formulasearchengine - User contributions [en]

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2014-12-11T13:36:20Z

132.199.34.102:

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2014-11-25T15:30:28Z

132.199.131.4: /* Example */ missing and

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2014-09-08T14:05:18Z

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2014-03-04T08:39:47Z

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Hurewicz theorem

2014-03-03T19:14:42Z

132.199.33.191: The property "path-connected" is essential for the Hurewicz homomorphism to be the abelianization (cf. Hatcher: Algebraic Topology, §2.A, page 167). Regard (as a counterexample) a disjoint union of two 1-SPHERES. Then \pi_1=\Z and H_1=\Z\oplus\Z.

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2014-02-14T12:03:35Z

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2014-02-12T15:45:08Z

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2014-02-10T14:59:08Z

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2013-11-20T12:07:22Z

132.199.97.12: /* Trends in melting point */ rewrite for clarity

{{Infobox scientist
|name = John R. Stallings
|image = Stallings.jpg
|caption = 2006 photo of Stallings
|birth_date = {{Birth date|1935|7|22}}
|birth_place = [[Morrilton, Arkansas]], [[United States]]
|death_date = {{death date and age|2008|11|24|1935|7|22}}
|death_place = [[Berkeley, California]], [[United States]]
|residence =
|nationality = [[United States]]
|field = [[Mathematics]]
|work_institutions = [[University of California at Berkeley]]
|alma_mater = [[University of Arkansas]] [[Princeton University]]
|doctoral_advisor = [[Ralph Fox]]
|doctoral_students =
|known_for = proof of [[Generalized Poincaré conjecture|Poincaré Conjecture in dimensions greater than six]]; [[Stallings theorem about ends of groups]]
|prizes = [[Cole Prize|Frank Nelson Cole Prize in Algebra]] (1971)
|Erdős number =
|religion =
|footnotes =
}}
'''John Robert Stallings Jr.''' (July 22, 1935 – November 24, 2008) was a [[mathematician]] known for his seminal contributions to [[geometric group theory]] and [[Low-dimensional topology|3-manifold topology]]. Stallings was a Professor Emeritus in the Department of Mathematics at the [[University of California at Berkeley]]<ref name="UCBPR">[http://berkeley.edu/news/media/releases/2009/01/12_stallings.shtml Mathematician John Stallings died last year at 73.] [[UC Berkeley]] press release, January 12, 2009. Accessed January 26, 2009</ref> where he had been a faculty member since 1967.<ref name="UCBPR"/> He published over 50 papers, predominantly in the areas of [[geometric group theory]] and the topology of [[3-manifold]]s. Stallings' most important contributions include a proof, in a 1960 paper, of the [[Generalized Poincaré conjecture|Poincaré Conjecture in dimensions greater than six]] and a proof, in a 1971 paper, of the [[Stallings theorem about ends of groups]].

==Biographical data==

John Stallings was born on July 22, 1935 in [[Morrilton, Arkansas]].<ref name="UCBPR"/>

Stallings received his B.Sc. from [[University of Arkansas]] in 1956 (where he was one of the first two graduates in the university's Honors program)<ref>[http://libinfo.uark.edu/ata/v3no4/honorscollege.asp All things academic.] Volume 3, Issue 4; November 2002.</ref> and he received a Ph.D. in Mathematics from [[Princeton University]] in 1959 under the direction of [[Ralph Fox]].<ref name="UCBPR"/>

After completing his PhD, Stallings held a number of postdoctoral and faculty positions, including being an NSF postdoctoral fellow at [[Oxford University]] as well as and instructorship and a faculty appointment at Princeton. Stallings joined the University of California at Berkeley as a faculty member in 1967 where he remained until his retirement in 1994.<ref name="UCBPR"/> Even after his retirement, Stallings continued supervising UC Berkeley graduate students until 2005.<ref name="NYT"/> Stallings was an [[Sloan Fellowship|Alfred P. Sloan Research fellow]] from 1962–65 and a Miller Institute fellow from 1972-73.<ref name="UCBPR"/>
Over the course of his career, Stallings had 22 doctoral students including [[Marc Culler]] and [[J. Hyam Rubinstein|Hyam Rubinstein]] and 60 doctoral descendants. He published over 50 papers, predominantly in the areas of [[geometric group theory]] and the topology of [[3-manifold]]s.

Stallings delivered an invited address as the [[International Congress of Mathematicians]] in [[Nice]] in 1970<ref>John R. Stallings. ''Group theory and 3-manifolds.'' Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pp. 165–167. Gauthier-Villars, Paris, 1971.</ref> and a James K. Whittemore Lecture at [[Yale University]] in 1969.<ref name="SWM">John Stallings. ''Group theory and three-dimensional manifolds.''
A James K. Whittemore Lecture in Mathematics given at Yale University, 1969. Yale Mathematical Monographs, 4. [[Yale University Press]], New Haven, Conn.–London, 1971.</ref>

Stallings received the [[Cole Prize|Frank Nelson Cole Prize in Algebra]] from the [[American Mathematical Society]] in 1970.<ref>[http://www.ams.org/prizes/cole-prize-algebra.html Frank Nelson Cole Prize in Algebra.] [[American Mathematical Society]].</ref>

The conference "Geometric and Topological Aspects of Group Theory", held at the [[Mathematical Sciences Research Institute]] in Berkeley in May 2000, was dedicated to the 65th birthday of Stallings.<ref>[http://atlas-conferences.com/cgi-bin/calendar/d/faam71 Geometric and Topological Aspects of Group Theory, conference announcement], atlas-conferences.com</ref>
In 2002 a special issue of the journal [[Geometriae Dedicata]] was dedicated to Stallings on the occasion of his 65th birthday.<ref>[http://www.springerlink.com/content/acnlhf5dylu1/?p=36fc0e096ab34a99bf226a2b5cd5ca0a&pi=0 Geometriae Dedicata], vol. 92 (2002). Special issue dedicated to John Stallings on the occasion of his 65th birthday. Edited by R. Z. Zimmer.</ref> Stallings died from [[prostate cancer]] on November 24, 2008.<ref name="NYT">{{citation|title=John R. Stallings Jr., 73, California Mathematician, Is Dead|journal=[[New York Times]]|date=January 18, 2009|url=http://www.nytimes.com/2009/01/19/us/19stallings.html|first=Kenneth|last=Chang}}. Accessed January 26, 2009.</ref><ref>[http://math.berkeley.edu/home.html Professor Emeritus John Stallings of the UC Berkeley Mathematics Department has died.] Announcement at the website of the Department of Mathematics of the [[University of California at Berkeley]]. Accessed December 4, 2008</ref>

==Mathematical contributions==

Most of Stallings' mathematical contributions are in the areas of [[geometric group theory]] and [[low-dimensional topology]] (particularly the topology of [[3-manifold]]s) and on the interplay between these two areas.

