Collected Papers - Gesammelte Abhandlungen Series: Springer Collected Works in Mathematics

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H. Hopf B. Eckmann (Ed.) Collected Papers - Gesammelte Abhandlungen Series: Springer Collected Works in Mathematics From the preface: "Hopf algebras, Hopf fibration of spheres, Hopf-Rinow complete Riemannian manifolds, Hopf theorem on the ends of groups - can one imagine modern mathematics without all this? Many other concepts and methods, fundamental in various mathematical disciplines, also go back directly or indirectly to the work of Heinz Hopf: homological algebra, singularities of vector fields and characteristic classes, group-like spaces, global differential geometry, and the whole algebraisation of topology with its influence on group theory, analysis and algebraic geometry. It is astonishing to realize that this oeuvre of a whole scientific life consists of only about 70 writings. Astonishing also the transparent and clear style, the concreteness of the problems, and how abstract and far-reaching the methods Hopf invented." 2001. Reprint 2013 of the 2001 edition, XIII, 1271 p. 1 illus. Printed book Softcover ▶ 64,99 € | £54.99 | $79.99 ▶ *69,54 € (D) | 71,49 € (A) | CHF 77.00 Order online at springer.com ▶ or for the Americas call (toll free) 1-800-SPRINGER ▶ or email us at: [email protected]. ▶ For outside the Americas call +49 (0) 6221-345-4301 ▶ or email us at: [email protected]. The first € price and the £ and $ price are net prices, subject to local VAT. Prices indicated with * include VAT for books; the €(D) includes 7% for Germany, the €(A) includes 10% for Austria. Prices indicated with ** include VAT for electronic products; 19% for Germany, 20% for Austria. All prices exclusive of carriage charges. Prices and other details are subject to change without notice. All errors and omissions excepted..

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Mathematics People
Mathematics People symmetric spaces, L²-cohomology, arithmetic groups, and MacPherson Awarded Hopf the Langlands program. The list is by far not complete, and Prize we try only to give a representative selection of his contri- bution to mathematics. He influenced a whole generation Robert MacPherson of the Institute for Advanced Study of mathematicians by giving them new tools to attack has been chosen the first winner of the Heinz Hopf Prize difficult problems and teaching them novel geometric, given by ETH Zurich for outstanding scientific work in the topological, and algebraic ways of thinking.” field of pure mathematics. MacPherson, a leading expert Robert MacPherson was born in 1944 in Lakewood, in singularities, delivered the Heinz Hopf Lectures, titled Ohio. He received his B.A. from Swarthmore College in “How Nature Tiles Space”, in October 2009. The prize also 1966 and his Ph.D. from Harvard University in 1970. He carries a cash award of 30,000 Swiss francs, approximately taught at Brown University from 1970 to 1987 and at equal to US$30,000. the Massachusetts Institute of Technology from 1987 to The following quotation was taken from a tribute to 1994. He has been at the Institute for Advanced Study in MacPherson by Gisbert Wüstholz of ETH Zurich: “Singu- Princeton since 1994. His work has introduced radically larities can be studied in different ways using analysis, new approaches to the topology of singular spaces and or you can regard them as geometric phenomena. For the promoted investigations across a great spectrum of math- latter, their study demands a deep geometric intuition ematics.
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Differential Geometry in the Large Seminar Lectures New York University 1946 and Stanford University 1956 Series: Lecture Notes in Mathematics
springer.com Heinz Hopf Differential Geometry in the Large Seminar Lectures New York University 1946 and Stanford University 1956 Series: Lecture Notes in Mathematics These notes consist of two parts: Selected in York 1) Geometry, New 1946, Topics University Notes Peter Lax. by Differential in the 2) Lectures on Stanford Geometry Large, 1956, Notes J. W. University by Gray. are here with no essential They reproduced change. Heinz was a mathematician who mathema- Hopf recognized important tical ideas and new mathematical cases. In the phenomena through special the central idea the of a or difficulty problem simplest background is becomes clear. in this fashion a crystal Doing geometry usually lead serious allows this to to - joy. Hopf's great insight approach for most of the in these notes have become the st- thematics, topics I will to mention a of further try ting-points important developments. few. It is clear from these notes that laid the on Hopf emphasis po- differential 2nd ed. 1989, VIII, 192 p. Most of the results in smooth differ- hedral geometry. whose is both t1al have understanding geometry polyhedral counterparts, works I wish to mention and recent important challenging. Printed book Among those of Robert on which is much in the Connelly rigidity, very spirit R. and in - of these notes (cf. Connelly, Conjectures questions open International of Mathematicians, H- of Softcover gidity, Proceedings Congress sinki vol. 1, 407-414) 1978, . 29,95 € | £26.99 | $49.95 [1]32,05 € (D) | 32,95 € (A) | CHF 43,04 eBook 24,60 € | £20.99 | $34.99 [2]24,60 € (D) | 24,60 € (A) | CHF 34,00 Available from your library or springer.com/shop MyCopy [3] Printed eBook for just € | $ 24.99 springer.com/mycopy Error[en_EN | Export.Bookseller.
