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Random algebraic geometry - lecture 1
Multi angle
Authors : Lerario, Antonio (Author of the conference)
CIRM (Publisher )
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 (Publisher )
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Abstract :
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
MSC Codes : 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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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 18/11/2022 Conference Date : 26/10/2022 Subseries : Research School arXiv category : Algebraic Geometry Mathematical Area(s) : Geometry ; Algebraic & Complex Geometry Format : MP4 (.mp4) - HD Video Time : 00:57:16 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2022-10-24_Lerario_Part1.mp4 
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Information on the Event
Event Title : Real Algebraic Geometry / Géometrie algébrique réelleEvent Organizers : Bihan, Frédéric ; Brugallé, Erwan ; Dickenstein, Alicia ; Dutertre, Nicolas Dates : 26/10/2022 - 30/10/2022 Event Year : 2022 Event URL : https://conferences.cirm-math.fr/2626.html
Citation Data
DOI : 10.24350/CIRM.V.19978403Cite this video as: 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 |
See Also
- [Multi angle] Random algebraic geometry - lecture 4 / Author of the conference Lerario, Antonio.
- [Multi angle] Random algebraic geometry - lecture 3 / Author of the conference Lerario, Antonio.
- [Multi angle] Random algebraic geometry - lecture 2 / Author of the conference Lerario, Antonio.
- [Multi angle] Poincaré-Reeb graphs of real algebraic domains / Author of the conference Sorea, Miruna-Stefana.
Bibliography
- 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
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- 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
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- 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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