An early significant result of Stallings is his 1960 proof<ref>John Stallings. [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183523757 ''Polyhedral homotopy spheres.''] [[Bulletin of the American Mathematical Society]], vol. 66 (1960), pp. 485–488.</ref> of the [[Generalized Poincaré conjecture|Poincaré Conjecture in dimensions greater than six]]. (Stallings' proof was obtained independently from and shortly after the different proof of [[Steve Smale]] who established the same result in dimensions bigger than four<ref>S. Smale. ''Generalized Poincaré's conjecture in dimensions greater than four''. [[Annals of Mathematics]] (2nd Ser.), vol. 74 (1961), no. 2, pp. 391–406</ref>).

Using "engulfing" methods similar to those in his proof of the Poincaré Conjecture for ''n'' > 6, Stallings proved that ordinary Euclidean ''n''-dimensional space has a unique piecewise linear, hence also smooth, structure, if ''n'' is not equal to 4. This took on added significance when, as a consequence of work of [[Michael Freedman]] and [[Simon Donaldson]] in 1982, it was shown that 4-space has [[exotic R4|exotic smooth structures]], in fact uncountably many such.

In a 1963 paper<ref>John Stallings.
''A finitely presented group whose 3-dimensional integral homology is not finitely generated.''
[[American Journal of Mathematics]], vol. 85 (1963), pp. 541–543</ref> Stallings constructed an example of a [[finitely presented group]] with infinitely generated 3-dimensional integral [[Group homology|homology group]] and, moreover, not of the type <math>\mathcal F_3 </math>, that is, not admitting a [[classifying space]] with a finite 3-skeleton. This example came to be called the ''Stallings group'' and is a key example in the study of homological finiteness properties of groups. Bieri later showed<ref>Robert Bieri. ''Homological dimension of discrete groups.''
Queen Mary College Mathematical Notes. [[Queen Mary, University of London|Queen Mary College]], Department of Pure Mathematics, London, 1976.</ref> that the Stallings group is exactly the kernel of the homomorphism from the direct product of three copies of the [[free group]] ''F''2 to the additive group '''Z''' of integers that sends to 1 ∈ '''Z''' the six elements coming from the choice of free bases for the three copies of ''F''2. Bieri also showed that the Stallings group fits into a sequence of examples of groups of type <math>\mathcal F_n </math> but not of type <math>\mathcal F_{n+1} </math>. The Stallings group is a key object in the version of discrete [[Morse theory]] for cubical complexes developed by [[Mladen Bestvina|Bestvina]] and Brady<ref>Mladen Bestvina, and Noel Brady. [http://www.springerlink.com/content/nhj24dgb0vb7bx5p/?p=62f8c742e1c64076994f8b151392c1f6&pi=1 ''Morse theory and finiteness properties of groups''.] [[Inventiones Mathematicae]], vol. 129 (1997), no. 3, pp. 445–470</ref> and in the study of subgroups of direct products of [[limit group]]s.<ref>Martin R. Bridson, James Howie, Charles F. Miller, and Hamish Short. [http://www.springerlink.com/content/l7653623q4205434/ ''The subgroups of direct products of surface groups''.] [[Geometriae Dedicata]], vol. 92 (2002), pp. 95–103.</ref><ref>Martin R. Bridson, and James Howie. [http://www.springerlink.com/content/w34016g8379x5q47/ ''Subgroups of direct products of elementarily free groups.''] [[Geometric and Functional Analysis]], vol. 17 (2007), no. 2, pp. 385–403</ref><ref>Martin R. Bridson, and James Howie. [http://www.mrlonline.org/mrl/2007-014-004/2007-014-004-001.pdf ''Subgroups of direct products of two limit groups.''] Mathematical Research Letters, vol. 14 (2007), no. 4, 547–558</ref>

Stallings' most famous theorem in [[group theory]] is an algebraic characterization of groups with more than one [[End (topology)|end]] (that is, with more than one "connected component at infinity"), which is now known as [[Stallings theorem about ends of groups|Stallings' theorem about ends of groups]]. Stallings proved that a [[finitely generated group]] ''G'' has more than one end if and only if this group admits a nontrivial splitting as an [[free product with amalgamation|amalgamated free product]] or as an [[HNN-extension]] over a finite group (that is, in terms of [[Bass-Serre theory]], if and only if the group admits a nontrivial action on a [[tree (graph theory)|tree]] with finite edge stabilizers). More precisely, the theorem states that a [[finitely generated group]] ''G'' has more than one end if and only if either ''G'' admits a splitting as an [[free product with amalgamation|amalgamated free product]] <math>\scriptstyle G=A\ast_C B</math>, where the group ''C'' is finite and ''C'' ≠ ''A'', ''C'' ≠ ''B'', or ''G'' admits a splitting as an [[HNN-extension]] <math>\scriptstyle G=\langle H, t | t^{-1}Kt=L\rangle</math> where ''K'',''L'' ≤ ''H'' are finite [[subgroup]]s of ''H''.

