A STRUCTURES day

in Computational Topology

Monday, November 19, 2018

Venue: Heidelberg Institute for Theoretical Studies – Studio Villa Bosch

Schedule:

10:00-11:00 Registration and coffee
11:00-12:00 Herbert Edelsbrunner (Institute of Science and Technology Austria)
12:00-13:30 Lunch
13:30-14:30 Heather Harrington (University of Oxford)
14:30-15:15 Coffee Break
15:15-16:15 Ulrich Bauer (Technische Universität München)
16:30-17:30 Egor Shelukhin (University of Montreal)

Due to space constraints we ask participants to register for this workshop with name and affiliation at persistence@mathi.uni-heidelberg.de .

Titel and Abstracts:

Herbert Edelsbrunner – Stochastic geometry with topological flavor

Abstract: We study classical questions in stochastic geometry, such as the expected density of p-simplices in the Delaunay mosaic of a Poisson point process in d-dimensional Euclidean space. Using a discrete Morse theory approach, we distinguish between critical and non-critical of the radius function and determine their expected densities dependent on a radius threshold. We generalize the analytic results to weighted Delaunay mosaics and to order-k Delaunay mosaics, and we present experimental result for wrap complexes and for weighted Voronoi tessellations.

Joint work with Anton Nikitenko, Katharina Oelsboeck, and Peter Synak.

Heather Harrington – Comparing models and data using computational algebraic geometry and topology

Abstract: Many biological problems, such as tumor-induced angiogenesis (the growth of blood vessels to nourish a tumor), or signaling pathways involved in the dysfunction of cancer (sets of molecules that interact that turn genes on/off and ultimately determine whether a cell lives or dies), can be modeled using differential equations. The challenge with analyzing these types of mathematical models is that the rate constants, often referred to as parameter values, are difficult to measure or estimate from available data.

I will present mathematical methods we have developed to enable us to compare mathematical models with experimental data. Depending on the type of data available, and the type of model constructed, we have combined techniques from computational algebraic geometry and topology, with statistics, networks and optimization to compare and classify models without necessarily estimating parameters. Specifically, I will introduce our methods that use computational algebraic geometry (e.g., Gröbner bases) and computational algebraic topology (e.g., persistent homology). I will present applications of our methodology on datasets involving cancer. Time permitting, I will conclude with our current work for analyzing spatio-temporal datasets with multiple parameters using computational algebraic topology. Mathematically, this is studying a module over a multivariate polynomial ring, and finding discriminating and computable invariants.

Ulrich Bauer - Persistent homology: from theory to computation

In this talk, I will survey some recent results on theoretical and computational aspects of applied topology. I will focus on three aspects of topological descriptors: their use for the inference and simplification of topological features, their stability with respect to perturbations of the data, and efficient methods for their computation on a large scale.

These questions will be motivated and illustrated by concrete examples and problems, such as faithful simplification of contours lines in cartography, denoising of isosurfaces for visualization of medical images, and reconstruction of a shape and its topological properties from a point cloud.

Egor Shelukin - On persistence modules in symplectic topology

Abstract: To resolve V. Arnol'd's famous conjecture from the 1960's on the number of fixed points of a Hamiltonian diffeomorphism of a symplectic manifold, A. Floer has associated in the late 1980's a homology theory to the Hamiltonian action functional on the loop space of the manifold. It has long been known that Floer homology can be filtered by the values of the action functional, yielding information about metric invariants in symplectic topology (Hofer's metric, for example). We discuss recent interactions between this filtered version of Floer theory and persistent homology, providing examples of new results.