The slice rank polynomial method and its limitations
Event details
| Date | 08.02.2023 |
| Hour | 16:00 › 17:00 |
| Speaker | Lisa Sauermann, MIT |
| Location | Online |
| Category | Conferences - Seminars |
| Event Language | English |
Seminar in Mathematics
Abstract: The slice rank polynomial method is a fairly new polynomial method, developed by Tao (building on work of Croot-Lev-Pach and Ellenberg-Gijswijt) in the context of the famous cap-set problem. This problem asks about the maximum possible size of a subset of F_3^n not containing a three-term arithmetic progression, and Ellenberg-Gijswijt proved that any such subset has size at most 2.756^n. The slice rank polynomial method also lead to progress on various other problems in extremal combinatorics and additive number theory. However, the method is not very flexible, and even small changes to the setting can bring the method to fail. This talk discusses several problems where the slice rank polynomial method can be combined with combinatorial and probabilistic tools in order to overcome some of its limitations. Specifically, the talk discusses results on the Erdös-Ginzburg-Ziv problem in discrete geometry, on extremal questions in additive number theory about subsets of F_p^n without distinct-variable solutions to certain (systems of) linear equations, and on bounds for arithmetic removal lemmas (which are closely connected to property testing problems in theoretical computer science).
Abstract: The slice rank polynomial method is a fairly new polynomial method, developed by Tao (building on work of Croot-Lev-Pach and Ellenberg-Gijswijt) in the context of the famous cap-set problem. This problem asks about the maximum possible size of a subset of F_3^n not containing a three-term arithmetic progression, and Ellenberg-Gijswijt proved that any such subset has size at most 2.756^n. The slice rank polynomial method also lead to progress on various other problems in extremal combinatorics and additive number theory. However, the method is not very flexible, and even small changes to the setting can bring the method to fail. This talk discusses several problems where the slice rank polynomial method can be combined with combinatorial and probabilistic tools in order to overcome some of its limitations. Specifically, the talk discusses results on the Erdös-Ginzburg-Ziv problem in discrete geometry, on extremal questions in additive number theory about subsets of F_p^n without distinct-variable solutions to certain (systems of) linear equations, and on bounds for arithmetic removal lemmas (which are closely connected to property testing problems in theoretical computer science).
Practical information
- Informed public
- Free
- This event is internal
Organizer
- Institute of Mathematics
Contact
- Prof. Maryna Viazovska, Director