Combinatorial and computational algebra of toric degenerations of varieties
Lead Research Organisation:
University of Bristol
Department Name: Mathematics
Abstract
Toric varieties play an important role in algebraic geometry. There are several methods developed in polyhedral and combinatorial geometry to compute the invariants of a toric variety.
The student will work on toric varieties which are at the interface of algebra, combinatorics and geometry. Recently there has been a breakthrough in algebraic geometry making a direct connection between the theory of Newton-Okounkov bodies and tropical geometry, and in particular the toric varieties arising in both contexts. The main component of the student's project is to apply algebraic and combinatorial methods to further explore such connections.
The student will work on toric varieties which are at the interface of algebra, combinatorics and geometry. Recently there has been a breakthrough in algebraic geometry making a direct connection between the theory of Newton-Okounkov bodies and tropical geometry, and in particular the toric varieties arising in both contexts. The main component of the student's project is to apply algebraic and combinatorial methods to further explore such connections.
Organisations
Publications
Clarke O
(2021)
Conditional probabilities via line arrangements and point configurations
in Linear and Multilinear Algebra
Clarke Oliver
(2019)
Conditional independence ideals with hidden variables
in arXiv e-prints
Clarke Oliver
(2019)
Toric degenerations of flag varieties from matching field tableaux
in arXiv e-prints
Clarke Oliver
(2019)
Toric degenerations of Grassmannians and Schubert varieties from matching field tableaux
in arXiv e-prints
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509619/1 | 01/10/2016 | 30/09/2021 | |||
| 1953188 | Studentship | EP/N509619/1 | 01/10/2017 | 30/04/2021 | Oliver Clarke |
| Description | Toric degenerations are an important tool which connected algebraic geometry and combinatorics. We have performed calculations to compute toric degenerations of a number of objects including families of Grassmannians, Flag varieties and Schubert variteties. From these calculations we have been able to produce new tools which allow us to give classifications of the types of toric degnerations obtained by our process. The toric degenerations obtained by our procedure are of particular interesst to the mathematical community as exhibit many interesting properties that connect to other areas of combinatorics. |
| Exploitation Route | Our research provides a wealth of examples of toric degenerations which serve as a test bed for further research. Our results also give rise to fascinating polytopes and combinatroial objects which can be explored separately. |
| Sectors | Other |