Geometry of multidimensional continued fractions
| Thesis title in Czech: | Geometrie vícerozměrných řetězových zlomků |
|---|---|
| Thesis title in English: | Geometry of multidimensional continued fractions |
| Key words: | geometrické řetězové zlomky|celočíselná geometrie|totálně reálná číselná tělesa |
| English key words: | continued fractions|integer geometry|totally real number fields |
| Academic year of topic announcement: | 2023/2024 |
| Thesis type: | diploma thesis |
| Thesis language: | angličtina |
| Department: | Department of Algebra (32-KA) |
| Supervisor: | doc. Mgr. Vítězslav Kala, Ph.D. |
| Author: | hidden - assigned and confirmed by the Study Dept. |
| Date of registration: | 26.03.2024 |
| Date of assignment: | 01.04.2024 |
| Confirmed by Study dept. on: | 01.04.2024 |
| Date and time of defence: | 13.06.2024 09:00 |
| Date of electronic submission: | 02.05.2024 |
| Date of submission of printed version: | 02.05.2024 |
| Opponents: | Pavlo Yatsyna, Ph.D. |
| Guidelines |
| The geometric approach to multidimensional continued fractions is based on the notion of sail, defined as the boundary of the convex hull of lattice points in a cone.
The student will develop basic theory of sails for general lattices and general cones, including some facts about empty integer simplices. In particular, he will focus on the case of the ring of integers in a number field under Minkowski embedding, and of sails attached to a matrix [Kar, Chapter 22]. He will also consider the question of which polyhedra can be realized as faces of sails. |
| References |
| [Ill] A. A. Illarionov (2013). On the statistical properties of Klein polyhedra in three-dimensional lattices. Sb. Math. 204 801.
[Kar] Karpenkov, O. (2022). Geometry of Continued Fractions. Algorithms and Computation in Mathematics, vol 26. Springer Berlin, Heidelberg. [Neu] Neukirch, J. (1999). Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften, vol 332. Springer Berlin, Heidelberg. |