Algebraic Tools in Combinatorial Geometry and Topology
| Thesis title in Czech: | Algebraické nástroje v kombinatorické geometrii a topologii |
|---|---|
| Thesis title in English: | Algebraic Tools in Combinatorial Geometry and Topology |
| Key words: | kombinatorika|topologie|geometrie|komutativní algebra|homologická algebra|Stanley-Reisnerův okruh|těžká Lefschetzova věta |
| English key words: | Combinatorics|Topology|Geometry|Commutative Algebra|Homological Algebra|Stanley-Reisner Ring|Hard Lefschetz Theorem |
| Academic year of topic announcement: | 2018/2019 |
| Thesis type: | dissertation |
| Thesis language: | angličtina |
| 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: | 17.07.2019 |
| Date of assignment: | 17.07.2019 |
| Confirmed by Study dept. on: | 04.10.2019 |
| Date and time of defence: | 28.02.2024 12:20 |
| Date of electronic submission: | 29.09.2023 |
| Date of submission of printed version: | 29.09.2023 |
| Date of proceeded defence: | 28.02.2024 |
| Opponents: | Bruno Benedetti |
| Andreas Holmsen | |
| Guidelines |
| The applicant will work on resolving some important open questions in combinatorial geometry and topology, using the aid of algebra. The applicant will build on established tools such as Stanley's work on face numbers of polytopes/simplicial complexes via commutative algebra as well as very recently introduced tools such as those appearing in the breakthrough result of Adiprasito---a proof of a certain combinatorial hard Lefschetz theorem, which resolved, among others, the g-conjecture and the Grünbaum-Kalai-Sarkaria conjecture. Possible directions of applications of these tools include improving bounds for Helly-type results, e.g., (p,q)-theorem, or improving various non-embeddability results (such as non-embeddability of buildings). |
| References |
| K. Adiprasito: Combinatorial Lefschetz theorems beyond positivity, Preprint https://arxiv.org/abs/1812.10454
D. Eisenbud: Commutative Algebra: with a view toward algebraic geometry. Vol. 150. Springer Science & Business Media, 2013. A. Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002. E. Miller, B. Sturmfels: Combinatorial commutative algebra. Vol. 227. Springer Science & Business Media, 2004. R. Stanley: The number of faces of a simplicial convex polytope, Adv. in Math. 35 (1980), no. 3, 236–238. R. Stanley: Combinatorics and commutative algebra, second ed., Progress in Mathematics, vol. 41, Birkhäuser Boston Inc., Boston, MA, 1996 |