DOCSLIB.ORG

Remembering John Stallings Danny Calegari and Benson Farb, Coordinating Editors

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Remembering John Stallings Danny Calegari and Benson Farb, Coordinating Editors Remembering John Stallings Danny Calegari and Benson Farb, coordinating editors Introduction/Authors’ Note This document collects reminiscences of John Stallings by some of us who knew him. The contribu- tions have been minimally edited and are supplemented by Stallings’s own “Items from my unknown autobiography”. Photo credits are given where known. Danny Calegari Bestvina-Brady’s discrete Morse theory), and clari- John Robert Stallings Jr. was born fying the work of Serre on trees, especially as it in Morrilton (which John pro- applied to Haken 3-manifolds. In 1983 he wrote nounced “Marrrlton”, rolling the a brief paper, “Topology of finite graphs”, which, r’s), Arkansas, on the 22nd of July besides giving astonishingly short and illuminating 1935, and died at his home in proofs of notoriously opaque algebraic theorems, Berkeley, California, on the 24th anticipated to some extent the theory of automatic of November 2008. He achieved groups and structures developed by Cannon, Thur- early fame as a postdoc with his ston, Epstein et al. Later work was more sporadic alternative proof of the Poincaré but included such important notions as triangles Conjecture in high dimensions of groups and angles between subgroups and con- (obtained only days after Smale’s tinued to be guided by the interaction between (es- breakthrough) using an original pecially 3-dimensional) topology and group theory. method known as engulfing. John John is remembered as much for his anti- Photo credit: Enrico Le Donne, 2006. then switched his attention to John Robert Stallings Jr. establishment attitudes and distinctive personal- 3-manifold topology and geomet- ity as for his mathematics (and sometimes for the ric group theory, and transformed interaction between the two, such as in his famous both. He brought combinatorial methods from paper “How not to prove the Poincaré Conjecture”); group theory to 3-manifold theory, which at the some of this is captured in the reminiscences time was preoccupied with the point-set methods below. He was always a champion of the original- of Bing and Wilder — giving new and cleaner proofs ity and brilliance of teenagers and young people, of Papakyriakopoulos’s fundamental discoveries, keeping in touch with many young correspondents characterizing fibered 3-manifolds by using the even in his seventies and taking great delight in fundamental group, and introducing important the music and antics of modern enfants terrible new methods (e.g., plumbing) in knot theory. Con- such as Eminem. Both aspects of his personality versely, he brought geometry and topology back are revealed in the following excerpt from his to group theory, reinvigorating a field that had introduction to a talk at Barry Mazur’s birthday forgotten the perspective of Dehn and Nielsen, conference in 1998: and laid the groundwork for the later radical in- novations of Thurston and Gromov: characterizing There are some lessons to be learned groups with infinitely many ends (an achievement from this story. One that appeals to me for which he was recognized with the Cole Prize is that Barry broke the rules; he didn’t in 1965), constructing an example of a finitely have a Bachelor’s degree, he didn’t stay presented group with infinite-dimensional H3 quietly in Princeton for three years. (an example which became the cornerstone of Even Douady broke the unspoken rule that one is supposed to wear clothes Danny Calegari is Richard Merkin Professor of Math- under the gown. All those rules, BA ematics at Caltech. His email address is dannyc@its. diploma, residence in Princeton, wear- caltech.edu. ing shoes, being nice to chairmen and 1410 NOTICES OF THE AMS VOLUME 56, NUMBER 11 deans, being a team player—basically I a Ph.D. from Princeton. His thesis gave a new hate those rules, even though I give in topological proof of an old important theorem of and adjust to them. It makes me feel Grushko in combinatorial group theory. This was good to see a rule-breaker succeed! the first indication of how Stallings would trans- form combinatorial group theory by introducing And, teenagers are creative and amaz- powerful topological methods. ing. Barry’s theorem could be compared Stallings definitely followed his own path in to Opus One of some great composer. mathematics. He was known for his great origi- For instance, Gustav Mahler, at the nality; he most often came up with completely age of 17, wrote a long poem, and original ideas rather than following up on the ideas set it to music by the age of 19. This of others. His work often inspired a great flurry cantata, “Das