Stallings proved this result in a series of works, first dealing with the torsion-free case (that is, a group with no nontrivial elements of finite [[Order (group theory)|order]])<ref>John R. Stallings. ''On torsion-free groups with infinitely many ends.'' [[Annals of Mathematics]] (2), vol. 88 (1968), pp. 312–334.</ref> and then with the general case.<ref name="SWM"/><ref>John Stallings. ''Groups of cohomological dimension one.'' Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVIII, New York, 1968) pp. 124–128. [[American Mathematical Society]], Providence, R.I, 1970.</ref> Stalling's theorem yielded a positive solution to the long-standing open problem about characterizing finitely generated groups of cohomological dimension one as exactly the [[free group]]s.<ref>John R. Stallings. [http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.bams/1183529548&page=record ''Groups of dimension 1 are locally free.''] Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 361–364</ref> Stallings' theorem about ends of groups is considered one of the first results in [[geometric group theory]] proper since it connects a geometric property of a group (having more than one end) with its algebraic structure (admitting a splitting over a finite subgroup). Stallings' theorem spawned many subsequent alternative proofs by other mathematicians (e.g.<ref>M. J.Dunwoody. [http://www.springerlink.com/content/yp22n46n40813lwr/ ''Cutting up graphs.''] Combinatorica 2 (1982), no. 1, pp. 15–23.</ref><ref>Warren Dicks, and M. J. Dunwoody. [http://books.google.com/books?id=xgsM5BvkMvIC&printsec=frontcover&dq=Warren+Dicks,+and+M.+J.+Dunwoody.+Groups+acting+on+graphs ''Groups acting on graphs.''] Cambridge Studies in Advanced Mathematics, 17. [[Cambridge University Press]], Cambridge, 1989. ISBN 0-521-23033-0</ref>) as well as many applications (e.g.<ref>Peter Scott. [http://www.jstor.org/pss/2374238 ''A new proof of the annulus and torus theorems.''] American Journal of Mathematics, vol. 102 (1980), no. 2, pp. 241–277</ref>). The theorem also motivated several generalizations and relative versions of Stallings' result to other contexts, such as the study of the notion of relative ends of a group with respect to a subgroup,<ref>G. A.Swarup. [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V0K-45FT7S1-2F&_user=571676&_coverDate=12%2F31%2F1977&_rdoc=11&_fmt=high&_orig=browse&_srch=doc-info(%23toc%235649%231977%23999889998%23298178%23FLP%23display%23Volume)&_cdi=5649&_sort=d&_docanchor=&_ct=28&_acct=C000029040&_version=1&_urlVersion=0&_userid=571676&md5=96e9b99d8411df349be7999d21503ca9 ''Relative version of a theorem of Stallings.'']
Journal of Pure and Applied Algebra, vol. 11 (1977/78), no. 1–3, pp. 75–82</ref><ref>M. J. Dunwoody, and E. L. Swenson. [http://www.springerlink.com/content/892hyejtew7h2vg5/ ''The algebraic torus theorem.''] [[Inventiones Mathematicae]], vol. 140 (2000), no. 3, pp. 605–637</ref><ref>G. P. Scott, and G. A. Swarup. [http://nyjm.albany.edu:8000/PacJ/p/2000/196-2-13.pdf ''An algebraic annulus theorem.''] Pacific Journal of Mathematics, vol. 196 (2000), no. 2, pp. 461–506</ref> including a connection to [[CAT(0) space|CAT(0) cubical complexes]].<ref>Michah Sageev. [http://plms.oxfordjournals.org/cgi/content/abstract/s3-71/3/585 ''Ends of group pairs and non-positively curved cube complexes.''] [[Proceedings of the London Mathematical Society]] (3), vol. 71 (1995), no. 3, pp. 585–617</ref> A comprehensive survey discussing, in particular, numerous applications and generalizations of Stallings' theorem, is given in a 2003 paper of [[Terry Wall|Wall]].<ref>C. T. Wall. ''The geometry of abstract groups and their splittings.''
Revista Matemática Complutense vol. 16 (2003), no. 1, pp. 5–101.</ref>

Another influential paper of Stalling is his 1983 article "Topology on finite graphs".<ref>John R. Stallings. [http://www.springerlink.com/content/mn2h645qw2058530/ ''Topology of finite graphs.''] [[Inventiones Mathematicae]], vol. 71 (1983), no. 3, pp. 551–565</ref> Traditionally, the algebraic structure of [[subgroup]]s of [[free group]]s has been studied in [[combinatorial group theory]] using combinatorial methods, such as the [[Schreier's subgroup lemma|Schreier rewriting method]] and [[Nielsen transformation]]s.<ref>[[Roger Lyndon|Roger C. Lyndon]] and Paul E. Schupp. [http://books.google.com/books?id=aiPVBygHi_oC&printsec=frontcover&dq=Roger+C.+Lyndon+and+Paul+E.+Schupp.+Combinatorial+Group+Theory ''Combinatorial Group Theory.''] Springer–Verlag, New York, 2001. "Classics in Mathematics" series, reprint of the 1977 edition. ISBN 978-3-540-41158-1</ref> Stallings' paper put forward a topological approach based on the methods of [[covering space|covering space theory]] that also used a simple [[graph theory|graph-theoretic]] framework. The paper introduced the notion of what is now commonly referred to as ''Stallings subgroup graph'' for describing subgroups of free groups, and also introduced a foldings technique (used for approximating and algorithmically obtaining the subgroup graphs) and the notion of what is now known as a ''Stallings folding''. Most classical results regarding subgroups of free groups acquired simple and straightforward proofs in this set-up and Stallings' method has become the standard tool in the theory for studying the subgroup structure of free groups, including both the algebraic and algorithmic questions (see <ref name="KWM">Ilya Kapovich, and Alexei Myasnikov. ''Stallings foldings and subgroups of free groups.'' [[Journal of Algebra]], vol. 248 (2002), no. 2, 608–668</ref>). In particular, Stallings subgroup graphs and Stallings foldings have been the used as a key tools in many attempts to approach the [[Hanna Neumann conjecture]].<ref>J. Meakin, and P. Weil. ''Subgroups of free groups: a contribution to the Hanna Neumann conjecture.''
Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000).
Geometriae Dedicata, vol. 94 (2002), pp. 33–43.</ref><ref>Warren Dicks. ''Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture.'' [[Inventiones Mathematicae]], vol. 117 (1994), no. 3, pp. 373–389.</ref><ref>Warren Dicks, and [[Edward Formanek]]. ''The rank three case of the Hanna Neumann conjecture''. Journal of Group Theory, vol. 4 (2001), no. 2, pp. 113–151</ref><ref>Bilal Khan. ''Positively generated subgroups of free groups and the Hanna Neumann conjecture.'' Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), pp. 155–170, Contemp. Math., 296, Amer. Math. Soc., Providence, RI, 2002; ISBN 0-8218-2822-3</ref>

Stallings subgroup graphs can also be viewed as [[finite state automata]]<ref name="KWM"/> and they have also found applications in [[semigroup|semigroup theory]] and in [[computer science]].<ref>Jean-Camille Birget, and Stuart W. Margolis. ''Two-letter group codes that preserve aperiodicity of inverse finite automata.'' Semigroup Forum, vol. 76 (2008), no. 1, pp. 159–168</ref><ref>D. S. Ananichev, A. Cherubini, M. V. Volkov. ''Image reducing words and subgroups of free groups.'' Theoretical Computer Science, vol. 307 (2003), no. 1, pp. 77–92.</ref><ref>J. Almeida, and M. V. Volkov. ''Subword complexity of profinite words and subgroups of free profinite semigroups.'' International Journal of Algebra and Computation, vol. 16 (2006), no. 2, pp. 221–258.</ref><ref>Benjamin Steinberg. ''A topological approach to inverse and regular semigroups.'' Pacific Journal of Mathematics, vol. 208 (2003), no. 2, pp. 367–396</ref>

