Random algebraic geometry - lecture 2 | Vidéo | Carmin.tv

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Random algebraic geometry - lecture 2

By Antonio Lerario

Appears in collection : Real Algebraic Geometry / Géometrie algébrique réelle

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.

Information about the video

Date of recording 26/10/2022
Date of publication 18/11/2022
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19980003
Cite this video Lerario, Antonio (26/10/2022). Random algebraic geometry - lecture 2. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19980003
URL https://dx.doi.org/10.24350/CIRM.V.19980003

Domain(s)

Algebraic Geometry

Bibliography

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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
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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
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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