Minicourses

Ulrich Bauer

An introduction to persistent homology: inference, stability, and computation

Abstract: Persistent homology, the homology of a filtration, is described up to isomorphism by the persistence diagram (barcode), which encodes the structure of its indecomposable summands.

This course will give an overview of several theoretical and computational aspects of persistence. I will focus on three aspects of persistent homology as a topological descriptor:

- One of the initial motivations for studying persistent homology is given by the problem of inferring the homology of a compact set from a finite sample of points, which has a simple solution circumventing the need for topologically reconstructing the shape.

- A fundamental theorem states that small changes in the input data lead only to small changes in the persistence barcode. We will study a constructive algebraic approach to this result that admits a high level of generality.

- The efficient computation of persistent homology is key to its applicability for data analysis. We will discuss efficient algorithms and optimizations that lead to massive improvements from the first to the latest generation of implementations.

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

- homology inference from synthetic and real-world data,

- 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.

Daniel Rosen

Symplectic geometry and Hamiltonian dynamics

Abstract: These lectures will provide an introduction to modern symplectic geometry. Symplectic geometry grew out of the mathematical formalism of classical mechanics, where symplectic manifolds play the role of phase spaces of mechanical systems, and Hamiltonian diffeomorphisms are admissible mechanical motions of the phase space. The group of all Hamiltonian diffeomorphisms carries a famous conjugation invariant norm, the Hofer norm, which measures, roughly speaking, the minimal energy required to generate a given mechanical motion. In the 1980s Floer homology, a powerful tool for the study of Hamiltonians diffeomorphisms, was developed by Andreas Floer. We will introduce the Hamiltonian group and the Hofer norm, and present applications of Floer homology to Hamiltonian dynamics and Hofer geometry.

Michael Usher

Persistence modules in symplectic topology

Abstract: This mini-course will describe some applications of ideas related to persistence modules to questions about automorphisms and embeddings of symplectic manifolds. After a general introduction to the relevant notions, I will explain how the Hamiltonian Floer persistence module is used to understand the Hofer distance between Hamiltonian diffeomorphisms, and the symplectic homology persistence module gives information about symplectic isotopy classes of embeddings between domains in Euclidean space. In both cases some extensions of the standard algebraic persistence theory are required to obtain the sharpest or most general results, leading to the introduction of a version of persistence barcodes for certain chain complexes over Novikov fields, and to consideration of an asymmetrical version of the standard notion of interleaving between persistence modules.

Additional material

Topological Persistence in Geometry and Analysis

by Leonid Polterovich, Daniel Rosen, Karina Samvelyan, Jun Zhang

Link to the book

Exercises

by Ulrich Bauer, Fabian Roll

sheet and solutions

Lecture Notes

by Daniel Rosen

(pdf)

Titels and abstracts

Crawley-Boevey - Classification of persistence and other modules

Abstract: As background, I shall discuss the classification of modules for finite-dimensional algebras, Drozd's Tame and Wild Theorem and some cases where a classification is known. I shall then discuss implications for the classification of persistence modules, including joint work with Magnus Bakke Botnan giving a new approach to the classification of middle exact persistence modules on the plane, originally studied by Cochoy and Oudot.

Biswas - Revisiting Alexander Duality with Tessellations and Mosaics

Edelsbrunner - Stochastic geometry with topological flavor

Abstract: Motivated by the use of persistent diagrams to analyze random and non-random datasets, we study classical questions in stochastic geometry, such as the expected density of critical p-simplices in the Delaunay mosaic of a Poisson point process in d-dimensional Euclidean space. We present analytic results for Delaunay, weighted Delaunay, and order-k Delaunay mosaics, and experimental result for alpha complexes and wrap complexes.

Haim Kislev - The EHZ capacity of polytopes

Mescher - Oriented robot motion planning in Riemannian manifolds

Abstract: We consider the problem of robot motion planning in an oriented Riemannian manifold by investigating its oriented frame bundle. For this purpose, we study the topological complexity of oriented frame bundles, a homotopy invariant introduced by M. Farber to model motion planning problems from robotics in a topological framework. I will describe differences between topological motion planning with and without orientations and discuss upper and lower bounds in the oriented case.

If time permits, we will additionally talk about a result obtained with M. Grant on the topological complexity of symplectic manifolds.

Slides of the talk

Patel - Multiparameter Persistent Homology

Abstract: The 2009 paper of Carlsson and Zomorodian established the current algebraic model for persistent homology. In this model, the barcode or, equivalently, the persistence diagram is defined as the list of indecomposables of a persistence module. This model works well for the setting of vector spaces arranged in a 1-parameter pattern, but it fails for more general abelian objects and for the n-parameter setting. We have been developing an alternative algebraic model for persistent homology. In this talk, I will present our latest results. Given a n-parameter filtration of a chain complex, we use lattice theory to define its persistence diagram. We prove that our n-parameter persistence diagram satisfies bottleneck stability. https://arxiv.org/abs/1905.13220

Shelukhin - Smith theory in Floer persistence, and dynamics

Abstract: We describe results related to the Smith inequality in filtered Floer homology, and their applications to questions in dynamics. In particular, we show that for a class of symplectic manifolds including the complex projective spaces, a Hamiltonian diffeomorphism with more fixed points, counted suitably, than the dimension of the ambient rational homology, must have an infinite number of simple periodic points. This is a higher-dimensional homological generalization of a celebrated result of Franks from 1992, as conjectured by Hofer and Zehnder in 1994. This talk is partially based on joint work with Jingyu Zhao.

Stojisavljevic - Symplectic Banach-Mazur distance and stability of barcodes

Abstract: Let $M$ be a closed manifold and denote by $\mathcal{C}_M$ the set of all domains in the cotangent bundle $T^*M$ which are star-shaped in every fiber. On $\mathcal{C}_M$ one may define a distance called symplectic Banach-Mazur distance. I will explain how to a "nice" domain from $\mathcal{C}_M$ we may associate a pointwise finite-dimensional persistence module called filtered symplectic homology. The corresponding barcode will be stable with respect to symplectic Banach-Mazur distance and will carry information about the Reeb dynamics on the boundary of the domain. We will use these properties to study the large-scale geometry of $\mathcal{C}_M$ as well as to obtain certain results about stability of closed geodesics. The talk is based on a joint work with Jun Zhang.

Registration:

The registration is complete. We are fully booked and the waiting list is full.

Arrival:

August 5, before lunch

Shuttle Bus: For Monday, August 5, we organised a shuttle service which commutes between Marburg main station and the castle. The shuttle stops at the central bus station, opposite to the main entrance of the station building (timetable: 2 times 10:00 am, 11:30 am, 11:45 am, 12:30 am).

Departure:

August 9, after lunch

On Friday, August 9, we drive together at 1:45 pm with a bus to the main train station Marburg, which we will reach by 2:20 pm at the latest.

Observe:

At the castle we have booked rooms for you; all meals are organised at the castle; all local costs are covered by us; all lectures will take place at the castle.

Excursion to Marburg:

14:00 Departure – Bus: castle to Marburg

14:30 Meeting point: Stadthalle Marburg

14:45 Start - city ​​tour

16:15 End - city ​​tour

16:45 Meeting point: Stadthalle Marburg

17:00 Departure – Bus: Marburg to the castle