Stallings' foldings method has been generalized and applied to other contexts, particularly in [[Bass-Serre theory]] for approximating group actions on [[tree (graph theory)|trees]] and studying the subgroup structure of the [[Bass-Serre theory|fundamental groups of graphs of groups]]. The first paper in this direction was written by Stallings himself,<ref>John R. Stallings. ''Foldings of G-trees.'' Arboreal group theory (Berkeley, CA, 1988), pp. 355–368, Math. Sci. Res. Inst. Publ., 19, Springer, New York, 1991; ISBN 0-387-97518-7</ref> with several subsequent generalizations of Stallings' folding methods in the [[Bass-Serre theory]] context by other mathematicians.<ref>Mladen Bestvina and Mark Feighn. ''Bounding the complexity of simplicial group actions on trees'', [[Inventiones Mathematicae]], vol. 103, (1991), no. 3, pp. 449–469</ref><ref>M. J. Dunwoody.
[http://msp.warwick.ac.uk/gtm/1998/01/p007.xhtml ''Folding sequences.''] The Epstein birthday schrift, pp. 139–158 (electronic),
Geometry and Topology Monographs, 1, Geom. Topol. Publ., Coventry, 1998.</ref><ref>Ilya Kapovich, Richard Weidmann, and Alexei Miasnikov. ''Foldings, graphs of groups and the membership problem.'' International Journal of Algebra and Computation, vol. 15 (2005), no. 1, pp. 95–128.</ref><ref>Yuri Gurevich, and Paul E. Schupp. ''Membership problem for the modular group.'' SIAM Journal on Computing, vol. 37 (2007), no. 2, pp. 425–459</ref>

Stallings' 1991 paper ''"Non-positively curved triangles of groups"''<ref>John R. Stallings. ''Non-positively curved triangles of groups.'' Group theory from a geometrical viewpoint (Trieste, 1990), pp. 491–503, World Sci. Publ., River Edge, NJ, 1991; ISBN 981-02-0442-6</ref> introduced and studied the notion of a [[Orbifold#Triangles of groups|triangle of groups]]. This notion was the starting point for the theory of [[Orbifold#Complexes of groups|complexes of groups]] (a higher-dimensional analog of [[Bass-Serre theory]]), developed by Haefliger<ref>[[André Haefliger]]. Complexes of groups and orbihedra. in: "Group theory from a geometrical viewpoint (Trieste, 1990)", pp. 504–540, World Sci. Publ., River Edge, NJ, 1991. ISBN 981-02-0442-6</ref> and others.<ref>Jon Corson. ''Complexes of groups.'' [[Proceedings of the London Mathematical Society]] (3) 65 (1992), no. 1, pp. 199–224.</ref><ref>Martin R. Bridson, and André Haefliger. Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319. Springer-Verlag, Berlin, 1999. ISBN 3-540-64324-9</ref> Stallings' work pointed out the importance of imposing some sort of "non-positive curvature" conditions on the complexes of groups in order for the theory to work well; such restrictions are not necessary in the one-dimensional case of [[Bass-Serre theory]].

Among Stallings' contributions to [[3-manifold|3-manifold topology]], the most well-known is the ''Stallings fibration theorem''.<ref>John R. Stallings. ''On fibering certain 3-manifolds.'' 1962 Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) pp. 95–100. Prentice-Hall, Englewood Cliffs, N.J</ref> The theorem states that if ''M'' is a compact irreducible [[3-manifold]] whose [[fundamental group]] contains a [[normal subgroup]], such that this subgroup is [[finitely generated group|finitely generated]] and such that the [[quotient group]] by this subgroup is [[infinite cyclic group|infinite cyclic]], then ''M'' [[Fibration|fibers]] over a circle. This is an important structural result in the theory of [[Haken manifold]]s that engendered many alternative proofs, generalizations and applications (e.g.<ref>John Hempel, and William Jaco. ''3-manifolds which fiber over a surface.'' [[American Journal of Mathematics]], vol. 94 (1972), pp. 189–205</ref><ref>Alois Scharf. ''Zur Faserung von Graphenmannigfaltigkeiten.'' (in German)
Mathematische Annalen, vol. 215 (1975), pp. 35–45.</ref><ref>Louis Zulli. ''Semibundle decompositions of 3-manifolds and the twisted cofundamental group.''
Topology and its Applications, vol. 79 (1997), no. 2, pp. 159–172</ref><ref>Nathan M. Dunfield, and Dylan P. Thurston. [http://www.msp.warwick.ac.uk/gt/2006/10/p055.xhtml ''A random tunnel number one 3-manifold does not fiber over the circle.''] [[Geometry & Topology]], vol. 10 (2006), pp. 2431–2499</ref> ), including a higher-dimensional analog.<ref>W. Browder, and J. Levine.
[http://www.springerlink.com/content/6832937w73322373/ ''Fibering manifolds over a circle.''] [[Commentarii Mathematici Helvetici]], vol. 40 (1966), pp. 153–160</ref>

A 1965 paper of Stallings ''"How not to prove the Poincaré conjecture"''<ref name="HNTPPC">John R. Stallings. [http://math.berkeley.edu/~stall/notPC.pdf ''How not to prove the Poincaré conjecture''.] Topology Seminar, Wisconsin, 1965.
Edited by R. H. Bing and R. J. Bean. Annals of Mathematics Studies, No. 60. [[Princeton University Press]], Princeton, N.J. 1966</ref> gave a [[group theory|group-theoretic]] reformulation of the famous [[Poincaré conjecture]]. The paper began with a humorous admission: "I have committed the sin of falsely proving Poincare's Conjecture. But that was in another country; and besides, until now, no one has known about it."<ref name="UCBPR"/><ref name="HNTPPC"/> Despite its ironic title, Stallings' paper informed much of the subsequent research on exploring the algebraic aspects of the [[Poincaré Conjecture]] (see, for example,<ref>Robert Myers. [http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=57081&fulltextType=RA&fileId=S0305004100004631 ''Splitting homomorphisms and the geometrization conjecture.''] Mathematical Proceedings of the Cambridge Philosophical Society, vol. 129 (2000), no. 2, pp. 291–300</ref><ref>Tullio Ceccherini-Silberstein. [http://www.springerlink.com/content/un4rk51at6r1cr3h/ ''On the Grigorchuk-Kurchanov conjecture.'']
Manuscripta Mathematica 107 (2002), no. 4, pp. 451–461</ref><ref>V. N. Berestovskii. ''Poincaré's conjecture and related statements.'' (in Russian) Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika. vol. 51 (2000), no. 9, pp. 3–41; translation in Russian Mathematics (Izvestiya VUZ. Matematika), vol. 51 (2007), no. 9, 1–36</ref><ref>V. Poenaru. ''Autour de l'hypothèse de Poincaré''. in: "Géométrie au XXe siècle, 1930–2000 : histoire et horizons". Montréal, Presses internationales Polytechnique, 2005. ISBN 2-553-01399-X, 9782553013997.</ref>).

