michio Yoshiwaki (Kyoto)

An algebraic stability theorem for the derived category of persistence modules

The algebraic stability theorem is an important part of the stability theorem in the theory of persistent homology. It guarantees that the persistence diagram is robust to changes in the given persistence module.

Our motivation is to derive an algebraic stability theorem for zigzag persistence modules. Botnan-Lesnick proved such a theorem by embedding the category of zigzag persistence modules into that of 2D block decomposable persistence modules. On the other hand, Meehan-Meyer introduced the Auslander-Reiten graph distance and proved a similar theorem using this distance. Here, we adopt a different approach. It is known that the derived categories of persistence modules and of zigzag persistence modules are equivalent. Thus, our strategy is to derive an algebraic stability theorem for zigzag persistence modules from the known case by passing through the derived category.

In this talk, we will discuss an algebraic stability theorem for the derived category of persistence modules.

This talk is based on joint work with Hiraoka.