From Hilbert's 13th problem to essential dimension and back
type: Item
instance of: scholarly article
title: From Hilbert's 13th problem to essential dimension and back
zbMATH Open document ID: 1492.14085
DOI: 10.4171/MAG-57
author: Zinovy Reichstein
published in: European Mathematical Society Magazine
publication date: 2022-02-17T00:00:00Z
review text: Summary: This paper has two goals. The first is to survey some of the research on essential dimension in Sections 2--7. This survey is not comprehensive; it is only intended to convey the flavor of the subject and sample some of its highlights. My second goal for this paper is to define the notion of resolvent degree of an algebraic group in Section 8, building on the work of Farb and Wolfson [\textit{B. Farb} and \textit{J. Wolfson}, Enseign. Math. (2) 65, No. 3--4, 303--376 (2019; Zbl 1460.14104); \textit{B. Farb} and \textit{J. Wolfson}, Enseign. Math. (2) 65, No. 3--4, 303--376 (2019; Zbl 1460.14104)] but focusing on connected, rather than finite groups. The quantity \(\mathrm{rd}(n)\) defined above is recovered in this setting as \(\mathrm{rd}(S_n)\). For more comprehensive surveys of essential dimension and resolvent degree, see [\textit{A. S. Merkurjev}, Transform. Groups 18, No. 2, 415--481 (2013; Zbl 1278.14066); \textit{Z. Reichstein}, in: Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19--27, 2010. Vol. II: Invited lectures. Hackensack, NJ: World Scientific; New Delhi: Hindustan Book Agency. 162--188 (2011; Zbl 1232.14030)] and [\textit{B. Farb} and \textit{J. Wolfson}, Enseign. Math. (2) 65, No. 3--4, 303--376 (2019; Zbl 1460.14104)], respectively.
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 11E72
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 12-03
Mathematics Subject Classification ID: 01A60
zbMATH DE Number: 7475657
MaRDI profile type: Publication
cites work: Q3263814
cites work: From Hilbert's Superposition Problem to Dynamical Systems
cites work: Kirillov's orbit method and polynomiality of the faithful dimension of $p$-groups
cites work: Finite Simple Groups of Small Essential Dimension
cites work: Fields of Definition for Representations of Associative Algebras
cites work: Essential dimension: A functorial point of view (after A. Merkurjev)
cites work: Singularities of linear systems and boundedness of Fano varieties
cites work: On the resolvent problem
cites work: Essential dimension, spinor groups, and quadratic forms
cites work: Essential dimension in mixed characteristic
cites work: Essential dimension of abelian varieties over number fields
cites work: Q4090273
cites work: Q4345956
cites work: Q5834527
cites work: Resolving \(G\)-torsors by abelian base extensions.
cites work: Essential dimension of spinor and Clifford groups
cites work: Lower bounds for essential dimensions via orthogonal representations.
cites work: Groups with essential dimension one
cites work: Q4282462
cites work: Finite Subgroups of the Plane Cremona Group
cites work: Essential dimensions of \(A_7\) and \(S_7\)
cites work: Finite groups of essential dimension 2
cites work: Resolvent degree, Hilbert's 13th problem and geometry
cites work: On the essential dimension of cyclic \(p\)-groups
cites work: Cohomological invariants: exceptional groups and spin groups
cites work: Spinors and essential dimension
cites work: A lower bound on the essential dimension of a connected linear group
cites work: A numerical invariant for linear representations of finite groups
cites work: Essential dimension of finite \(p\)-groups.
cites work: Minimal models and the Kodaira dimension of algebraic fiber spaces.
cites work: Q3235831
cites work: Q3232381
cites work: Q4817221
cites work: Finite groups of essential dimension one
cites work: Essential dimension: a survey
cites work: Some consequences of the Karpenko-Merkurjev theorem.
cites work: Q3762473
cites work: The moduli space of commutative algebras of finite rank
cites work: On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties
cites work: Three plots about the Cremona groups
cites work: Q5602178
cites work: Simple finite subgroups of the Cremona group of rank 3
cites work: Jordan property for Cremona groups
cites work: Q3096595
cites work: On the notion of essential dimension for algebraic groups
cites work: Essential dimension of inseparable field extensions
cites work: Essential dimension of finite groups in prime characteristic
cites work: Essential Dimensions of Algebraic Groups and a Resolution Theorem for G-Varieties
cites work: Essential dimension of representations of algebras
cites work: Q4416651
cites work: Q3076568
cites work: Essential dimension of the spin groups in characteristic 2
cites work: A method for removing all intermediate terms from a given equation
cites work: Tschirnhaus transformations after Hilbert
full work available at URL: mag-57
OpenAlex ID: W4200508239
description: scientific article; zbMATH DE number 7475657