Real Algebraic Geometry / Géometrie algébrique réelle

Collection Real Algebraic Geometry / Géometrie algébrique réelle

Organisateur(s) Bihan, Frédéric ; Brugallé, Erwan ; Dickenstein, Alicia ; Dutertre, Nicolas

Date(s) 26/10/2022 - 30/10/2022

URL associée https://conferences.cirm-math.fr/2626.html

00:00:00 / 00:00:00

4 5

Random algebraic geometry - lecture 3

De Antonio Lerario

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 $\sqrt{d \log d}$ ).

Informations sur la vidéo

Date de captation 26/10/2022
Date de publication 18/11/2022
Institut CIRM
Licence CC BY NC ND
Langue Anglais
Audience Chercheurs
Réalisateur(s) Jean Petit
Format MP4

Données de citation

DOI 10.24350/CIRM.V.19980103
Citer cette vidéo Lerario, Antonio (26/10/2022). Random algebraic geometry - lecture 3. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19980103
URL https://dx.doi.org/10.24350/CIRM.V.19980103

Domaine(s)

Géométrie algébrique

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

Codes MSC

Dernières questions liées sur MathOverflow

Pour poser une question, votre compte Carmin.tv doit être connecté à mathoverflow

Poser une question sur MathOverflow

Toutes les vidéos de la collection

57:16

publiée le 18 novembre 2022

Random algebraic geometry - lecture 1

De Antonio Lerario

01:00:39

publiée le 18 novembre 2022

Random algebraic geometry - lecture 2

De Antonio Lerario

58:34

publiée le 18 novembre 2022

Poincaré-Reeb graphs of real algebraic domains

De Miruna-Stefana Sorea

52:23

publiée le 18 novembre 2022

Random algebraic geometry - lecture 3

De Antonio Lerario

57:17

publiée le 18 novembre 2022

Random algebraic geometry - lecture 4

De Antonio Lerario

Copyright Carmin.tv 2024

Donner son avis