Schedule and Index

In the first lecture of the course we will introduce simplicial complexes, a model for topological spaces intensively used in topological data analysis. Then, we will describe the computation of homology groups of simplicial complexes as a tool that provides information on the shape of the spaces and data. Finally, we will describe the use of simplicial complexes and homology in TDA as a motivation for the following lectures.

Slides

Recording

This will gently introduce persistent homology, which is the main tool in topological data analysis. We will focus on discrete persistence modules; these consist of finite sequences of vector spaces and linear maps, obtained by taking homology of filtered simplicial complexes with coefficients in a field. The ultimate goal is to describe how every discrete persistence module can be recovered from purely combinatorial data called a barcode. 

Slides

Recording

Here we view persistence modules with a more geometric lens, namely as vector spaces and linear maps indexed by real numbers. With this shift in perspective comes the celebrated stability theorem, which shows how small perturbations of a given filtration lead to small perturbations of the associated barcode.

Slides - Lectures 2 and 3

Recording

Having seen the basics of persistent homology and barcodes/diagrams, this talk will turn to the larger questions in persistence related to functoriality, parameters, and coefficients, in which very difficult algebraic challenges arise. We will quickly point out several new ideas in the literature to make persistence work in broader cases. This will involve a mixture of some very theoretical work along with a few pointers to practical implementation.

Slides

Recording

This lecture will focus on one direction for future work in TDA related to distributed computation and distributed systems. The mathematical topics will cover cellular sheaves and cosheaves. Applications will begin in basic TDA and persistence but will broaden out to a number or related areas in which the ideas behind TDA have broader applicability than simply characterizing point-cloud data.

Slides

Recording

This lecture will focus on computational aspects of Topological Data Analysis (TDA). During the first part, we will discuss the main libraries and languages to code TDA and explain some fundamental tools to perform data analysis using persistent homology. In the second part, we will address more advanced topics such as the efficiency of different packages, the construction of custom filtrations in GUDHI, and the use of TDA in deep learning. 

Python Notebooks from Aina 

Python Notebooks from Rubén

Slides

Recording

The first part of this two-part lecture series will gently introduce some aspects of modern machine learning research, followed by an overview of how topological features can be integrated into deep learning models. A particular focus will be given to methods working with unstructured data like point clouds.

Slides - Part 1

Slides - Part  2

Recording

The second part of this two-part lecture series will discuss recent advances in the integration of geometrical and topological aspects into deep learning models. Starting from hybrid models for graph learning tasks, we will subsequently leave traditional contemporary machine learning methods behind and present a new class of models that are specifically poised to leverage intrinsic geometrical-topological properties of data.

Slides - Part 1

Slides - Part  2

Recording