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Quasirandomness of definable subsets of definable groups in finite fields
Multi angle
Authors : Pillay, Anand (Author of the conference)
CIRM (Publisher )
03C45 - Classification theory, stability and related concepts [See also 03C48]
05C75 - Structural characterization of types of graphs
11B30 - Arithmetic combinatorics; higher degree uniformity
Additional resources :
https://www.cirm-math.fr/RepOrga/3112/Slides/Anand-Pillay.pdf
CIRM (Publisher )
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Abstract :
We give an arithmetic version of Tao's algebraic regularity lemma (which was itself an improved Szemerédi regularity lemma for graphs uniformly definable in finite fields). In the arithmetic regime the objects of study are pairs $(G, D)$ where $G$ is a group and $D$ an arbitrary subset, all uniformly definable in finite fields. We obtain optimal results, namely that the algebraic regularity lemma holds for the associated bipartite graph $(G, G, E)$ where $E(x, y)$ is $x y^{-1} \in D$, witnessed by a the decomposition of $G$ into cosets of a uniformly definable small index normal subgroup $H$ of $G$.
Keywords : Arithmetic regularity; finite fields
MSC Codes : Keywords : Arithmetic regularity; finite fields
03C45 - Classification theory, stability and related concepts [See also 03C48]
05C75 - Structural characterization of types of graphs
11B30 - Arithmetic combinatorics; higher degree uniformity
Additional resources :
https://www.cirm-math.fr/RepOrga/3112/Slides/Anand-Pillay.pdf
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Information on the Video
Film maker : Recanzone, LucaLanguage : English Available date : 28/10/2024 Conference Date : 03/10/2024 Subseries : Research talks arXiv category : Algebraic Geometry ; Logic Mathematical Area(s) : Algebra ; Algebraic & Complex Geometry ; Logic and Foundations Format : MP4 (.mp4) - HD Video Time : 00:53:59 Targeted Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students Download : https://videos.cirm-math.fr/2024-10-03_Pillay.mp4 
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Information on the Event
Event Title : Model Theory and Applications to Groups and Combinatorics / Théorie des modèles et applications en théorie des groupes et en combinatoireEvent Organizers : Ben Yaacov, Itaï ; Hempel, Nadja ; Hils, Martin ; Zou, Tingxiang Dates : 30/09/2024 - 04/10/2024 Event Year : 2024 Event URL : https://conferences.cirm-math.fr/3112.html
Citation Data
DOI : 10.24350/CIRM.V.20252803Cite this video as: Pillay, Anand (2024). Quasirandomness of definable subsets of definable groups in finite fields. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20252803 URI : http://dx.doi.org/10.24350/CIRM.V.20252803 |
See Also
- [Multi angle] Pseudofinite omega-categorical groups / Author of the conference Tent, Katrin.
- [Multi angle] A Borovik-Cherlin bound for primitive pseudo-finite permutation groups / Author of the conference Ramsey, Nicholas.
- [Multi angle] Almost periodicity from a model-theoretic perspective / Author of the conference Martin-Pizarro, Amador.
- [Multi angle] Finite-dimensional pseudofinite groups of small dimension, without CFSG / Author of the conference Karhumäki, Ulla.
Bibliography
- TAO, Terence. Expanding polynomials over finite fields of large characteristic, and a regularity lemma for definable sets. arXiv preprint arXiv:1211.2894, 2012. - https://arxiv.org/abs/1211.2894
- GOWERS, W. T. Quasirandom groups, eprint. arXiv, 2007, vol. 710. - https://arxiv.org/abs/0710.3877
- GREEN, Ben. A Szemerédi-type regularity lemma in abelian groups, with applications. Geometric & Functional Analysis GAFA, 2005, vol. 15, no 2, p. 340-376.
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