klagende Lied”, is rarely of activity by other mathematicians, who would performed in its original form, because develop Stallings’s methods. There are important Mahler “improved” it later on. I recently cases, though, including his Ph.D. thesis, where heard the original work in Berkeley, Stallings reunderstood someone else’s theorem in played by the Berkeley Symphony Or- a completely new way, which would open the door chestra under Kent Nagano, with a to new developments on an old topic. large chorus, six vocal soloists, two Stallings made fundamental contributions to a of which were from the San Francisco number of different areas of mathematics, from Girls Choir, and an offstage band from combinatorial and geometric group theory to knot Hayward State University that played theory to the theory of 3-dimensional manifolds. in a different key and time-signature. Indeed, he was one of the main forces responsible It had its adolescent moments, but it is for tying these areas together. truly a great work of art. Just as Barry’s Stallings’s ideas were usually remarkably simple sphere theorem was. but remarkably deep. They seemed to grow out of his incessant doodling. He doodled all the time, Those of us who are antiestablishmentarians especially when he attended talks. One example at heart or who met him later in his life probably of a simple but powerful idea is Stallings’s beauti- remember John more for his laconic wit, his pre- ful paper “Topology of finite graphs”. How many dilection for Thai food or for beer and pizza after people could write a deep paper with that title? In seminars (at La Val’s or Jupiter), his fondness for this paper Stallings took our understanding of free reading “Zits” in the morning paper (usually in groups to a new level, along the way making older, Brewed Awakenings), his impossible turn of speed difficult theorems look almost obvious. around about the fourth mile of a hike to Point Stallings was greatly respected as a mathemati- Reyes, his long gossipy monologues while driving cian. One acknowledgment came in the form of his out to a seminar at Stanford, his provocations and being awarded the Cole Prize for his now famous his unconventional behavior (tearing up a Bulletin theorem on ends of groups: a group splits as an article on “Making proofs without modus ponens” amalgamated product over a finite group if and with mock rage; speculating on the likelihood that only if it has at least two ends. Here is a paradig- Alex Gottlieb would fall off the balcony outside matic example of how fine algebraic structure can 1015 Evans) than for his mathematical insights, be determined by coarse, large-scale information. which were rare in his later years but seemed rarer No wonder this theorem is a cornerstone of geo- still because of his modesty and high standards. The truth is, we have internalized his mathematics metric group theory. so that it seems less like a reflection of John and Everyone loved Stallings. He was always gener- more like a fact of nature. He felt more like a fellow ous with his time and with his ideas. He treated graduate student than one of the exalted faculty. students with the same respect he did colleagues, For generations of graduate students at Princeton, indeed with more respect. Stallings had a dislike of Berkeley, and elsewhere, Stallings was larger than authority and made a point of playing by his own life. It is hard to believe that he is gone. rules. He often made fun of authority, but not in a mean way. He made fun of himself all the time. Benson Farb Actually, I can’t think of anything or any person John Stallings was a one-of-a-kind mathematician. (including himself) that Stallings took seriously. Perhaps this is due to the fact that he came out of Stallings was widely known not just for his ge- a small town in Arkansas. He loved to play up his nius but for his sense of humor. As one example, roots, often portraying himself as a kind of naive he wrote an entire paper in the universal language country bumpkin. Of course, he ended up getting Interlingua. One of his famous papers on one of the most famous mathematical problems, “How not Benson Farb is professor of mathematics at the University to prove the Poincaré Conjecture”, begins: “I have of Chicago. His email address is [email protected]. committed—the sin of falsely proving Poincaré’s edu. Conjecture. But that was in another country; and DECEMBER 2009 NOTICES OF THE AMS 1411 besides, until now beautiful paper on the loop theorem. By the time I no one has known was done I had a sense of what 3-manifold theory about it.”1 was about and how closely connected to group Stallings was in- theory it was, and I felt that this was my area of credibly generous research, even though I had not yet made contribu- with his time. He tions of my own.
