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Details of Grant 

EPSRC Reference: EP/Y008642/1
Title: Topological Phase Transitions In Stochastic Geometry
Principal Investigator: Bobrowski, Dr O
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Sch of Mathematical Sciences
Organisation: Queen Mary University of London
Scheme: Standard Research
Starts: 01 January 2024 Ends: 31 December 2026 Value (£): 396,018
EPSRC Research Topic Classifications:
Algebra & Geometry Mathematical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
06 Sep 2023 EPSRC Mathematical Sciences Prioritisation Panel September 2023 Announced
Summary on Grant Application Form
This project is at the intersection of stochastic topology and geometry. Phase transitions are common phenomena in probability and statistical models, where the behaviour of a system changes abruptly. Our goal is to study various types of topological phase transitions in stochastic geometry models, generalising well-known lower dimensional phenomena. The phenomena we study are also related to statistical challenges in topological data analysis (TDA). To analyse high-dimensional structures we use the language of homology from algebraic topology. Briefly, in addition to connected components, homology can describe "holes" (closed loops), "cavities" (closed surfaces), and higher dimensional structures known as k-cycles.

The first part of this proposal deals with high-dimensional generalisations of connectivity. From a homological perspective, connectivity is the "convergence" of 0-cycles to their ultimate (trivial) limit. Analogously, we can study phase transitions where higher-dimensional k-cycle converge. We plan to study several random models related to both stochastic geometry and TDA. Our goal is to prove sharp transitions for homological connectivity and analyse the obstructions to connectivity in the critical window. Our analysis relies on Morse theory - a mathematical framework linking between topology and differentiable functions. Specifically, we use critical points of the distance function to analyse changes in homology. In terms of TDA, the outcomes will be useful to prove consistency for topological inference methods, and to reduce their computational costs.

The second part addresses the well-established area of coverage theory. Problems in this field are concerned with the ability of random small sets to cover a bigger region. We propose a novel topological approach linking Morse theory to coverage. This allows us to translate questions about m-coverage (where each point is covered by m or more small sets) into questions about the critical points of a new type of distance functions. Our new approach enables us to provide new and simpler proofs for well-known results, extend these results to more generic settings, and provide functional limit theorems for the vacant regions. In addition, we can go beyond coverage, and study homological connectivity for m-coverage objects.

The third part deals with the formation of large-scale topological structures. This is a high-dimensional generalisation for percolation theory - the study of large connected components in random media. Our goal is to extend this study from components to large k-cycles, i.e. large "loops" or "closed sheets". We refer to this study as homological percolation. The main challenge is that the techniques used in percolation theory do not naturally extend to higher dimensions. As a first step, we will study homological percolation in compact spaces and the formation of "giant" k-cycles. Once the theory matures, we will shift our focus to infinite spaces, which is the most well-studied setting in classical percolation. This will present a new line of challenges, as topological definitions and dualities do not extend naturally to non-compact spaces. Finally, we plan to study the link between homological percolation and the Euler characteristic (EC). The connection we seek is highly non-obvious, as the EC is dominated by many small local objects, and a-priori should not be related to the appearance of a few large global structures. Nevertheless, recent experimental work suggests a strong link between the critical values for homological percolation and the zeros of the EC curve. The contribution of this study is twofold. In TDA, it will provide important insights into the detectability of meaningful topological features in data. In percolation theory, it represents new high-dimensional generalisations for the theory and methods used. Ultimately, this generalised theory will provide significant novel insight into classical special cases.

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