Fridays at 14:15
|Functional PCA of persistent homology rank functions||Kate Turner
|27.02.15||Computational topology and shape comparison||Claudia Landi
|20.03.15||Multi persistence and topological data analysis||Wojciech Chachólski
|01.05.15||Invariants for multidimensional persistence||Martina Scolamiero
|08.05.15||Spatial graphs and minimally knottedness||Senja Barthel
|Topological complexity: a product formula||Paul-Eugène Parent
|Notions of distance in persistent homology||Jonathan Scott
|Networks of continuous time open systems||Eugene Lerman
|Simplicial manifolds with small valence||Florian Frick
Turner: Persistent homology provides a method to capture topological and geometrical information in data. I will discuss the topological summary statistics called persistent homology rank functions. Under reasonable assumptions, satisfied in almost all applications, the persistent homology rank functions of interest will lie in an affine subspace. This means we can perform PCA. I will look at examples using point processes and real world data involving colloids and sphere packings. Joint work with Vanessa Robins. This talk is meant to be an introduction to Topological Data Analysis and no background in algebraic topology nor statistics is required..
Landi: In the last twenty years, computational topology has contributed with a number of methods to deal with the task of describing and comparing shapes. The ultimate goal of this research area is to provide topological ground for classification and retrieval of digital shapes contained in a database based on content. The general pipeline in this field is to associate with each shape a shape descriptor and then recast the shape comparison problem as a metric problem on shape descriptors. The talk will report on the state of the art of some popular topological shape descriptors coming from the Persistent Homology Theory. In particular, persistence diagrams with the bottleneck distance and persistence modules with the interleaving distance will be reviewed and shown to satisfy stability and optimality properties.
Chachólski: In this talk I will present a new method for extracting persistence information out of multi graded modules. This method is part of the so called topological data analysis (TDA for short) and builds on understanding mathematical properties of what one might regard as noise. Denoising can be then thought as the localization away from the noise. I will take this opportunity to summarise what TDA is and argue that it is an interesting mathematical subject. This is a report on joint work of the topological data analysis group at KTH that consists of Anders Lundman, Ryan Ramanujam, Martina Scolamiero, Sebastian Öberg and myself.
Scolamiero: Multidimensional Persistence is a method in topological data analysis which allows to study several properties of a dataset contemporarily. It is important to identify discrete invariants for multidimensional persistence in order to compare properties of different datasets. Furthermore such invariants should be stable, i.e., data sets which are considered to be close should give close values of the invariant. We will introduce a framework that allows to compute a new class of stable discrete invariants for multidimensional persistence. In doing this, we will generalize the notion of interleaving topology on multi- dimensional persistence modules and consequently the notion of closeness for datasets. A filter function is usually chosen to highlight properties we want to examine from a dataset. Similarly, our new topology allows some features of datasets to be considered as noise. (Joint work with Chachólski, Lundman, and Öberg.)
Barthel: We introduce spatial graph theory by defining some properties of spatial graphs that can be considered as generalisation of unknottedness and investigating relations between them. Everything is illustrated by giving examples. Finally, we show that there exist no minimally knotted planar spatial graphs on the torus. Minimally knotted entanglements of θ-n graphs are called ravels in chemistry.
Parent: In 2003, Michael Farber introduced the notion of topological complexity TC(X) for the motion planning problem in robotics. In his words this none negative integer TC(X) measures discontinuity of the process of motion planning in the configuration space X. Very rapidly one notices that it is in fact a homotopy invariant which can be very effectively studied using tools developed for the computation of another homotopy invariant: the Lusternik-Schnirelmann category of a space X. In this talk, we will give a small survey of recent results. Moreover, while working over the rational numbers we will exhibit new computational examples (joint work with my student Gabrielle Poirier) and give a product formula for an approximation to TC(X).
Scott: In this talk, we will discuss how the interleaving distance for persistence diagrams can be generalized, by using first the “future equivalences” of Grandis, and then further, using modified co-spans. The utility of our approach is demonstrated by revealing shift equivalences of dynamical systems as generalized interleavings. This is joint work with P. Bubenik (CSU) and V. de Silva (Pomona College).
Lerman: (This talk is based on collaborations with Lee DeVille and with Dmitry Vagner and David Spivak) Dynamics on networks has been studied from many different points of view. I prefer to think of a complex continuous time system (“a network”) as a collection of interacting subsystems. These subsystems are open (i.e., control) systems. There are two rather different aspects of these collections. Given a collection of open systems one can interconnect them to obtain a new open system. The interconnection process can be iterated: collections of open systems obtained by interconnecting smaller open systems can be interconnected again. The iterative aspect of interconnection of open systems is captured well by viewing the collection of all open systems as an algebra over a colored operad. This is an instance of an approach advocated by Spivak. On the other hand one of the fundamental problems in the theory of (closed) dynamical systems is constructing maps between dynamical systems or, failing that, proving their existence. For example, a map from a point to our favorite closed system is an equilibrium, periodic orbits are maps from circles and so on. Thus it is desirable to have a systematic way of constructing maps of dynamical systems out of appropriate maps between collections of open systems. It is these kinds of considerations that underlie the development of the groupoid formalism for coupled cell networks of Golubitsky, Steward and their collaborators and its reinterpretation and extension by DeVille and Lerman. The two considerations/viewpoints suggest that networks of open systems in general should be an algebra over some sort of a double monoidal category. I will outline one such possible construction.
Frick: We will investigate combinatorial restrictions on manifold triangulations that have an interpretation in terms of sectional curvature: the valence of a face of codimension two is the number of facets it is contained in. We give a topological as well as a combinatorial classification of triangulations that are positively curved in the sense that they have small valence. We simplify the proof of a result of Brady, McCammond, and Meier that any closed and orientable 3-manifold has a triangulation with valence at most six and improve a result of Cooper and Thurston on the number of combinatorial types of vertex links needed to triangulate any closed orientable 3-manifold, which was independently observed by Kevin Walker. We will remark on applications to physics and chemistry, in particular, to crystallographic periodic foams. This is joint work with Frank Lutz and John M. Sullivan.