Load more

Recommended publications
  • Samuel Eilenberg Assistant Professorships
    Faculty of Mathematics, Informatics and Mechanics Dean, Professor Paweł Strzelecki Samuel Eilenberg Assistant Professorships Four academic positions in the group of researchers and lecturers are vailable at the Faculty of Mathematics, Informatics and Mechanics Pursuant to the Law on Higher Education and Science, the Univeristy of Warsaw invites the applications for four positions of Samuel Eilenberg Assistant Professorships. Samuel Eilenberg (1913-1998) obtained his PhD degree in Mathematics at the University of Warsaw in 1936 under the supervision of Kazimierz Kuratowski and Karol Borsuk. He spent most of his career in the USA as a professor at Columbia Univeristy. He exerted a critical influence on contemporary mathematics and theoretical computer science; he was a co- founder of modern topology, category theory, homological algebra, and automata theory. His scientific development epitomizes openness to new ideas and readiness to face demanding intellectual challenges. Terms of employment: starting date October 1, 2020; full time job, competitive salary, fixed term employment contract for 2 or 4 years (to be decided with successful applicants). The candidates will concentrate on intensive research and will have reduced teaching duties (120 teaching hours per academic year). Requirements. Successful candidates for Samuel Eilenberg Professorships should have: 1. a PhD degree in mathematics, computer science or related fields obtained in the past 5 years; 2. significant papers in mathematics or computer science, published in refereed, globally recognized journals or confer- ences; 3. teaching experience and willingness to undertake organizational activities related to teaching; 4. significant international experience (internships or post-doctoral fellowships, projects etc.). Candidates obtain extra points for: • research fellowships outside Poland within the last two years; • high pace of scientific development that is confirmed by high quality papers; a clear concept of maintaining this development is required.
    [Show full text]
  • Cohomology Theory of Lie Groups and Lie Algebras
    COHOMOLOGY THEORY OF LIE GROUPS AND LIE ALGEBRAS BY CLAUDE CHEVALLEY AND SAMUEL EILENBERG Introduction The present paper lays no claim to deep originality. Its main purpose is to give a systematic treatment of the methods by which topological questions concerning compact Lie groups may be reduced to algebraic questions con- cerning Lie algebras^). This reduction proceeds in three steps: (1) replacing questions on homology groups by questions on differential forms. This is accomplished by de Rham's theorems(2) (which, incidentally, seem to have been conjectured by Cartan for this very purpose); (2) replacing the con- sideration of arbitrary differential forms by that of invariant differential forms: this is accomplished by using invariant integration on the group manifold; (3) replacing the consideration of invariant differential forms by that of alternating multilinear forms on the Lie algebra of the group. We study here the question not only of the topological nature of the whole group, but also of the manifolds on which the group operates. Chapter I is concerned essentially with step 2 of the list above (step 1 depending here, as in the case of the whole group, on de Rham's theorems). Besides consider- ing invariant forms, we also introduce "equivariant" forms, defined in terms of a suitable linear representation of the group; Theorem 2.2 states that, when this representation does not contain the trivial representation, equi- variant forms are of no use for topology; however, it states this negative result in the form of a positive property of equivariant forms which is of interest by itself, since it is the key to Levi's theorem (cf.
    [Show full text]
  • Lectures on Quasi-Isometric Rigidity Michael Kapovich 1 Lectures on Quasi-Isometric Rigidity 3 Introduction: What Is Geometric Group Theory? 3 Lecture 1
    Contents Lectures on quasi-isometric rigidity Michael Kapovich 1 Lectures on quasi-isometric rigidity 3 Introduction: What is Geometric Group Theory? 3 Lecture 1. Groups and Spaces 5 1. Cayley graphs and other metric spaces 5 2. Quasi-isometries 6 3. Virtual isomorphisms and QI rigidity problem 9 4. Examples and non-examples of QI rigidity 10 Lecture 2. Ultralimits and Morse Lemma 13 1. Ultralimits of sequences in topological spaces. 13 2. Ultralimits of sequences of metric spaces 14 3. Ultralimits and CAT(0) metric spaces 14 4. Asymptotic Cones 15 5. Quasi-isometries and asymptotic cones 15 6. Morse Lemma 16 Lecture 3. Boundary extension and quasi-conformal maps 19 1. Boundary extension of QI maps of hyperbolic spaces 19 2. Quasi-actions 20 3. Conical limit points of quasi-actions 21 4. Quasiconformality of the boundary extension 21 Lecture 4. Quasiconformal groups and Tukia's rigidity theorem 27 1. Quasiconformal groups 27 2. Invariant measurable conformal structure for qc groups 28 3. Proof of Tukia's theorem 29 4. QI rigidity for surface groups 31 Lecture 5. Appendix 33 1. Appendix 1: Hyperbolic space 33 2. Appendix 2: Least volume ellipsoids 35 3. Appendix 3: Different measures of quasiconformality 35 Bibliography 37 i Lectures on quasi-isometric rigidity Michael Kapovich IAS/Park City Mathematics Series Volume XX, XXXX Lectures on quasi-isometric rigidity Michael Kapovich Introduction: What is Geometric Group Theory? Historically (in the 19th century), groups appeared as automorphism groups of some structures: • Polynomials (field extensions) | Galois groups. • Vector spaces, possibly equipped with a bilinear form| Matrix groups.
