- Frederic Chazal (INRIA-Saclay)
- Christian M. Reidys (University of Southern Denmark)
- Mikael Vejdemo-Johansson
- Joao Pita Costa
- Giovanni Petri
- Paul Expert
09:00 – 09:30 Welcome and Introduction
09:30 – 10:45 Frederic Chazal
10:45 – 11:30 Coffee
11:45 – 12:15 Giovanni Petri
12:15 – 12:45 Joao Pita Costa
13:00 – 14:00 Lunch
14:00 – 15:15 Christian M. Reidys
15:15 – 15:45 Mikael Vejdemo-Johansson
15:45 – 16:15 Paul Expert
16:30 – 17:30 Coffee
17:30 – 17:45 Discussion
09:30-10:45 Stability and convergence of persistence diagrams in Topological Data Analysis
Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field. In this talk, we will show that this tool allows to robustly infer topological information from metric data. We will also show that the use of persistent homology can be naturally considered in general statistical frameworks and persistence diagrams can be used as statistics with interesting convergence properties.
14:00 – 15:15 Topology and combinatorics of RNA and DNA
Christian M. Reidys
In this talk we discuss the concepts of combinatorial RNA structures
as well as DNA. We show how to pass from combinatorics to topology by constructing topological cell complexes from combinatorial data. This allows to identify building principles in topological RNA structures as well as insights into the organization of DNA. In both cases a natural notion of shape appears, which contains the key information about
the data. For fixed topological genus these shape spaces are finite and we shall discuss their associated generating polynomials.
11:45 – 12:15 Topology and mechanical manifolds at critical points of classical Hamiltonian system
The different Hamiltonian dynamics associated with the appearance of different order phase transitions in classical, planar, Heisenberg XY and φ^4 models on various regular lattices has been linked to the major topological change in the structure of the underlying mechanical manifold. Showing such change in the topological structure is however difficult, since, in the general case, the mechanical manifold is known imperfectly through appropriate samples. Here we employ persistent homology to overcome this limitation and explicitly show the different topologies on both sides of the phase transition. In particular, crossing the transition in the XY model we and that -as expected. the topology shifts from a complicated, holes-filled structure below the critical energy to a trivial topology above the critical energy. At the same time, no significant topological change is present in the 3D φ^4 model, where the nature of the phase transition is not tied to a topological shift.
12:15 – 12:45 Order structures for Topological Data Analysis
Joao Pita Costa
In the past 20 years Topological Data Analysis has been a vibrant area of research a lot due to the developments in applied and computational algebraic topology. Essentially it applies the qualitative methods of topology to problems of machine learning, data mining or computer vision. Under this topic, persistent homology is an area of mathematics interested in identifying a global structure by inferring high-dimensional structure from low-dimensional representations and studying properties of a often continuous space by the analysis of a discrete sample of it, assembling discrete points into global structure. Lattices are ever present in the everyday life of a working mathematician being distributive lattices some of the most important varieties of these algebras. A recent approach to the study of persistent homology using techniques of lattice theory is presented in this talk where we will also look at several algorithmic applications that imply the impact of these strategies.
15:15 – 15:45 A survey of tools for topological data analysis
Topological Data Analysis is a family of analysis tools that have emerged in the past decade to enable analysis of data sets of high levels of complexity. The techniques draw from classical algebraic topology to produce estimation procedures for topological features of data sets, producing a highly compressed, coordinate invariant and transformation invariant perspective on the data sets. We aim to give an overview of the techniques that have been developed and the software tools currently available for doing these kinds of analyses.
15:45 – 16:15 Homological backbone of the psychedelic brain
Traditionally, the structure of complex networks is studied through the statistical properties of the network. Here we present an alternative approach that uses the homological structure found via persistent homology. We introduce the homological backbones, two secondary networks designed to represent compactly the homological features of the original network under study and make them amenable to network theoretical tools. We apply these tools to compare the resting-state functional activity of volunteers after the infusion of either placebo or psilocybin, a psychedelic com- pound found in magic mushrooms, finding that the mesoscopic network structures undergo a dramatic change showing greater integration in subjects in psychedelic state.