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Details of Grant 

EPSRC Reference: EP/X029387/1
Title: Minimal Models of Foliations
Principal Investigator: Spicer, Dr C
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Mathematics
Organisation: Kings College London
Scheme: New Investigator Award
Starts: 01 April 2023 Ends: 31 March 2026 Value (£): 408,751
EPSRC Research Topic Classifications:
Algebra & Geometry Mathematical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
01 Nov 2022 EPSRC Mathematical Sciences Prioritisation Panel November 2022 Announced
Summary on Grant Application Form

My research is an interdisciplinary project focused at the interface of two active fields of

pure mathematics: dynamical systems and algebraic geometry.

I am particularly interested in the case of foliations on algebraic varieties.

Much of the study of foliations began in the early 20th century as a way to understand

the solutions of differential equations. Often it may not be possible to find

a simple closed form solution to some differential equation of interest, however,

it may be possible to say something about the general behaviour of the flows

or orbits of the differential equation.

These flows partition the underlying space into disjoint

immersed submanifolds, called leaves, and this decomposition is referred to as a foliation.

The study of the differential equation is therefore replaced by the study

of the qualitative (e.g., geometric, topological, etc.)

properties of the foliation.

This is a powerful idea and foliations have arisen

in a very wide range of contexts, for instance,

topology, geometry, number theory, and mathematical physics.

Understanding foliations is increasingly being understood as an essential research direction

in a wide range of fields.

My particular interests are in foliations in the context of algebraic geometry

where, in the past few years, the study of foliations has been at the heart of several recent

major developments.

The general idea of much of my research is to understand the qualitative properties

of foliations from a relatively recent perspective: by developing techniques and ideas in foliation theory

inspired by (and extending) the study of the birational geometry of varieties, in particular, the ideas of

the Minimal Model Program.

The key insight that this interdisciplinary fusion brings to the study of foliations is

that it should be possible to tweak a foliation in a controlled way which simplifies some of its geometric properties,

but which does not alter the aspects of the foliation we are most interested in, for instance its dynamical properties.

Moreover, by performing these alterations we expect to transform an arbitrary foliation into one which decomposes into ``atomic"

foliations. If one could realize this decomposition strategy then one be able to reduce the study of foliation geometry and

dynamics in general

to the study of these properties on these atomic foliations, where we expect this study to be much more feasible.

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