    [Show full text]
  • 3-Manifold Groups
    3-Manifold Groups Matthias Aschenbrenner Stefan Friedl Henry Wilton University of California, Los Angeles, California, USA E-mail address: [email protected] Fakultat¨ fur¨ Mathematik, Universitat¨ Regensburg, Germany E-mail address: [email protected] Department of Pure Mathematics and Mathematical Statistics, Cam- bridge University, United Kingdom E-mail address: [email protected] Abstract. We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise. Contents Introduction 1 Chapter 1. Decomposition Theorems 7 1.1. Topological and smooth 3-manifolds 7 1.2. The Prime Decomposition Theorem 8 1.3. The Loop Theorem and the Sphere Theorem 9 1.4. Preliminary observations about 3-manifold groups 10 1.5. Seifert fibered manifolds 11 1.6. The JSJ-Decomposition Theorem 14 1.7. The Geometrization Theorem 16 1.8. Geometric 3-manifolds 20 1.9. The Geometric Decomposition Theorem 21 1.10. The Geometrization Theorem for fibered 3-manifolds 24 1.11. 3-manifolds with (virtually) solvable fundamental group 26 Chapter 2. The Classification of 3-Manifolds by their Fundamental Groups 29 2.1. Closed 3-manifolds and fundamental groups 29 2.2. Peripheral structures and 3-manifolds with boundary 31 2.3. Submanifolds and subgroups 32 2.4. Properties of 3-manifolds and their fundamental groups 32 2.5. Centralizers 35 Chapter 3. 3-manifold groups after Geometrization 41 3.1. Definitions and conventions 42 3.2. Justifications 45 3.3. Additional results and implications 59 Chapter 4. The Work of Agol, Kahn{Markovic, and Wise 63 4.1.
    [Show full text]
  • Algebraic Topology - Wikipedia, the Free Encyclopedia Page 1 of 5
    Algebraic topology - Wikipedia, the free encyclopedia Page 1 of 5 Algebraic topology From Wikipedia, the free encyclopedia Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Contents 1 The method of algebraic invariants 2 Setting in category theory 3 Results on homology 4 Applications of algebraic topology 5 Notable algebraic topologists 6 Important theorems in algebraic topology 7 See also 8 Notes 9 References 10 Further reading The method of algebraic invariants An older name for the subject was combinatorial topology , implying an emphasis on how a space X was constructed from simpler ones (the modern standard tool for such construction is the CW-complex ). The basic method now applied in algebraic topology is to investigate spaces via algebraic invariants by mapping them, for example, to groups which have a great deal of manageable structure in a way that respects the relation of homeomorphism (or more general homotopy) of spaces. This allows one to recast statements about topological spaces into statements about groups, which are often easier to prove. Two major ways in which this can be done are through fundamental groups, or more generally homotopy theory, and through homology and cohomology groups.
    [Show full text]
  • On the Kinematic Formula in the Lives of the Saints Danny Calegari
    SHORT STORIES On the Kinematic Formula in the Lives of the Saints Danny Calegari Saint Sebastian was an early Christian martyr. He served as a captain in the Praetorian Guard under Diocletian un- til his religious faith was discovered, at which point he was taken to a field, bound to a stake, and shot by archers “till he was as full of arrows as an urchin1 is full of pricks2.” Rather miraculously, he made a full recovery, but was later executed anyway for insulting the emperor. The trans- pierced saint became a popular subject for Renaissance painters, e.g., Figure 1. The arrows in Mantegna’s painting have apparently ar- rived from all directions, though they are conspicuously grouped around the legs and groin, almost completely missing the thorax. Intuitively, we should expect more ar- rows in the parts of the body that present a bigger cross section. This intuition is formalized by the claim that a subset of the surface of Saint Sebastian has an area pro- portional to its expected number of intersections with a random line (i.e., arrow). Since both area and expectation are additive, we may reduce the claim (by polygonal ap- proximation and limit) to the case of a flat triangular Saint Sebastian, in which case it is obvious. Danny Calegari is a professor of mathematics at the University of Chicago. His Figure 1. Andrea Mantegna’s painting of Saint Sebastian. email address is [email protected]. 1 hedgehog This is a 3-dimensional version of the classical Crofton 2quills formula, which says that the length of a plane curve is pro- For permission to reprint this article, please contact: portional to its expected number of intersections with a [email protected].
