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Random algebraic geometry - lecture 3
Multi angle
Authors : ... (Author of the conference)
... (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
... (Publisher )
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Abstract :
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
MSC Codes : 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
Language : EnglishAvailable 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 : 00:52:23 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2022-10-27_Lerario_Part3.mp4 
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Information on the Event
Event Title : Real Algebraic Geometry / Géometrie algébrique réelleDates : 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.19980103Cite this video as: (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 |
See Also
- [Multi angle] Random algebraic geometry - lecture 4 / Author of the conference ....
- [Multi angle] Random algebraic geometry - lecture 2 / Author of the conference ....
- [Multi angle] Random algebraic geometry - lecture 1 / Author of the conference ....
- [Multi angle] Poincaré-Reeb graphs of real algebraic domains / Author of the conference ....
Bibliography
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- 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. 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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- 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
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- LERARIO, Antonio. What is... random algebraic geometry ? - https://drive.google.com/file/d/1qZ2ebeV4gv7TTEyFRdebGhWZpEp8XY5V/view
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