Vrije Universiteit Amsterdam, 22th September 2023
NU-building, 9th floor (Math department), seminar room next to the common area.
10:00 – 10:50 Coffee, arriving
10:50 – 11:00 Welcome
11:00 – 11:45 Tim Ophelders (TU Eindhoven)
11:45 – 12:30 Pepijn Roos Hoefgeest (VU Amsterdam)
12:30 – 14:00 Lunch in the building
14:00 – 14:45 Walter van Suijlekom (Radboud Nijmegen)
14:45 – 15:30 Fernando Nobrega Santos (UvA)
15:30 – 15:45 Coffee
16:30 – 18:00 Discussions over drinks
(There will not be a hybrid or online version of the event.)
1) Tim Ophelders, TU Eindhoven
Title: Shortest Paths in General Polyhedral Surfaces
Abstract: Any surface that is intrinsically polyhedral can be represented by a collection of simple polygons (which we call fragments), glued along pairs of equally long oriented edges, where each fragment is endowed with the geodesic metric arising from its Euclidean metric. We refer to such a representation as a portalgon, and we call two portalgons equivalent if the surfaces they represent are isometric.
We analyze the complexity of shortest paths. We call a fragment happy if any shortest path on the portalgon visits it at most a constant number of times. A portalgon is happy if all of its fragments are happy. We present an efficient algorithm to compute shortest paths on happy portalgons.
The number of times that a shortest path visits a fragment is unbounded in general. We contrast this by showing that the intrinsic Delaunay triangulation of any polyhedral surface corresponds to a happy portalgon. Since computing the intrinsic Delaunay triangulation may be inefficient, we provide an efficient algorithm to compute happy portalgons for a restricted class of portalgons.
This talk is based on joint work with Maarten Löffler, Rodrigo Silveira, and Frank Staals
2) Pepijn Roos Hoefgeest, VU Amsterdam
Title: The Christoffel-Darboux Kernel for Topological Data Analysis
Abstract: In topological data analysis, persistent homology has been widely used to study the topology of point clouds in R^n. Unfortunately, standard methods are very sensitive to outliers, and their computational complexity depends badly on the number of data points. In this talk we will present a novel persistence module, based on recent applications of Christoffel-Darboux kernels in the context of statistical data analysis and geometric inference. Our approach is robust to outliers and can be computed in time linear in the number of data points. We illustrate the benefits and limitations of our new module with various numerical examples in R^n, n=1,2,3.
3) Walter van Suijlekom, Radboud Nijmegen
Title: Geometric spaces at finite resolution
Abstract: After a gentle introduction to the spectral approach to geometry, we extend the framework in order to deal with two types of approximation of metric spaces. On the one hand, we consider spectral truncations of geometric spaces, while on the other hand, we consider metric spaces up to finite resolution. In our approach, the traditional role played by operator algebras is taken over by so-called operator systems. Essentially, this is the minimal structure required on a space of operators to be able to speak of positive elements, states, pure states, etc.
We illustrate our methods in concrete examples obtained by spectral truncations of the circle and of metric spaces up to finite resolution. The former yield operator systems of finite-dimensional Toeplitz matrices, and the latter give suitable subspaces of the compact operators. We also analyze the cones of positive elements and the pure-state spaces for these operator systems, which turn out to possess a very rich structure.
4) Fernando Nobrega Santos, Uv Amsterdam
Title: Navigating High-Order Interactions in the Brain: A Topological Perspective with Hypergraphs.
Abstract: In functional brain networks, the traditional approach of quantifying correlations between time series of brain region activities often results in networks with nodes, edges, triangles, and even higher-dimensional objects. The topological structure of these networks offers a rich playground for applying Topological Data Analysis (TDA) in multiple ways. Our first study delves into the intricate interplay between TDA, topology, geometry, physics, and network theory to uncover topological phase transitions in functional brain networks. These transitions, marked by singularities in the Euler entropy and zeros of the Euler characteristics, coincide with the emergence of multidimensional topological holes in the brain network. Drawing parallels with percolation theory, we interpret these geometric transitions and highlight their potential as markers for differences in brain network organization.
However, the traditional network theory, predominantly based on pairwise relationships, often needs to catch up in capturing the complexity of systems like the human brain. In our subsequent work, we venture beyond the conventional Vietoris-Rips complex to introduce a multivariate signal processing pipeline for constructing high-order networks from time series. Applying this to resting-state fMRI signals, we characterize high-order communication between brain regions, developing uniform hypergraphs. By investigating three-point interactions in the human brain, we identify high-order "hubs" and discern patterns of integration and segregation in these high-order functional brain networks.
Together, these studies underscore the versatility of topological techniques in understanding the brain's complex network structure and pave the way for innovative representations and analyses in the broader context of complex systems. Join us as we bridge the gap between neuroscience and algebraic topology, offering insights and challenges for the applied topology community.
5) Casper Gyurik, U Leiden
Title: Quantum algorithms for topological data analysis
Abstract: In this talk I will overview the current landscape of quantum algorithms for topological data analysis. In particular, we review existing quantum algorithms for the estimation of (persistent) Betti numbers of simplicial complexes. We highlight regimes where they outperform classical methods, and we point out their limitations. Finally, we survey some results on the complexity of the Betti number problem that are inspired by connections to quantum computing.