English
Random algebraic geometry - lecture 1
Multi angle
Auteurs : Lerario, Antonio (Auteur de la Conférence)
CIRM (Editeur )
14P05 - Real algebraic sets, See also {12Dxx}
14P25 - Topology of real algebraic varieties
52A22 - Random convex sets and integral geometry
14N15 - Classical problems, Schubert calculus
CIRM (Editeur )
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Résumé :
In the last years there has been an increasing interest into the statistical behaviour of algebraic sets over non-algebraically closed fields: when the notion of 'generic' is no longer available, one seeks for a 'random' study of the objects of interest. In this course, divided into four lectures, I will present the major ideas in the subject (lecture notes will be made available):
1. Generic and random. In the first lecture I will discuss how to switch from the notion of generic, from classical algebraic geometry, to the notion of random. Of course, this depends on the choice of the probability distribution on the 'moduli space' of the objects of interest. I will discuss what are the reasonable choices in the case $\mathbb{K}=\mathbb{C}$ (where still these questions make sense, and 'random' and 'generic' are synonymous) and in the case $\mathbb{K}=\mathbb{R}$ (where spherical harmonics play a crucial role).
Keywords : real algebraic geometry; Schubert calculus
Codes MSC : 1. Generic and random. In the first lecture I will discuss how to switch from the notion of generic, from classical algebraic geometry, to the notion of random. Of course, this depends on the choice of the probability distribution on the 'moduli space' of the objects of interest. I will discuss what are the reasonable choices in the case $\mathbb{K}=\mathbb{C}$ (where still these questions make sense, and 'random' and 'generic' are synonymous) and in the case $\mathbb{K}=\mathbb{R}$ (where spherical harmonics play a crucial role).
Keywords : real algebraic geometry; Schubert calculus
14P05 - Real algebraic sets, See also {12Dxx}
14P25 - Topology of real algebraic varieties
52A22 - Random convex sets and integral geometry
14N15 - Classical problems, Schubert calculus
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Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de publication : 18/11/2022 Date de captation : 26/10/2022 Sous collection : Research School arXiv category : Algebraic Geometry Domaine : Geometry ; Algebraic & Complex Geometry Format : MP4 (.mp4) - HD Durée : 00:57:16 Audience : Researchers Download : https://videos.cirm-math.fr/2022-10-24_Lerario_Part1.mp4 
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Informations sur la Rencontre
Nom de la rencontre : Real Algebraic Geometry / Géometrie algébrique réelleOrganisateurs de la rencontre : Bihan, Frédéric ; Brugallé, Erwan ; Dickenstein, Alicia ; Dutertre, Nicolas Dates : 26/10/2022 - 30/10/2022 Année de la rencontre : 2022 URL Congrès : https://conferences.cirm-math.fr/2626.html
Données de citation
DOI : 10.24350/CIRM.V.19978403Citer cette vidéo: Lerario, Antonio (2022). Random algebraic geometry - lecture 1. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19978403 URI : http://dx.doi.org/10.24350/CIRM.V.19978403 |
Voir aussi
- [Multi angle] Random algebraic geometry - lecture 4 / Auteur de la Conférence Lerario, Antonio.
- [Multi angle] Random algebraic geometry - lecture 3 / Auteur de la Conférence Lerario, Antonio.
- [Multi angle] Random algebraic geometry - lecture 2 / Auteur de la Conférence Lerario, Antonio.
- [Multi angle] Poincaré-Reeb graphs of real algebraic domains / Auteur de la Conférence Sorea, Miruna-Stefana.
Bibliographie
- BREIDING, Paul, BÜRGISSER, Peter, LERARIO, Antonio, et al. The zonoid algebra, generalized mixed volumes, and random determinants. Advances in Mathematics, 2022, vol. 402, p. 108361. - https://doi.org/10.1016/j.aim.2022.108361
- BÜRGISSER, Peter et LERARIO, Antonio. Probabilistic schubert calculus. Journal für die reine und angewandte Mathematik (Crelles Journal), 2020, vol. 2020, no 760, p. 1-58. - https://doi.org/10.1515/crelle-2018-0009
- DIATTA, Daouda Niang et LERARIO, Antonio. Low-degree approximation of random polynomials. Foundations of Computational Mathematics, 2022, vol. 22, no 1, p. 77-97. - http://dx.doi.org/10.1007/s10208-021-09506-y
- EDELMAN, Alan et KOSTLAN, Eric. How many zeros of a random polynomial are real?. Bulletin of the American Mathematical Society, 1995, vol. 32, no 1, p. 1-37. - http://dx.doi.org/10.1090/S0273-0979-1995-00571-9
- GAYET, Damien et WELSCHINGER, Jean-Yves. Lower estimates for the expected Betti numbers of random real hypersurfaces. Journal of the London Mathematical Society, 2014, vol. 90, no 1, p. 105-120. - https://doi.org/10.1112/jlms/jdu018
- GAYET, Damien et WELSCHINGER, Jean-Yves. What is the total Betti number of a random real hypersurface?. Journal für die reine und angewandte Mathematik (Crelles Journal), 2014, vol. 2014, no 689, p. 137-168. - https://doi.org/10.1515/crelle-2012-0062
- GAYET, Damien et WELSCHINGER, Jean-Yves. Betti numbers of random real hypersurfaces and determinants of random symmetric matrices. Journal of the European Mathematical Society, 2016, vol. 18, no 4, p. 733-772. - https://doi.org/10.4171/jems/601
- HOWARD, Ralph. The kinematic formula in Riemannian homogeneous spaces. American Mathematical Soc., 1993. - https://bookstore.ams.org/memo-106-509
- KAC, Mark. On the average number of real roots of a random algebraic equation. Bulletin of the American Mathematical Society, 1943, vol. 49, no 4, p. 314-320. - https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-49/issue-4/On-the-average-number-of-real-roots-of-a-random/bams/1183505112.full?tab=ArticleLink
- KOSTLAN, Eric. On the distribution of roots of random polynomials. In : From Topology to Computation: Proceedings of the Smalefest. Springer, New York, NY, 1993. p. 419-431. - http://dx.doi.org/10.1007/978-1-4612-2740-3_38
- LERARIO, Antonio. Lecture notes on metric algebraic geometry, 2022. - https://www.mn.uio.no/math/english/research/groups/algebra/events/conferences/nordfjordeid2022/metrics.html
- LERARIO, Antonio et LUNDBERG, Erik. Statistics on Hilbert's 16th problem. International Mathematics Research Notices, 2015, vol. 2015, no 12, p. 4293-4321. - https://doi.org/10.1093/imrn/rnu069
- SARNAK, Peter. Letter to B. Gross and J. Harris on ovals of random plane curves, 2011, p. 1-14. - https://publications.ias.edu/node/510
- SHUB, Michael et SMALE, Steve. Complexity of Bezout's theorem II Volumes and probabilities. In : Computational algebraic geometry. Birkhäuser, Boston, MA, 1993. p. 267-285. - http://dx.doi.org/10.1007/978-1-4612-2752-6_19
- LERARIO, Antonio. What is... random algebraic geometry ? - https://drive.google.com/file/d/1qZ2ebeV4gv7TTEyFRdebGhWZpEp8XY5V/view
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