Celebratio Mathematica — Simons

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Works connected to Jean Pierre Bourguignon

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J.-P. Bourguignon, H. B. Lawson, and J. Simons: “Stability and gap phenomena for Yang–Mills fields,” Proc. Natl. Acad. Sci. U.S.A. 76 : 4 (April 1979), pp. 1550–1553. MR 526178 Zbl 0408.53023 article

It is shown that any weakly stable Yang–Mills field of type \( \mathit{SU}(2) \) or \( \mathit{SU}(3) \) on the four-sphere must be self-dual or anti-self-dual. Any Yang–Mills field on \( \mathbb{S}^n \), \( n \geq 5 \), is unstable. Examples of stable fields on \( \mathbb{S}^4 \) and \( \mathbb{S}^n/\Gamma \) for \( n \geq 5 \) and \( \Gamma \neq \{e\} \) are given. It is also shown that, for any Yang–Mills field \( R \) on \( \mathbb{S}^4 \), the pointwise condition \( \| R^-\|^2 < 3 \) (or \( \| R^+\|^2 < 3 \)) implies that \( R^- = 0 \) (or respectively that \( R^+ = 0 \)). In general, any Yang–Mills field \( R \) on \( \mathbb{S}^n \), \( n \geq 3 \), that satisfies the pointwise condition \( \| R\|^2 < \frac{1}{2}{2 \choose n} \) is trivial. If \( n = 3 \) or 4, the condition \( \| R\|^2 < \frac{1}{2}{2 \choose n} \) implies that either \( R \) is the trivial field or it is the direct sum of a trivial field with a field of tangent spinors carrying the standard connection.

H. Blaine Lawson, Jr.	Related
Jean Pierre Bourguignon	Related

J. Simons and D. Sullivan: “Structured bundles define differential K-theory,” pp. 1–3 in Géométrie différentielle, physique mathématique, mathématiques et société, I: Volume en l’honneur de Jean Pierre Bourguignon [Differential geometry, mathematical physics, mathematics and society, I: Volume in honor of Jean Pierre Bourguignon]. Edited by O. Hijazi. Astérisque 321–322. Société Mathématique de France (Paris), 2008. MR 2521641 Zbl 1204.57021 incollection

Oussama Hijazi	Related
Jean Pierre Bourguignon	Related
Dennis P. Sullivan	Related

@incollection {key2521641m,
	AUTHOR = {Simons, James and Sullivan, Dennis},
	TITLE = {Structured bundles define differential
		 {K}-theory},
	BOOKTITLE = {G\'eom\'etrie diff\'erentielle, physique
		 math\'ematique, math\'ematiques et soci\'et\'e,
		 {I}: {V}olume en l'honneur de {J}ean
		 {P}ierre {B}ourguignon [Differential
		 geometry, mathematical physics, mathematics
		 and society, {I}: {V}olume in honor
		 of {J}ean {P}ierre {B}ourguignon]},
	EDITOR = {Hijazi, Oussama},
	SERIES = {Ast\'erisque},
	NUMBER = {321--322},
	PUBLISHER = {Soci\'et\'e Math\'ematique de France},
	ADDRESS = {Paris},
	YEAR = {2008},
	PAGES = {1--3},
	URL = {http://smf4.emath.fr/en/Publications/Asterisque/2008/321/html/smf_ast_321_1-3.php},
	NOTE = {MR:2521641. Zbl:1204.57021.},
	ISSN = {0303-1179},
	ISBN = {9782856292587},
}