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Details of Grant 

EPSRC Reference: EP/V008250/1
Title: Systoles and eigenvalues in non-positive curvature
Principal Investigator: Fortier Bourque, Mr M
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: School of Mathematics & Statistics
Organisation: University of Glasgow
Scheme: New Investigator Award
Starts: 01 June 2021 Ends: 31 May 2023 Value (£): 249,020
EPSRC Research Topic Classifications:
Algebra & Geometry Logic & Combinatorics
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
24 Nov 2020 EPSRC Mathematical Sciences Prioritisation Panel November 2020 Announced
Summary on Grant Application Form
This project will investigate which shapes optimize certain geometric problems. A classical example is the isoperimetric inequality: among all closed curves of a given length, the circle is the one which encloses the largest area. Another example is the sphere packing problem asking for the most efficient way to pack balls of equal radii in Euclidean space. Back in 1611, Kepler conjectured that the usual pyramidal configuration one sees in grocery stores was optimal, but in took until 1998 for a proof to be found by Thomas Hales. Surprisingly, the optimal configurations of spheres are also known in dimensions 8 and 24, thanks to recent breakthroughs by Maryna Viazovska and her co-authors.

The isosystolic problem, popularized by Marcel Berger in the 1970's, asks which Riemannian metrics of unit volume on a given manifold maximize the systole (defined as the length of the shortest non-contractible closed curve in the manifold). There very few cases in which the answer is known: for the 2-dimensional torus, the projective plane, and the Klein bottle. More progress can be made by restricting the classes of metrics under consideration. For example, more optimizers are known among constant curvature metrics on some other surfaces and on higher-dimensional tori. Although this is not immediately obvious, there is a close connection between the isosystolic problem for tori and the sphere packing problem. Indeed, any torus gives rise to a periodic packing of Euclidean space with spheres of radius equal to half the systole. The larger the systole is, the denser the associated sphere packing is.

The closely related kissing number problem asks what is the largest possible number of homotopically distinct systoles (shortest closed curves) among a given family of Riemannian metrics. The name comes from the fact that for a flat torus, this number counts how many spheres are tangent to (or kiss) any given sphere in the corresponding sphere packing of Euclidean space.

These two problems have analogues in the world of spectral geometry, which studies the frequencies at which a given planar membrane, or drum, can vibrate. The lowest of these frequencies, or the bass note, plays the role of the systole while its multiplicity, or the number of distinct vibration modes producing that note, plays the role of the kissing number. Mathematically, these frequencies are the eigenvalues of the Laplacian operator on the membrane. In particular, these concepts make sense not only for planar membranes, but for any Riemannian manifold.

Both the isosystolic problem and the kissing number problem, as well as their spectral counterparts, can be formulated for regular graphs. There again, a small collection of optimizers is known. However, the main unresolved questions concern the growth of these invariants in terms of the size of the graphs.

The goal of this project is improve the best known bounds for the isosystolic and kissing number problems and their spectral analogues for hyperbolic surfaces. We also aim to generalize results known for flat tori, hyperbolic manifolds and regular graphs to the setting of non-positively curved spaces.
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