== Selected works ==
* {{Citation | last1=Stallings | first1=John R. | author1-link=John R. Stallings | title=Polyhedral homotopy spheres | mr=0124905 | year=1960 | journal=Bulletin of the American Mathematical Society | volume=66 | pages=485–488 | url=http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183523757}}
* {{Citation | last1=Stallings | first1=John R. | author1-link=John R. Stallings | title=The piecewise-linear structure of Euclidean space | mr=0149457 | year=1962 | journal=Proceedings of the Cambridge Philosophical Society | volume=58 | issue=03 | pages=481–488 | doi=10.1017/S0305004100036756 | last2=Zeeman | first2=E. C.}}
*{{Citation | last1=Stallings | first1=John R. | author1-link=John R. Stallings | title=Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) | publisher=[[Prentice Hall]] | mr=0158375 | year=1962 | chapter=On fibering certain 3-manifolds | pages=95–100}}
* {{Citation | last1=Stallings | first1=John R. | author1-link=John R. Stallings | title=Homology and central series of groups | doi=10.1016/0021-8693(65)90017-7 | mr=0175956 | year=1965 | journal=Journal of Algebra | volume=2 | issue=2 | pages=170–181 | url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WH2-4D7K7V7-1ND&_user=10&_coverDate=06%2F30%2F1965&_rdoc=2&_fmt=high&_orig=browse&_srch=doc-info(%23toc%236838%231965%23999979997%23518386%23FLP%23display%23Volume)&_cdi=6838&_sort=d&_docanchor=&_ct=4&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=8ab7d96d2a5bc830377ddb428f668224}}
* {{Citation | last1=Stallings | first1=John | author1-link=John R. Stallings | title=A finitely presented group whose 3-dimensional integral homology is not finitely generated | mr=0158917 | year=1963| journal=[[American Journal of Mathematics]] | volume=85 | pages=541–543 | doi=10.2307/2373106 | jstor=2373106 | issue=4 | publisher=The Johns Hopkins University Press}}
* {{Citation | last1=Stallings | first1=John R. | author1-link=John R. Stallings | title=On torsion-free groups with infinitely many ends | doi=10.2307/1970577 | mr=0228573 | year=1968 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | volume=88 | pages=312–334 | issue=2 | publisher=Annals of Mathematics | jstor=1970577}}
*{{Citation | last1=Stallings | first1=John R. | author1-link=John R. Stallings | title=Group theory and three-dimensional manifolds | publisher=[[Yale University Press]] | mr=0415622 | year=1971|isbn=0-300-01397-3}}
*{{Citation | last1=Stallings | first1=John R. | author1-link=John R. Stallings | title=Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Proc. Sympos. Pure Math., XXXII | mr=520522 | year=1978 | chapter=Constructions of fibred knots and links | pages=55–60}}
* {{Citation | last1=Stallings | first1=John R. | author1-link=John R. Stallings | title=Topology of finite graphs | doi=10.1007/BF02095993 | mr=695906 | year=1983 | journal=[[Inventiones Mathematicae]] | volume=71 | issue=3 | pages=551–565 | url=http://www.springerlink.com/content/mn2h645qw2058530/}}, with over 100 recent citations
*{{Citation | last1=Stallings | first1=John R. | author1-link=John R. Stallings | title=Arboreal group theory (Berkeley, CA, 1988)| publisher=Springer | location=New York | series=Mathematical Sciences Research Institute Publications|volume =19 | mr=1105341 | year=1991 | chapter=Folding ''G''-trees| pages=355–368| isbn=0-387-97518-7}}
*{{Citation | last1=Stallings | first1=John R. | author1-link=John R. Stallings | title=Group theory from a geometrical viewpoint (Trieste, 1990) | publisher=World Scientific | location=River Edge, NJ| mr=1170374 | year=1991 | chapter=Non-positively curved triangles of groups | pages=491–903|isbn=981-02-0442-6}}

==Notes==
{{reflist}}

== External links ==
* {{MathGenealogy|id=452}}
*[http://math.berkeley.edu/~stall/ home page] of John Stallings.
*[http://www.ams.org/notices/200911/rtx091101410p.pdf Remembering John Stallings,] [[Notices of the American Mathematical Society]], vol. 56 (2009), no. 11, pp. 1410 1417

{{Authority control|VIAF=93821690}}
{{Persondata 
| NAME = Stallings, John R. Jr.
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = American mathematician
| DATE OF BIRTH = July 22, 1935
| PLACE OF BIRTH = [[Morrilton, Arkansas]], [[United States]]
| DATE OF DEATH = November 24, 2008
| PLACE OF DEATH = [[Berkeley, California]], [[United States]]
}}

{{DEFAULTSORT:Stallings, John R. Jr.}}
[[Category:1935 births]]
[[Category:2008 deaths]]
[[Category:American mathematicians]]
[[Category:Group theorists]]
[[Category:Topologists]]
[[Category:20th-century mathematicians]]
[[Category:21st-century mathematicians]]
[[Category:University of Arkansas alumni]]
[[Category:Princeton University alumni]]
[[Category:University of California, Berkeley faculty]]

Chevalley–Warning theorem

2013-10-18T14:25:11Z

132.199.96.177: fixed a typo

{{ShouldBeSVG}}
==Summary==
This is an illustration of a three-dimensional hyperslice of a simple example of static spherically symmetric perfect fluid solution, with one dimension suppressed, embedded in a flat euclidean spacetime for ease of visualizing the geometry.

The figure shows a perfect fluid ''interior region'' (gold) matched (with continuous metric and extrinsic curvature tensor) across a spherical surface (represented here as a circle) to a vacuum ''exterior region'' (khaki), which is a portion of the Schwarzschild vacuum solution.

In this example, the interior region is Schwarzschild's fluid, in which the hyperslices have the geometry of a ''three-spherical cap''. In this example, the embedding is
:<math>z(r) = 10-\sqrt{125-r^2}, \; 0 < r < 5 </math>
:<math>z(r) = -4 + 2 \sqrt{r-1}, \; 5 < r < \infty</math>

All static spherically symmetric stellar models resemble the Schwarzschild solution near the center, and are matched to a Schwarzschild vacuum exterior, but other choices of fluid solution for the interior region may differ from the geometry shown here in ways difficult to depict without restoring the suppressed dimension.
== Licensing ==
{{cc-by-sa-2.5}}
{{Copy to Wikimedia Commons|bot=Svenbot|priority=true}}

Constructive dilemma

2013-06-25T15:14:14Z

132.199.98.150: /* Variable English */

{{Other uses}}
{{redirect2|RPM|rpm}}

'''Revolutions per minute''' (abbreviated '''rpm''', '''RPM''', '''r/min''', or '''r·min−1''') is a measure of the [[frequency]] of a rotation. It annotates the number of [[Turn (geometry)|turns]] completed in one minute around a [[rotation around a fixed axis|fixed axis]]. It is used as a measure of [[rotational speed]] of a mechanical component.

