English
Random algebraic geometry - lecture 3
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 :
3. The square-root law and the topology of random hypersurfaces. In the third lecture I will focus on the case $\mathbb{K}=\mathbb{R}$ and explain in which sense random real algebraic geometry behaves as the 'square root' of complex algebraic geometry. I will discuss a probabilistic version of Hilbert's Sixteenth Problem, following the work of Gayet & Welschinger (introducing a local random version of Nash and Tognoli's Theorem and of Morse theory for the study of Betti numbers of random hypersurfaces) and of Diatta $\&$ Lerario (showing that 'most' hypersurfaces of degree $d$ are isotopic to hypersurfaces of degree $\sqrt{d \log d}$ ).
Keywords : real algebraic geometry; Schubert calculus
Codes MSC : 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 : Petit, JeanLangue : 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 : 00:52:23 Audience : Researchers Download : https://videos.cirm-math.fr/2022-10-27_Lerario_Part3.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.19980103Citer cette vidéo: Lerario, Antonio (2022). Random algebraic geometry - lecture 3. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19980103 URI : http://dx.doi.org/10.24350/CIRM.V.19980103 |
Voir aussi
- [Multi angle] Random algebraic geometry - lecture 4 / Auteur de la Conférence Lerario, Antonio.
- [Multi angle] Random algebraic geometry - lecture 2 / Auteur de la Conférence Lerario, Antonio.
- [Multi angle] Random algebraic geometry - lecture 1 / 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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