Research seminar Applied and Computational Topology

This seminar will normally take place on Thursdays 10:15–12:00 in room 02.12.020. For questions contact Magnus Botnan

List of upcoming seminars

Finite dimensional path algebras of quivers are a handy example for quasi-hereditary algebras; their module categories are well-understood. Notably, they admit the Jordan-Hölder property and have a finite number of indecomposable projectives or simples. Quivers indexed by Coxeter groups arise in the context of representation theory of Lie algebras and category 𝓞. We shall see how to use this connection to obtain an easy to employ description of Irving's shuffling functor.

Past seminars

We consider the approximability of maximization and minimization variants of the Morse matching problem, posed as open problems by Joswig and Pfetsch. We establish hardness results for Max-Morse matching and Min-Morse matching. In particular, we show that, for a simplicial complex with n simplices and dimension d ≥ 3, it is NP-hard to approximate Min-Morse matching within a factor of O(n1−ε), for any ε > 0. Moreover, using an L-reduction from Degree 3 Max-Acyclic Subgraph to Max-Morse matching, we show that it is both NP-hard and UGC-hard to approximate Max-Morse matching for simplicial complexes of dimension d ≥ 2 within certain explicit constant factors.

We start with a concrete version of the interleaving distance of Reeb graphs. Then we connect this approach to the interleaving distance of merge trees. Afterwards we discuss Reeb graphs and merge trees coming from Morse functions in more detail. Then we talk about some ideas for using branch decomposition trees to "approximate" the interleaving distance of merge trees. Afterwards we discuss how this might help with contour trees and some problems with that.

We start with a concrete version of the interleaving distance of Reeb graphs. Then we connect this approach to the interleaving distance of merge trees. Afterwards we discuss Reeb graphs and merge trees coming from Morse functions in more detail. Then we talk about some ideas for using branch decomposition trees to "approximate" the interleaving distance of merge trees. Afterwards we discuss how this might help with contour trees and some problems with that.

I will introduce the concept of filtered covers. I show that the persistent homology of a bounded metric space obtained from the Cech complex is the persistent homology of what I call the filtered nerve of the filtered Cech cover. Given a parameter δ with 0<δ≤1 I introduce the concept of a δ-filtered cover and show that its filtered nerve is interleaved with the Cech complex. Finally, I introduce a particular δ-filtered cover, the divisive cover. The special feature of the divisive cover is that it is constructed top-down. If we disregard fine scale structure and X is a finite subspace of euclidean space, then we obtain a filtered simplicial complex whose size is bounded by an upper bound independent of the cardinality of X. The time needed to compute this filtered simplicial complex depends linearly on the cardinality of X.

We consider Reeb graphs in a general topological setting, and the construction of distances between Reeb graphs that are stable, meaning that similar functions in the supremum norm have similar Reeb graphs. We define a graph edit distance and show that it is universal, providing an upper bound to any other stable distance.

Mapper is probably the most widely used TDA (Topological Data Analysis) tool in the applied sciences and industry. Its main application is in exploratory analysis, where it provides novel data representations that allow for a higher-level understanding of the geometric structures underlying the data. The output of Mapper takes the form of a graph, whose vertices represent homogeneous subpopulations of the data, and whose edges represent certain types of proximity relations. Nevertheless, the inherent instability of the output and the difficult parameter tuning make the method rather difficult to use in practice. This talk will focus on the study of the structural properties of the graphs produced by Mapper, together with their partial stability properties, with a view towards the design of new tools to help users set up the parameters and interpret the outputs.

Some recent results on persistent Betti numbers.

Computational topology has recently seen an important development toward data analysis, giving birth to Topological Data Analysis. Persistent homology appears as a fundamental tool in this field. It is usually computed from filtrations built on top of data sets sampled from some unknown (metric) space, providing "topological signatures" revealing the structure of the underlying space. When the size of the sample is large, direct computation of persistent homology often suffers two issues. First, it becomes prohibitive due to the combinatorial size of the considered filtrations and, second, it appears to be very sensitive to noise and outliers. The goal of the talk is twofold. First, we will briefly introduce the notion of persistent homology and show how it can be used to infer relevant topological information from metric data through stability properties. Second, we will present a method to overcome the above mentioned computational and noise issues by computing persistent diagrams from several subsamples and combining them in order to efficiently infer robust and relevant topological information.

The theory of interleavings lies at the very core of the theoretical foundations of persistent homology. With rising interest in more generalized indexing categories (zigzag, commutative ladders, circle valued maps, etc. ) comes the desire to extend this theory. In the first part of the talk I will survey interleavings in ordinary persistent homology, and, in particular, the celebrated algebraic stability theorem. Then I will report on recent work with Michael Lesnick (Princeton U.) where we extend this to zigzag persistence. Lastly I will show how the notion of interleavings can be generalized to general posets by means of cosheaves. The last part is joint work with Justin Curry (Duke) and Elizabeth Munch (U Albany).

I'll give a survey about my recent research at TUM.

I'll give an elementary introduction to the decomposition of representations of (infinite) linear quivers.

During this talk I will look into two different problems related to data analysis. The first one is the denoising of the data. Usual methods rely either on a precise knowledge of the noise model or several choices of parameter. I will present a new parameter free methods to denoise data assuming some minimal assumption. On a second, I will look into a difficult problem from the user point of view when working with persistence diagram. What does the persistence diagram means in the data? I will present new work on how to build good representatives for one-dimensional homology features and associate them with barcodes.

I will outline a novel computational method for persistent homology of finite metric spaces, which outperforms the previous state of the art by an order of magnitude. After that, I will survey several surprising connections and applications of persistent homology to various areas relevant to our collaborative research center.

Summer 2016

An algorithm is presented that constructs an acyclic partial matching on the cells of a simplicial complex from a vector-valued function defined on the vertices. The resulting acyclic partial matching is used to construct a reduced complex with the same multidimensional persistent homology as the original complex filtered by sublevel sets of the function. Numerical tests on triangle meshes show that the achieved rate of reduction is substantial.

Alexandrov spaces (with curvature bounded below) are a synthetic generalization of Riemannian manifolds with sectional curvature bounded below. In this talk I will discuss the geometric and topological properties of these metric spaces and how they arise in the context of topological data analysis.

2-stratifolds are the simplest 2-complexes which can be good models for topological data analysis. A 2-stratifold X contains a collection X1 of finitely many simple closed curves such that X –X1 is a 2-manifold and a neighborhood of each component C of X1 consists of sheets intersecting in C. In contrast to 2-manifolds there is no known classification of 2-stratifolds in terms of algebraic invariants. In this talk we will describe 2-stratifolds and their graphs and present efficient algorithms that detect whether certain 2-stratifolds have trivial first homology, whether they are simply connected, or whether they have the same homotopy type as a 2-sphere. This is joint work with F. González-Acuña, UNAM, Mexico and W. Heil, Florida State University.

An introduction to Auslander-Reiten Theory. This theory deals with the problem of understanding all indecomposable representations of a quiver with relations (or equivalently the indecomposable modules of certain algebras), as well as the "minimal" morphisms. In general, writing down all indecomposable modules turns out to be an impossible task, but one of the things AR-theory provides us is a way to find new indecomposable modules from old ones. In particular, AR-theory provides theoretical justification for topological data analysis.

Some recent results on persistent Betti numbers.

A summary of

We examine the flow space of proper CAT(0)-spaces and warped products of Riemannian manifolds and intrinsic metric spaces. Then we construct some embeddings of the flow space of non-constant generalized geodesics on Hadamard manifolds and proper CAT(0)-spaces using warped products.