[[Standards organization]]s generally recommend the symbol '''''r/min''''',{{citation needed|date=September 2013}} which is more consistent with the general use of unit symbols. This is not enforced as an international standard. In French for example, '''tr/mn''' ({{lang|fr|tours par minute}}) is commonly used, and the [[Germany|German]] equivalent reads '''U/min''' ({{lang|de|Umdrehungen pro Minute}}).

== International System of Units ==
According to the [[International System of Units]] (SI), rpm is not a unit. This is because the ''Revolution'' is a [[Semantics|semantic]] [[annotation]] rather than a unit. The annotation is instead done in the subscript of the formula sign if needed. Because of the measured [[physical quantity]], the formula sign has to be ''f'' for (rotational) [[frequency]] and ''ω'' or ''Ω'' for [[angular velocity]]. The corresponding basic SI unit is s−1 or [[Hertz|Hz]]. When measuring angular speed, [[radian per second|rad·s−1]] can also be used as unit.

<math>1~\text{rad/s} = 60/2\pi~~\text{rpm} = c\times9.55~\text{rpm} = 1/2\pi~\text{Hz}</math>

Even though angular velocity, angular frequency and [[hertz]] all have the dimensions of 1/s, angular velocity and angular frequency are not expressed in hertz, but rather in an appropriate angular unit such as radians per second. Thus a disc rotating at 60 revolutions per minute (rpm) is said to be rotating at either 2π rad/s or 1 Hz, where the former measures the angular velocity and latter reflects the number of complete revolutions per second. The conversion between a frequency f measured in hertz and an angular velocity ω measured in radians per second are: 
:<math>\omega = 2 \pi f\,\,\!</math> and <math>\,\,f = \frac {\omega} {2 \pi}\text{.}\,\!</math>

==Examples==
{{main|Orders of magnitude (angular velocity)}}
*On many kinds of disc recording media, the rotational speed of the medium under the read head is a standard given in rpm. [[Gramophone record|Gramophone (phonograph) records]], for example, typically rotate steadily at {{frac|16|2|3}}, {{frac|33|1|3}}, 45 or 78 rpm ({{frac|5|18}}, {{frac|5|9}}, {{frac|3|4}}, or 1.3 Hz respectively).
*Modern ultrasonic [[dental drill]]s can rotate at up to 800,000 rpm (13.3 kHz).
*The "second" hand of a conventional analogue clock rotates at 1 rpm.
*[[Red Book (audio CD standard)|Audio CD]] players read their discs at a precise, constant rate (4.3218 Mbit/s of raw physical data for 1.4112 Mbit/s (180.6 kB/s) of usable audio data) and thus must vary the disc's rotational speed from 8 Hz (480 rpm) when reading at the innermost edge, to 3.5 Hz (210 rpm) at the outer edge.<ref name="MPEG DVD Spec">{{cite web |url=http://www.mpeg.org/MPEG/DVD/Book_A/Specs.html |title=Physical parameters |work=DVD Technical Notes |publisher=Moving Picture Experts Group (MPEG) |date=1996-07-21 |accessdate=2008-05-30}}</ref>
*[[DVD]] players also usually read discs at a constant linear rate. The disc's rotational speed varies from 25.5 Hz (1530 rpm) when reading at the innermost edge, to 10.5 Hz (630 rpm) at the outer edge.<ref name="MPEG DVD Spec" />
*A [[washing machine]]'s drum may rotate at 500 to 2000 rpm (8–33 Hz) during the spin cycles.
*A power generation turbine ([[Alternator#Synchronous_speeds|with a 2 pole alternator]]) rotates at 3000 rpm (50 Hz) or 3600 rpm (60 Hz), depending on country – see [[AC power plugs and sockets]].
* Modern [[Automobile]] [[engine]]s are typically operated around 2000–3000 rpm (33–50 Hz) when cruising, with a minimum (idle) speed around 750–900 rpm (12.5–15 Hz), and an upper limit anywhere from 4500 to 10,000 rpm (75–166 Hz) for a road car or nearly 20,000 rpm for racing engines such as [[Formula One engines|those in Formula 1 cars]] (currently limited to 18,000 rpm).<ref>{{cite web |url=http://www.formula1.com/inside_f1/understanding_the_sport/5280.html |title=Engine / gearbox |publisher=Formula One |accessdate=2008-05-13}}</ref> The exhaust note of [[V8 engine|V8]] [[F1 car]]s have a much higher pitch than an [[straight engine|I4 engine]], because each of the cylinders of a [[four-stroke engine]] fires once for every two revolutions of the [[crankshaft]]. Thus an eight-cylinder engine turning 300 times per second will have an exhaust note of 1200 Hz.
*A piston [[aircraft engine]] typically rotates at a rate between 2000 and 3000 rpm (30–50 Hz).
*Computer [[hard drive]]s typically rotate at 5400 or 7200 rpm (90 or 120 Hz), the most common speeds for the [[Advanced Technology Attachment|ATA]] or [[Serial ATA|SATA]]-based drives in consumer models. High-performance drives (used in fileservers and enthusiast-gaming PCs) rotate at 10,000 or 15,000 rpm (160 or 250 Hz), usually with higher-level SATA, [[SCSI]] or [[Fibre Channel]] interfaces and smaller platters to allow these higher speeds, the reduction in storage capacity and ultimate outer-edge speed paying off in much quicker access time and average transfer speed thanks to the high spin rate. Until recently, lower-end and power-efficient laptop drives could be found with 4200 or even 3600 rpm spindle speeds (70 and 60 Hz), but these have fallen out of favour due to their lower performance, improvements in energy efficiency in faster models and the takeup of [[solid-state drive]]s for use in slimline and ultraportable laptops. Similar to CD and DVD media, the amount of data that can be stored or read for each turn of the disc is greater at the outer edge than near the spindle; however, hard drives keep a constant rotational speed so the effective data rate is faster at the edge (conventionally, the "start" of the disc, opposite to CD/DVD).
*[[Floppy disc]] drives typically ran at a constant 300 or occasionally 360 rpm (a relatively slow 5 or 6 Hz) with a constant per-revolution data density, which was simple and inexpensive to implement, though inefficient. Some designs such as those used with older Apple computers (Lisa, early Macintosh, later II's) were more complex and used variable rotational speeds and per-track storage density (at a constant read/record rate) to store more data per disc; for example, between 394 rpm (with 12 sectors per track) and 590 rpm (8 sectors) with the Mac's 800 KB double-density drive at a constant 39.4 KB/s (max) – vs. 300 rpm, 720 KB and 23 KB/s (max) for double-density drives in other machines.<ref>{{cite web |url=http://support.apple.com/kb/TA39910?viewlocale=en_US |title=Double-Density Versus High-Density Disks |publisher=Apple |accessdate=2012-05-05}}</ref>
*A [[Zippe-type centrifuge]] for enriching uranium spins at 90,000 rpm (1,500 Hz) or faster.<ref>{{cite web |url=http://www.electricityforum.com/news/mar04/centrifuge.html |title=Slender and Elegant, It Fuels the Bomb |publisher=The Electricity Forum |accessdate=2006-09-24}}</ref>
*[[Gas turbine]] engines rotate at tens of thousands of rpm. [[JetCat]] model aircraft turbines are capable of over 100,000 rpm (1,700 Hz) with the fastest reaching 165,000 rpm (2,750 Hz).<ref>{{cite web |url=http://www.jetcatusa.com/p60.html |title=P60-SE Special Edition |publisher=JetCat USA |accessdate=2006-07-19}}</ref>
*A [[Flywheel energy storage]] system works at 60,000–200,000 rpm (1–3 kHz) range using a passively magnetic levitated flywheel in vacuum.<ref>{{cite journal |title=A New Look at an Old Idea: The Electromechanical Battery |last=Post |first=Richard F. |work=Science & Technology Review |date=April 1996 |publisher=University of California |pages=12-19 |url=http://www.llnl.gov/str/pdfs/04_96.2.pdf |format=PDF |issn=10923055 |accessdate=2008-05-30}}</ref> The choice of the flywheel material is not the most dense, but the one that pulverises the most safely, at surface speeds about 7 times the speed of sound.
*A typical 80 mm, 30 CFM computer fan will spin at 2,600–3,000 rpm on 12 V DC power.
*A [[turbocharger]] can reach 290,000 rpm (4,800 Hz), while 80,000–200,000 rpm (1–3 kHz) is common.

