- Titel
- Computation and Physics in Algebraic Geometry
- Zitierfähige Url:
- https://nbn-resolving.org/urn:nbn:de:bsz:15-qucosa2-864704
- Datum der Einreichung
- 06.02.2023
- Datum der Verteidigung
- 21.06.2023
- Abstract (EN)
- Physics provides new, tantalizing problems that we solve by developing and implementing innovative and effective geometric tools in nonlinear algebra. The techniques we employ also rely on numerical and symbolic computations performed with computer algebra. First, we study solutions to the Kadomtsev-Petviashvili equation that arise from singular curves. The Kadomtsev-Petviashvili equation is a partial differential equation describing nonlinear wave motion whose solutions can be built from an algebraic curve. Such a surprising connection established by Krichever and Shiota also led to an entirely new point of view on a classical problem in algebraic geometry known as the Schottky problem. To explore the connection with curves with at worst nodal singularities, we define the Hirota variety, which parameterizes KP solutions arising from such curves. Studying the geometry of the Hirota variety provides a new approach to the Schottky problem. We investigate it for irreducible rational nodal curves, giving a partial solution to the weak Schottky problem in this case. Second, we formulate questions from scattering amplitudes in a broader context using very affine varieties and D-module theory. The interplay between geometry and combinatorics in particle physics indeed suggests an underlying, coherent mathematical structure behind the study of particle interactions. In this thesis, we gain a better understanding of mathematical objects, such as moduli spaces of point configurations and generalized Euler integrals, for which particle physics provides concrete, non-trivial examples, and we prove some conjectures stated in the physics literature. Finally, we study linear spaces of symmetric matrices, addressing questions motivated by algebraic statistics, optimization, and enumerative geometry. This includes giving explicit formulas for the maximum likelihood degree and studying tangency problems for quadric surfaces in projective space from the point of view of real algebraic geometry.
- Verweis
- KP Solitons from Tropical Limits
Link: https://www.sciencedirect.com/science/article/abs/pii/S0747717122000293 - Hirota Varieties and Rational Nodal Curves
Link: https://www.sciencedirect.com/science/article/abs/pii/S0747717123000536 - Likelihood Degenerations
Link: https://www.sciencedirect.com/science/article/abs/pii/S0001870823000063 - Vector Spaces of Generalized Euler Integrals
Link: https://arxiv.org/abs/2208.08967 - Pencils of Quadrics: Old and New
Link: https://www.sciencedirect.com/science/article/abs/pii/S0747717122000293 - Tangent Quadrics in Real 3-Space
Link: https://lematematiche.dmi.unict.it/index.php/lematematiche/article/view/2268 - Freie Schlagwörter (EN)
- Algebraic Geometry, Integrable Systems, Particle Physics
- Klassifikation (DDC)
- 500
- Den akademischen Grad verleihende / prüfende Institution
- Universität Leipzig, Leipzig
- Version / Begutachtungsstatus
- publizierte Version / Verlagsversion
- URN Qucosa
- urn:nbn:de:bsz:15-qucosa2-864704
- Veröffentlichungsdatum Qucosa
- 17.07.2023
- Dokumenttyp
- Dissertation
- Sprache des Dokumentes
- Englisch
- Lizenz / Rechtehinweis
CC BY 4.0