    [Show full text]
  • Groups of PL Homeomorphisms of Cubes
    GROUPS OF PL HOMEOMORPHISMS OF CUBES DANNY CALEGARI AND DALE ROLFSEN D´edi´e`aMichel Boileau sur son soixanti`eme anniversaire. Sant´e! Abstract. We study algebraic properties of groups of PL or smooth homeo- morphisms of unit cubes in any dimension, fixed pointwise on the boundary, and more generally PL or smooth groups acting on manifolds and fixing point- wise a submanifold of codimension 1 (resp. codimension 2), and show that such groups are locally indicable (resp. circularly orderable). We also give many examples of interesting groups that can act, and discuss some other algebraic constraints that such groups must satisfy, including the fact that a group of PL homeomorphisms of the n-cube (fixed pointwise on the boundary) contains no elements that are more than exponentially distorted. 1. Introduction We are concerned in this paper with algebraic properties of the group of PL homeomorphisms of a PL manifold, fixed on some PL submanifold (usually of codimension 1 or 2, for instance the boundary) and some of its subgroups (usually those preserving some structure). The most important case is the group of PL homeomorphisms of In fixed pointwise on ∂In; hence these are “groups of PL homeomorphisms of the (n-)cube”. The algebraic study of transformation groups (often in low dimension, or preserv- ing some extra structure such as a symplectic or complex structure) has recently seen a lot of activity; however, much of this activity has been confined to the smooth category. It is striking that many of these results can be transplanted to the PL category. This interest is further strengthened by the possibility of working in the PL category over (real) algebraic rings or fields (this possibility has already been exploited in dimension 1, in the groups F and T of Richard Thompson).
    [Show full text]
  • Hyperbolic Geometry
    Flavors of Geometry MSRI Publications Volume 31,1997 Hyperbolic Geometry JAMES W. CANNON, WILLIAM J. FLOYD, RICHARD KENYON, AND WALTER R. PARRY Contents 1. Introduction 59 2. The Origins of Hyperbolic Geometry 60 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65 5. Generalizing to Higher Dimensions 67 6. Rudiments of Riemannian Geometry 68 7. Five Models of Hyperbolic Space 69 8. Stereographic Projection 72 9. Geodesics 77 10. Isometries and Distances in the Hyperboloid Model 80 11. The Space at Infinity 84 12. The Geometric Classification of Isometries 84 13. Curious Facts about Hyperbolic Space 86 14. The Sixth Model 95 15. Why Study Hyperbolic Geometry? 98 16. When Does a Manifold Have a Hyperbolic Structure? 103 17. How to Get Analytic Coordinates at Infinity? 106 References 108 Index 110 1. Introduction Hyperbolic geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclid’s axiomatic basis for geometry. It is one type of non-Euclidean geometry, that is, a geometry that discards one of Euclid’s axioms. Einstein and Minkowski found in non-Euclidean geometry a This work was supported in part by The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc., by the Mathematical Sciences Research Institute, and by NSF research grants. 59 60 J. W. CANNON, W. J. FLOYD, R. KENYON, AND W. R. PARRY geometric basis for the understanding of physical time and space. In the early part of the twentieth century every serious student of mathematics and physics studied non-Euclidean geometry.
