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Hamm, Girtrude (2025) Growing Classifications: Widths, Ehrhart Theory and Spherical Geometry. PhD thesis, University of Nottingham.

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Abstract

This thesis focuses on three classifications of convex polytopes, which are separate, but the methods of each influences those that follow. There are links to combinatorial algebraic geometry throughout, particularly to toric and spherical geometry. This is most explicit in the third project, which is additionally a classification of certain spherical varieties.

In the first project we introduce the multi-width of a polytope, which is an extension of its lattice width. We study the classification of lattice simplices by their multi-width in dimensions two and three. This is motivated by computational questions in toric geometry. We completely classify lattice triangles by their multi-width and also classify lattice tetrahedra of small multi-width.

The second project concerns the Ehrhart theory of rational polygons. The Ehrhart theory of lattice polygons is already well understood and here we make steps towards a similar understanding of denominator two polygons. We classify denominator two polygons containing up to four lattice points, including a description of infinite families of polygons with no interior points. Using this data, we completely classify the Ehrhart polynomials of denominator two polygons with zero interior points and find three sharp bounds on the coefficients when there are interior points.

In the final project we study spherical varieties, which generalise toric and flag varieties. We discuss isomorphisms between spherical varieties and describe a class of lattice automorphisms which are induced by isomorphisms of spherical varieties. We define a normal form for lattice polytopes up to this group of automorphisms. This normal form is vital to our classification of spherical canonical Fano four-folds. Like toric Fano varieties, spherical Fano varieties correspond to polytopes. Therefore, we can classify the varieties by classifying the corresponding polytopes.

Item Type:	Thesis (University of Nottingham only) (PhD)
Supervisors:	Kasprzyk, Alexander Hofscheier, Johannes
Keywords:	convex polytopes, polytope classification, algebraic geometry
Subjects:	Q Science > QA Mathematics > QA440 Geometry
Faculties/Schools:	UK Campuses > Faculty of Science > School of Mathematical Sciences
Item ID:	80626
Depositing User:	Hamm, Girtrude
Date Deposited:	31 Jul 2025 04:40
Last Modified:	31 Jul 2025 04:40
URI:	https://eprints.nottingham.ac.uk/id/eprint/80626

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