CIRM - Videos & books Library - Random algebraic geometry

Multi angle

Authors : Lerario, Antonio (Author of the conference)
CIRM (Publisher )

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Abstract : 2. Degree and volume. In the second lecture I will try to explain to what extent the right notion of degree, in the probabilistic context, is the notion of volume. I will introduce the classical kinematic formula, over $\mathbb{R}$ and over $\mathbb{C}$, and explain the role of the Veronese variety in this context. In the complex case I will connect to the Bernstein-Khovanskii-Kouchnirenko Theorem.

Keywords : real algebraic geometry; Schubert calculus

MSC Codes :
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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Film maker :

Hennenfent, Guillaume

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https://videos.cirm-math.fr/2022-10-25_Lerario_Part2.mp4

Information on the Event

Event Title : Real Algebraic Geometry / Géometrie algébrique réelle
Event 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.19980003
Cite this video as: Lerario, Antonio (2022). Random algebraic geometry - lecture 2. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19980003
URI : http://dx.doi.org/10.24350/CIRM.V.19980003

[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 1 / 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

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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