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Jt3(S\ H. Hopf, W.K. Clifford, F. Klein
CHAPTER 20 jt3(S\ H. Hopf, W.K. Clifford, F. Klein H. Samelson* Department of Mathematics, Stanford University, Stanford, CA 94305, USA E-mail: [email protected] In 1931 there appeared the seminal paper [2] by Heinz Hopf, in which he showed that 7t2>{S^) (the third homotopy group of the two-sphere S^) is nontrivial, or more specifically that it contains an element of order oo. (His language was different. Homotopy groups had not been defined yet; E. Czech introduced them at the 1932 Congress in Zurich. Interest ingly enough, both Paul Alexandroff and Hopf persuaded him not to continue with these groups. They had different reasons for considering them as not fruitful; the one because they are Abehan, and the other (if I remember right) because there is no mechanism, like chains say, to compute them. It was not until 1936 that W. Hurewicz rediscovered them and made them respectable by proving substantial theorems with and about them.) There are two parts to the paper: The first one is the definition of what now is called the Hopf invariant and the proof of its homotopy invariance. The second consists in the presentation of an example of a map from S^ to S^ that has Hopf invariant 1 and thus represents an element of infinite order of 713(5^); it is what is now called the Hopf fibration; the inverse images of the points of S^ are great circles of S^. Taking S^ as the unit-sphere ki 1^ + k2p = 1 in C^, these circles are the intersections of S^ with the various complex lines through the origin.
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A Treatise on Quantum Clifford Algebras Contents
A Treatise on Quantum Clifford Algebras Habilitationsschrift Dr. Bertfried Fauser arXiv:math/0202059v1 [math.QA] 7 Feb 2002 Universitat¨ Konstanz Fachbereich Physik Fach M 678 78457 Konstanz January 25, 2002 To Dorothea Ida and Rudolf Eugen Fauser BERTFRIED FAUSER —UNIVERSITY OF KONSTANZ I ABSTRACT: Quantum Clifford Algebras (QCA), i.e. Clifford Hopf gebras based on bilinear forms of arbitrary symmetry, are treated in a broad sense. Five al- ternative constructions of QCAs are exhibited. Grade free Hopf gebraic product formulas are derived for meet and join of Graßmann-Cayley algebras including co-meet and co-join for Graßmann-Cayley co-gebras which are very efﬁcient and may be used in Robotics, left and right contractions, left and right co-contractions, Clifford and co-Clifford products, etc. The Chevalley deformation, using a Clif- ford map, arises as a special case. We discuss Hopf algebra versus Hopf gebra, the latter emerging naturally from a bi-convolution. Antipode and crossing are consequences of the product and co-product structure tensors and not subjectable to a choice. A frequently used Kuperberg lemma is revisited necessitating the def- inition of non-local products and interacting Hopf gebras which are generically non-perturbative. A ‘spinorial’ generalization of the antipode is given. The non- existence of non-trivial integrals in low-dimensional Clifford co-gebras is shown. Generalized cliffordization is discussed which is based on non-exponentially gen- erated bilinear forms in general resulting in non unital, non-associative products. Reasonable assumptions lead to bilinear forms based on 2-cocycles. Cliffordiza- tion is used to derive time- and normal-ordered generating functionals for the Schwinger-Dyson hierarchies of non-linear spinor ﬁeld theory and spinor electro- dynamics.
