Kombinatorické struktury v topologii a geometrii
| Thesis title in Czech: | Kombinatorické struktury v topologii a geometrii |
|---|---|
| Thesis title in English: | Combinatorial structures in topology and geometry |
| Academic year of topic announcement: | 2024/2025 |
| Thesis type: | dissertation |
| Thesis language: | |
| Department: | Department of Applied Mathematics (32-KAM) |
| Supervisor: | doc. RNDr. Martin Tancer, Ph.D. |
| Author: | hidden - assigned and confirmed by the Study Dept. |
| Date of registration: | 02.10.2024 |
| Date of assignment: | 02.10.2024 |
| Confirmed by Study dept. on: | 02.10.2024 |
| Guidelines |
| The applicant will work on resolving some important open questions on the boundary of geometry, topology and combinatorics. Together with resolving the problems, the applicant should also establish new methods and tools. The individual topics will include a work on geometric transversal theory via topological tools. A specific problem to work on is a recent Bárány-Kalai conjecture which generalizes (if true) the well-known Tverberg theorem to polytopes. The optimistic aim would be to resolve the conjecture, though it may be more realistic to resolve at least some important and currently open subcases. The applicant will also work on properties of combinatorial decompositions of topological spaces such as shellability, partitionability, or constructibility. An example of an open problem in the area is whether a skeleton of constructible complex is always constructible.
The initial part of the studies will be the learning phases devoted mainly to studying the literature and deepening the applicant's knowledge in the area. Later on, the applicant should work on his own results (probably with coauthors). |
| References |
| I. Bárány, G. Kalai. Helly-type problems. Bulletin of the American Mathematical Society 59.4(2022): 471-502.
M. Hachimori. Nonconstructible simplicial balls and a way of testing constructibility. Discrete & Computational Geometry 22 (1999): 223-230. A. Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002. V. Kaibel, M. E. Pfetsch. Some algorithmic problems in polytope theory. Algebra, geometry and software systems. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003. 23-47. J. Matoušek. Using the Borsuk-Ulam theorem. Springer-Verlag, Berlin, 2008. P. Soberón, S. Zerbib. The Bárány-Kalai conjecture for certain families of polytopes. ArXiv preprint arXiv:2404.11533, 2024. |