==See also==
*[[Constant angular velocity]], or '''CAV''', used when referring to the speed of gramophone (phonograph) records
*[[Constant linear velocity]], or '''CLV''', used when referring to the speed of audio CDs
*[[Orders of magnitude (angular velocity)]]
*[[Turn (geometry)]]

==References==
{{reflist}}

[[Category:Units of frequency]]

Pion decay constant

2012-08-06T15:41:45Z

132.199.99.7:

{{Expert-subject|Physics|date=February 2009}}
{{Lie groups |Physics}}

The connection between [[particle physics]] and [[representation theory]] is a natural connection, first noted in the 1930s by [[Eugene Wigner]],<ref>Wigner received the [[Nobel Prize in Physics]] in 1963 "for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles"; see also [[Wigner's theorem]], [[Wigner's classification]].</ref> between the properties of [[elementary particle]]s and the structure of [[Lie groups]] and [[Lie algebras]]. According to this connection, the different [[quantum state]]s of an elementary particle give rise to an [[irreducible representation]] of the [[Poincaré group]]. Moreover, the properties of the various particles, including their [[energy spectrum|spectra]], can be related to representations of Lie algebras, corresponding to "approximate symmetries" of the universe.

== General picture ==

In [[quantum mechanics]], any particular particle (with a given momentum distribution, location distribution, spin state, etc.) is written as a [[vector space|vector]] (or "[[bra-ket notation|ket]]") in a [[Hilbert space]] H. To help understand what types of particles can exist, it is important to classify the possibilities for H, and their properties. The particle is more precisely characterized by the associated ''[[projective space|projective]]'' Hilbert space '''P'''H, since two vectors that differ by a scalar factor (or in physics terminology, two "kets" that differ by a "[[phase factor]]") correspond to the same physical [[quantum state]].

Let ''G'' be the ''symmetry group of the universe'' – that is, the set of symmetries under which the laws of physics are invariant. (For example, one element of ''G'' is the simultaneous translation of all particles and fields forward in time by five seconds.) Starting with a particular particle in the state ket <math>|p_0\rangle</math>, and a symmetry transformation ''g'' in ''G'', it is possible to apply the symmetry transformation to the particle to get a new state ket <math>|p_g\rangle=g|p_0\rangle</math>. For this picture to be consistent, it is necessary that '''P'''H is a [[projective representation|projective group representation]] of ''G''. (For example, this condition guarantees that applying a symmetry transformation, then applying its inverse transformation, will restore the original quantum state.)

Therefore, any given particle is associated with a unique [[projective representation|representation]] of ''G'' on a projective vector space '''P'''H. (We say the particle "lies in", or "transforms as" the representation.) In many important cases, it can be shown that the particle is also (more specifically) associated with a [[group representation]] of ''G'' on the underlying (non-projective) space H.<ref name=WeinbergCh2Appendix/> [[Wigner's Theorem]] proves that it is a [[unitary representation]], or possibly anti-unitary.<ref name=WeinbergCh2Appendix>See Weinberg (1995), Chapter 2 appendix A and B.</ref>

So we conclude that each type of particle corresponds to a representation of ''G'', and if we can classify the group representations of ''G'', we will have much more information about the possibilities and properties of H, and hence what types of particles can exist.

== Poincaré group ==
The group of translations and [[Lorentz transformations]] form the [[Poincaré group]], and this group is certainly a subgroup of ''G'' (neglecting [[general relativity]] effects, or in other words, in [[flat space]]). Hence, any representation of ''G'' will in particular be a representation of the Poincaré group. [[Representation theory of the Poincaré group|Representations of the Poincaré group]] are in many cases characterized by a nonnegative [[mass]] and a half-integer [[spin (physics)|spin]] (see [[Wigner's classification]]); this can be thought of as the reason that particles have quantized spin. (Note that there are in fact other possible representations, such as [[tachyon]]s, [[infraparticle]]s, etc., which in some cases do not have quantized spin or fixed mass.)

== Other symmetries ==
[[File:Standard Model charges.svg|300px|right|thumb|The pattern of [[weak isospin]]s, [[weak hypercharge]]s, and [[color charge|color]] charges (weights) of all known elementary particles in the [[Standard Model]], rotated by the [[weak mixing angle]] to show electric charge roughly along the vertical.]]

While the [[spacetime symmetries]] in the Poincaré group are particularly easy to visualize and believe, there are also other types of symmetries, called [[internal symmetry|internal symmetries]]. One example is [[color charge|color]] [[SU(3)]], an exact symmetry corresponding to the continuous interchange of the three [[quark]] colors.

== Approximate symmetries ==

Although the above symmetries are believed to be exact, other symmetries are only approximate.