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1 Background and History 1.1 Classifying Exotic Spheres the Kervaire-Milnor Classification of Exotic Spheres
A historical introduction to the Kervaire invariant problem ESHT boot camp April 4, 2016 Mike Hill University of Virginia Mike Hopkins 1.1 Harvard University Doug Ravenel University of Rochester Mike Hill, myself and Mike Hopkins Photo taken by Bill Browder February 11, 2010 1.2 1.3 1 Background and history 1.1 Classifying exotic spheres The Kervaire-Milnor classification of exotic spheres 1 About 50 years ago three papers appeared that revolutionized algebraic and differential topology. John Milnor’s On manifolds home- omorphic to the 7-sphere, 1956. He constructed the first “exotic spheres”, manifolds homeomorphic • but not diffeomorphic to the stan- dard S7. They were certain S3-bundles over S4. 1.4 The Kervaire-Milnor classification of exotic spheres (continued) • Michel Kervaire 1927-2007 Michel Kervaire’s A manifold which does not admit any differentiable structure, 1960. His manifold was 10-dimensional. I will say more about it later. 1.5 The Kervaire-Milnor classification of exotic spheres (continued) • Kervaire and Milnor’s Groups of homotopy spheres, I, 1963. They gave a complete classification of exotic spheres in dimensions ≥ 5, with two caveats: (i) Their answer was given in terms of the stable homotopy groups of spheres, which remain a mystery to this day. (ii) There was an ambiguous factor of two in dimensions congruent to 1 mod 4. The solution to that problem is the subject of this talk. 1.6 1.2 Pontryagin’s early work on homotopy groups of spheres Pontryagin’s early work on homotopy groups of spheres Back to the 1930s Lev Pontryagin 1908-1988 Pontryagin’s approach to continuous maps f : Sn+k ! Sk was • Assume f is smooth.
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EMMY NOETHER and TOPOLOGY F. Hirzebruch1 My Knowledge
EMMY NOETHER AND TOPOLOGY F. Hirzebruch1 My knowledge about the role of Emmy Noether in Algebraic Topology stems from my encounters with Paul Alexandroff and Heinz Hopf. I studied in Zürich at the Eidgenössische Technische Hochschule for three terms, from the summer term 1949 to the summer term 1950. Heinz Hopf and Beno Eckmann were my teachers. For the first time I attended courses on Algebraic Topology and its applications in other fields. I also learnt some facts about the famous book Topologie I by Alexandroff and Hopf [A-H] and how the cooperation of Alexandroff and Hopf started. In May 1925 Hopf received his doctor’s degree at the University of Berlin un- der Erhard Schmidt. He spent the Academic Year 1925/26 at the University of Göttingen where Alexandroff and Hopf met for the first time in the summer term 1926. I quote from Alexandroff [A]: ”Die Bekanntschaft zwischen Hopf und mir wurde im selben Sommer zu einer en- gen Freundschaft. Wir gehörten beide zum Mathematiker-Kreis um Courant und Emmy Noether, zu dieser unvergeßlichen menschlichen Gemeinschaft mit ihren Musikabenden und ihren Bootsfahrten bei und mit Courant, mit ihren ”algebraisch- topologischen” Spaziergängen unter der Führung von Emmy Noether und nicht zuletzt mit ihren verschiedenen Badepartien und Badeunterhaltungen, die sich in der Universitäts-Badeanstalt an der Leine abspielten. ... Die Kliesche Schwimmanstalt war nicht nur ein Studentenbad, sie wurde auch von vielen Universitätsdozenten besucht, darunter von Hilbert, Courant, Emmy Noether, Prandtl, Friedrichs, Deuring, Hans Lewy, Neugebauer und vielen anderen. Von auswärtigen Mathematikern seien etwa Jakob Nielsen, Harald Bohr, van der Waerden, von Neumann, André Weil als Klies ständige Badegäste erwähnt.” Here one sees Göttingen as a mathematical world center.
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Hopf Fibrations and Hurwitz-Radon Numbers
Mathematical Gems and Curiosities Sergei Tabachnikov, Editor opf fibration is a beautiful topological construction Hopf Fibrations and representing a 3-dimensional sphere as a disjoint HHunion of pairwise linked great circles. Judging by its presence on YouTube, it is almost an object of mass Hurwitz-Radon culture; see [23] for a sampler. For visualization, we also strongly recommend the film ‘‘Dimensions’’ [3]. Hopf fibration was defined and studied by the famous Numbers German geometer and topologist Heintz Hopf in 1931 [16]. VALENTIN OVSIENKO AND SERGE TABACHNIKOV In a sense, contemporary algebraic topology has grown up with the Hopf fibration: the development of the theory of characteristic classes, homotopy theory, and K-theory was much influenced by the study of Hopf fibration; see [4, This column is a place for those bits of contagious 5, 8, 13, 14]. Hopf fibration appears in other areas of mathematics and physics, including fluid dynamics, gauge mathematics that travel from person to person in the theories, cosmology, and elementary particles [22]. In fact, there exist four fiber bundles, called the Hopf community, because they are so elegant, surprising, or fibrations, whose fibers, total spaces, and bases are appealing that one has an urge to pass them on. spheres: Contributions are most welcome. ð1Þ The vertical arrows denote projections whose fibers, that is, inverse images of points, are represented by the horizontal arrows. The term ