Schedule Topology of Data in Rome
Thursday, September 15
|09:30 am||Bastian Rieck, A Good Scale is Hard To Find: Shape Analysis Using Topology|
|10:30 am||coffee break|
|11:00 am||Barbara Giunti, Fantastic barcode algorithms and where to find them|
|12:00 am||Katherine Benjamin, Multiscale methods for spatial data in biology|
|01:00 pm||lunch break|
|02:30 pm||Wojtek Chacholski, Homological algebra and persistence|
|03:30 pm||Ryan Budney, Filtrations of smooth manifolds from maps to the plane|
|04:30 pm||coffee break|
|05:00 pm||poster session|
|08:00 pm||Conference dinner: Restaurant "Renato e Luisa", via dei Barbieri 25, www.renatoeluisa.it|
|09:30 am||Pawel Dlotko,Why should data scientists care about topology?|
|10:30 am||coffee break|
|11:00 am||Anibal Medina, Persistence Steenrod modules|
|12:00 am||Kelly Maggs, Morse Theoretic Signal Compression and Reconstruction on Chain Complexes|
********** Abstracts **********
It has long been envisioned that the strength of the barcode invariant of filtered cellular complexes could be increased using cohomology operations. Leveraging recent advances in the computation of Steenrod squares, we introduce a new family of computable invariants on mod 2 persistent cohomology termed Steenrod barcodes. We present a complete algorithmic pipeline for their computation and illustrate their real-world applicability using the space of conformations of the cyclo-octane molecule.
Our world is full of phenomena that happen at different spatial and temporal scales. If we pick the wrong scale, we might miss the forest for the trees---and vice versa! In recent years, methods from computational topology have started to emerge as one way to address the challenging task of picking the 'right' scale: Instead of enforcing one specific scale when analysing data, such methods afford an analysis of *all* scales inherent to data sets. In this talk, I will outline the general utility and expressivity of topological machine learning methods, i.e. methods combining a rigorous mathematical underpinning with the flexibility of modern deep learning architectures. I will provide both a theoretical as well as an applied view on this topic by showcasing how to employ topological methods to solve inverse problems, dealing with image reconstruction tasks in fluorescence microscopy.
Abstract: Spatial data in biology are often characterised by the presence of features at multiple scales. Topological data analysis and persistent homology are therefore natural choices for the study of these systems. Indeed, in recent years TDA has been applied to a broad range of biological spatial systems, from the structure of individual molecules to the spatial organisation of whole tissues. In this talk we will discuss some recent applications to biological spatial data, including the study of knots in proteins, that take advantage of the multiscale nature of TDA.
In this talk, I present an introduction to the standard barcode algorithm and an overview of some of its variants. I explain what the pivot pairing is and why it provides the barcode, and how it behaves together with the two classical optimizations called clear and compress. I provide some insight into how the sparseness of the matrix can affect efficiency. If there is time, I discuss the worst-case and average complexity for some special class of inputs.
Given a smooth map from a manifold to the plane, and some poset structure on the plane, one can take the pre-images of the planar subsets to get a poset structure on the manifold. In this talk I will outline how one can compute the homotopy-type of the poset structure (provided the poset on R^2 is smooth-enough) in terms of cellular attachments. In the case of the bi-filtration of the plane by “quadrants” (−∞, a] × (−∞, b] this gives us a description of the homotopy-type of the bi-filtration of the manifold, analogous to classical Morse theory.
TDA is becoming a mature dyscypline with a number of tools it can offer and a number of problems it has successfully solved. In this talk I will summarize some of the available techniques and successful applications of TDA. By doing so, I will hint to my biased selection of meta-nishes in which TDA is the tool to use in data science. We will see how the existing techniques and successful applications fit into those nishes. Lastly I will summarize the new work by my Dioscuri Centre in Topological Data Analysis to further explore new TDA tools and promote TDA in sciences.
There is a growing interest in TDA community regarding homological invariants of persistent modules. In my talk I will describe a set up for relative homological algebra with computationally effective methods based on Koszul complexes for calculating associated Betti diagrams. This is a join work with A. Guidolin, I. Ren, M. Scolamiero, F. Tombari
In this work, we provide an approach to signal compression and reconstruction on chain complexes that leverages the tools of algebraic discrete Morse theory. The main goal is to reduce and reconstruct a based chain complex together with a set of signals on its cells via deformation retracts, with the aim of preserving parts of the global topological structure of both the complex and the signals.
We first prove that any deformation retract of real degree-wise finite-dimensional based chain complexes is equivalent to a Morse matching. We then study how the signal changes under particular types of Morse matching, showing its reconstruction error is trivial on specific components of the Hodge decomposition. Furthermore, we provide an algorithm to compute Morse matchings which locally minimizes reconstruction error.