Topological Data Analysis
Topological Data Analysis (TDA) uses techniques from topology, a branch of mathematics, to study the shape and structure of data. A central tool in TDA is persistent homology, which, in its basic form, is similar to hierarchical clustering in statistics, but also captures complex, higher-dimensional topological features such as loops and voids.
Persistent homology produces a collection of intervals, known as a barcode, where each interval represents the size and scale of a topological feature relative to a single variable, such as spatial scale. Its applications span a wide range of scientific and mathematical fields, with notable advancements in areas like neuroscience and symplectic geometry. For example, recent research has shown that certain neurons organize themselves on a hollow torus, a topological space characterized by two independent loops enclosing a void.
Mathematical and Algorithmic Challenges
Magnus Bakke Botnan’s research focuses on developing new tools for multiparameter persistent homology, a method within TDA that allows for the analysis of data involving multiple variables. This is crucial because real-world data is often complex, involving factors such as time or interacting variables. However, transitioning from single-parameter to multiparameter persistent homology introduces significant mathematical and algorithmic challenges. This is a highly active area of research, with the potential to reshape TDA.
A Practical Tool
In this Vidi project, Botnan will tackle these challenges by leveraging recent advancements at the intersection of TDA and homological algebra. His goal is to develop a practical theory of multiparameter persistent homology by creating an efficient framework for generalized barcodes and designing computationally efficient and discriminative similarity measures.
All algorithms developed will be implemented in publicly accessible libraries, enabling broader use across disciplines, particularly in machine learning and related applications.
Vidi
The Dutch Research Council (NWO) has awarded 102 experienced researchers, seven of whom are from Vrije Universiteit Amsterdam and Amsterdam UMC, location VUmc, with a Vidi grant. This will enable them to develop their own innovative line of research over the next five years and set up a research group. Together with the Veni and Vici grants, Vidi is part of the NWO Talent Programme. They receive in total 86,7 million euros.