# Volume-collapsed manifolds in Riemannian geometry and geometric inference

Lead Research Organisation:
Cardiff University

Department Name: Sch of Mathematics

### Abstract

A piece of paper is a two-dimensional object and, if we roll it up into a cylinder, that cylinder is still two-dimensional, in the eyes of mathematicians. The more tightly we roll up the paper, the smaller the surface area of the resulting cylinder. If we could roll it up infinitely tightly it would (i) have zero surface area and (ii) not really be a cylinder any more, but rather a line.

This project studies how this behaviour generalises to more complicated shapes which take up more dimensions. These shapes are called manifolds, and once they have at least three dimensions the analogy to their area is always called the 'volume'. Manifolds which can be 'rolled up' as tightly as we like, such as the cylinder, can be called 'volume-collapsed manifolds'. I will study two aspects of their behaviour, with the aim of addressing long-standing questions in geometry. The new knowledge produced will both provide novel theoretical insights to support the work of other mathematicians, and have practical technological applications in sectors as diverse as healthcare, finance and industry

Firstly, it would seem that we can tell a lot about the manifold from the smaller limit space that it is approaching. The tightly wrapped cylinder is related to the line; we say that the cylinder is a 'fibre bundle' over the line. For every point on the line, there is a little circle (the 'fibre') on the cylinder which corresponds. All these circles 'bundled' up together add up to the cylinder. However, this example is a simple one. One simplifying aspect is the lack of curvature (a cylinder is made from a flat piece of paper). Another is that there are only two dimensions. Understanding how three-dimensional manifolds collapse while maintaining a curvature bound played a significant role in the proof of the Poincaré Conjecture, the only one of the Clay Mathematics Institute's 'Millennium Prize Problems' to have been solved so far.

Project A will aim to classify the volume-collapsed manifolds corresponding to a given limit space. This will contribute greatly to the goal of understanding how curvature and shape interact, which is one of the major fields of research in geometry.

The second aspect is statistical. Given a random sample of points from an unknown manifold, the more points we have the more confident we can be of identifying the manifold. However, when the manifold is volume-collapsed, it will clearly be very difficult to distinguish it from its limit space; in this example, to distinguish cylinder from line. In mathematical statistics, we seek to understand how well statistical procedures perform in the most challenging cases. When we are carrying out statistics on geometric objects, volume-collapsed manifolds clearly have a major role to play. Understanding these questions is the only way to bring the full rigour of mathematics to the exciting and successful new topological data analysis techniques being used on massive data sets.

Project B will develop statistical tools which come with strong mathematical guarantees and will search for procedures which are optimal, meaning that they perform as well as is theoretically possible under the worst case scenario. In this project I will also pursue more applied goals, developing procedures to test the validity of methods being used, integrating self-validating mechanisms into data analysis tools and developing links with business to use geometrically-aware data analysis.

Projects A and B will use very different methods, but as with any study of the same object from two points of view the aim is that each study will inform the other and that new connections will be revealed. Dedicating time and resources to the pursuit of both problems is the only way for us to discover those connections, and this Fellowship will enable exactly such an adventurous research programme.

This project studies how this behaviour generalises to more complicated shapes which take up more dimensions. These shapes are called manifolds, and once they have at least three dimensions the analogy to their area is always called the 'volume'. Manifolds which can be 'rolled up' as tightly as we like, such as the cylinder, can be called 'volume-collapsed manifolds'. I will study two aspects of their behaviour, with the aim of addressing long-standing questions in geometry. The new knowledge produced will both provide novel theoretical insights to support the work of other mathematicians, and have practical technological applications in sectors as diverse as healthcare, finance and industry

Firstly, it would seem that we can tell a lot about the manifold from the smaller limit space that it is approaching. The tightly wrapped cylinder is related to the line; we say that the cylinder is a 'fibre bundle' over the line. For every point on the line, there is a little circle (the 'fibre') on the cylinder which corresponds. All these circles 'bundled' up together add up to the cylinder. However, this example is a simple one. One simplifying aspect is the lack of curvature (a cylinder is made from a flat piece of paper). Another is that there are only two dimensions. Understanding how three-dimensional manifolds collapse while maintaining a curvature bound played a significant role in the proof of the Poincaré Conjecture, the only one of the Clay Mathematics Institute's 'Millennium Prize Problems' to have been solved so far.

Project A will aim to classify the volume-collapsed manifolds corresponding to a given limit space. This will contribute greatly to the goal of understanding how curvature and shape interact, which is one of the major fields of research in geometry.

The second aspect is statistical. Given a random sample of points from an unknown manifold, the more points we have the more confident we can be of identifying the manifold. However, when the manifold is volume-collapsed, it will clearly be very difficult to distinguish it from its limit space; in this example, to distinguish cylinder from line. In mathematical statistics, we seek to understand how well statistical procedures perform in the most challenging cases. When we are carrying out statistics on geometric objects, volume-collapsed manifolds clearly have a major role to play. Understanding these questions is the only way to bring the full rigour of mathematics to the exciting and successful new topological data analysis techniques being used on massive data sets.

Project B will develop statistical tools which come with strong mathematical guarantees and will search for procedures which are optimal, meaning that they perform as well as is theoretically possible under the worst case scenario. In this project I will also pursue more applied goals, developing procedures to test the validity of methods being used, integrating self-validating mechanisms into data analysis tools and developing links with business to use geometrically-aware data analysis.

Projects A and B will use very different methods, but as with any study of the same object from two points of view the aim is that each study will inform the other and that new connections will be revealed. Dedicating time and resources to the pursuit of both problems is the only way for us to discover those connections, and this Fellowship will enable exactly such an adventurous research programme.