    [Show full text]
  • Commentary on Thurston's Work on Foliations
    COMMENTARY ON FOLIATIONS* Quoting Thurston's definition of foliation [F11]. \Given a large supply of some sort of fabric, what kinds of manifolds can be made from it, in a way that the patterns match up along the seams? This is a very general question, which has been studied by diverse means in differential topology and differential geometry. ... A foliation is a manifold made out of striped fabric - with infintely thin stripes, having no space between them. The complete stripes, or leaves, of the foliation are submanifolds; if the leaves have codimension k, the foliation is called a codimension k foliation. In order that a manifold admit a codimension- k foliation, it must have a plane field of dimension (n − k)." Such a foliation is called an (n − k)-dimensional foliation. The first definitive result in the subject, the so called Frobenius integrability theorem [Fr], concerns a necessary and sufficient condition for a plane field to be the tangent field of a foliation. See [Spi] Chapter 6 for a modern treatment. As Frobenius himself notes [Sa], a first proof was given by Deahna [De]. While this work was published in 1840, it took another hundred years before a geometric/topological theory of foliations was introduced. This was pioneered by Ehresmann and Reeb in a series of Comptes Rendus papers starting with [ER] that was quickly followed by Reeb's foundational 1948 thesis [Re1]. See Haefliger [Ha4] for a detailed account of developments in this period. Reeb [Re1] himself notes that the 1-dimensional theory had already undergone considerable development through the work of Poincare [P], Bendixson [Be], Kaplan [Ka] and others.
    [Show full text]
  • Interview with Henri Cartan, Volume 46, Number 7
    fea-cartan.qxp 6/8/99 4:50 PM Page 782 Interview with Henri Cartan The following interview was conducted on March 19–20, 1999, in Paris by Notices senior writer and deputy editor Allyn Jackson. Biographical Sketch in 1975. Cartan is a member of the Académie des Henri Cartan is one of Sciences of Paris and of twelve other academies in the first-rank mathe- Europe, the United States, and Japan. He has re- maticians of the twen- ceived honorary doc- tieth century. He has torates from several had a lasting impact universities, and also through his research, the Wolf Prize in which spans a wide va- Mathematics in 1980. riety of areas, as well Early Years as through his teach- ing, his students, and Notices: Let’s start at the famed Séminaire the beginning of your Cartan. He is one of the life. What are your founding members of earliest memories of Bourbaki. His book Ho- mathematical inter- mological Algebra, writ- est? ten with Samuel Eilen- Cartan: I have al- berg and first published ways been interested Henri Cartan’s father, Élie Photograph by Sophie Caretta. in 1956, is still in print in mathematics. But Cartan. Henri Cartan, 1996. and remains a standard I don’t think it was reference. because my father The son of Élie Cartan, who is considered to be was a mathematician. I had no doubt that I could the founder of modern differential geometry, Henri become a mathematician. I had many teachers, Cartan was born on July 8, 1904, in Nancy, France. good or not so good.
    [Show full text]
  • Universal Circles for Quasigeodesic Flows 1 Introduction
    Geometry & Topology 10 (2006) 2271–2298 2271 arXiv version: fonts, pagination and layout may vary from GT published version Universal circles for quasigeodesic flows DANNY CALEGARI We show that if M is a hyperbolic 3–manifold which admits a quasigeodesic flow, then π1(M) acts faithfully on a universal circle by homeomorphisms, and preserves a pair of invariant laminations of this circle. As a corollary, we show that the Thurston norm can be characterized by quasigeodesic flows, thereby generalizing a theorem of Mosher, and we give the first example of a closed hyperbolic 3–manifold without a quasigeodesic flow, answering a long-standing question of Thurston. 57R30; 37C10, 37D40, 53C23, 57M50 1 Introduction 1.1 Motivation and background Hyperbolic 3–manifolds can be studied from many different perspectives. A very fruitful perspective is to think of such manifolds as dynamical objects. For example, a very important class of hyperbolic 3–manifolds are those arising as mapping tori of pseudo-Anosov automorphisms of surfaces (Thurston [24]). Such mapping tori naturally come with a flow, the suspension flow of the automorphism. In the seminal paper [4], Cannon and Thurston showed that this suspension flow can be chosen to be quasigeodesic and pseudo-Anosov. Informally, a flow on a hyperbolic 3–manifold is quasigeodesic if the lifted flowlines in the universal cover are quasi-geodesics in H3 , and a flow is pseudo-Anosov if it looks locally like a semi-branched cover of an Anosov flow. Of course, such a flow need not be transverse to a foliation by surfaces. See Thurston [24] and Fenley [6] for background and definitions.
    [Show full text]
  • Fundamental Theorems in Mathematics
    SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635].
    [Show full text]