===Hypothetical example===
As an example of what an approximate symmetry means, suppose we lived inside an infinite [[ferromagnet]], with magnetization in some particular direction. An experimentalist in this situation would find not one but two distinct types of electrons: one with spin along the direction of the magnetization, with a slightly lower energy (and consequently, a lower mass), and one with spin anti-aligned, with a higher mass. Our usual [[SO(3)]] rotational symmetry, which ordinarily connects the spin-up electron with the spin-down electron, has in this hypothetical case become only an ''approximate'' symmetry, relating ''different types of particles'' to each other.

===Lie algebras versus Lie groups===
Many (but not all) symmetries or approximate symmetries, for example the ones above, form [[Lie groups]]. Rather than study the [[Representation theory#Lie groups|representation theory]] of these Lie groups, it is often preferable to study the closely related [[Representation theory#Lie algebras|representation theory]] of the corresponding Lie algebras, which are usually simpler to compute.

===General definition===
In general, an approximate symmetry arises when there are very strong interactions that obey that symmetry, along with weaker interactions that do not. In the electron example above, the two "types" of electrons behave identically under the [[strong force|strong]] and [[weak force]]s, but differently under the [[electromagnetic force]].

===Example: isospin symmetry===
{{main|Isospin}}
An example from the real world is [[isospin|isospin symmetry]], an [[SU(2)]] group corresponding to the similarity between [[up quark]]s and [[down quark]]s. This is an approximate symmetry: While up and down quarks are identical in how they interact under the [[strong force]], they have different masses and different electroweak interactions. Mathematically, there is an abstract two-dimensional vector space
:<math>\text{up quark} \rightarrow \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \qquad \text{down quark} \rightarrow \begin{pmatrix} 0 \\ 1 \end{pmatrix}</math>
and the laws of physics are ''approximately'' invariant under applying a determinant-1 [[unitary transformation]] to this space:<ref name=Thomson>[http://www.hep.phy.cam.ac.uk/~thomson/partIIIparticles/handouts/Handout_7_2011.pdf Lecture notes by Prof. Mark Thomson]</ref>
:<math>\begin{pmatrix} x \\ y \end{pmatrix} \mapsto A \begin{pmatrix} x \\ y \end{pmatrix}, \quad \text{where } A \text{ is in } SU(2)</math>
For example, <math>A=\begin{pmatrix} 0&1 \\ -1&0 \end{pmatrix}</math> would turn all up quarks in the universe into down quarks and vice-versa. Some examples help clarify the possible effects of these transformations:
*When these unitary transformations are applied to a [[proton]], it can be transformed into a [[neutron]], or into a superposition of a proton and neutron, but not into any other particles. Therefore, the transformations move the proton around a two-dimensional space of quantum states. The proton and neutron are called an "[[isospin multiplet|isospin doublet]]", mathematically analogous to how a [[spin-½]] particle behaves under ordinary rotation.
*When these unitary transformations are applied to any of the three [[pion]]s ({{SubatomicParticle|Pion0}}, {{SubatomicParticle|Pion+}}, and {{SubatomicParticle|Pion-}}), it can change any of the pions into any other, but not into any non-pion particle. Therefore, the transformations move the pions around a three-dimensional space of quantum states. The pions are called an "[[isospin multiplet|isospin triplet]]", mathematically analogous to how a spin-1 particle behaves under ordinary rotation.
*These transformations have no effect at all on an [[electron]], because it contains neither up nor down quarks. The electron is called an isospin singlet, mathematically analogous to how a spin-0 particle behaves under ordinary rotation.

In general, particles form [[isospin multiplet]]s, which correspond to irreducible representations of the [[Special unitary group#Lie algebra|Lie algebra SU(2)]]. Particles in an isospin multiplet have very similar but not identical masses, because the up and down quarks are very similar but not identical.

===Example: Flavour symmetry===
Isospin symmetry can be generalized to [[flavour symmetry]], an [[SU(3)]] group corresponding to the similarity between [[up quark]]s, [[down quark]]s, and [[strange quark]]s.<ref name=Thomson/> This is, again, an approximate symmetry, violated by quark mass differences and electroweak interactions—in fact, it is a poorer approximation than isospin, because of the strange quark's noticeably higher mass.

Nevertheless, particles can indeed be neatly divided into groups that form irreducible representations of the [[Special unitary group#Lie algebra|Lie algebra SU(3)]], as first noted by [[Murray Gell-Mann]] and independently by [[Yuval Ne'eman]] (see [[eightfold way (physics)|the eightfold way]]).

== See also ==
*[[Lie algebra]]
*[[Lie group]]
*[[Poincaré group]]
*[[Representation theory]]:
**[[Representation theory#Lie algebras|Of Lie algebras]]
**[[Representation theory#Lie groups|Of Lie groups]]
**[[Representation theory of the Poincaré group|Of the Poincaré group]]
*[[Special unitary group]]
*[[Symmetry]]

== References ==
*Coleman, Sidney (1985) ''Aspects of Symmetry: Selected Erice Lectures of Sidney Coleman''. Cambridge Univ. Press. ISBN 0-521-26706-4.
*Georgi, Howard (1999) ''Lie Algebras in Particle Physics''. Reading, MA: Perseus Books. ISBN 0-7382-0233-9.
* Hall, Brian C., (2006) ''Lie Groups, Lie Algebras, and Representations: An Elementary Introduction''. Springer. ISBN 0-387-40122-9.
*Sternberg, Shlomo (1994) ''Group Theory and Physics''. Cambridge Univ. Press. ISBN 0-521-24870-1. Especially pp. 148–150.
*{{cite book | author=[[Steven Weinberg]] | title=The Quantum Theory of Fields, Volume 1: Foundations | publisher=Cambridge Univ. Press | year=1995 | isbn=0-521-55001-7}} Especially appendices A and B to Chapter 2.

== Notes ==
{{reflist}}

== External links ==
*John C. Baez & John Huerta, ''The Algebra of Grand Unified Theories'', {{arXiv|0904.1556}}

{{DEFAULTSORT:Particle Physics And Representation Theory}}
[[Category:Lie algebras]]
[[Category:Particle physics]]
[[Category:Representation theory of Lie groups]]
[[Category:Theoretical physics]]
[[Category:Conservation laws]]
[[Category:Quantum field theory]]

Functor category

2012-08-01T07:03:05Z

132.199.101.221: /* Definition */

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Concurrence (quantum computing)

2012-06-13T09:16:28Z

132.199.97.100: The concept is not specific to quantum computing

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Banach bundle (non-commutative geometry)

2012-06-11T13:33:51Z

132.199.101.221: /* Definition */

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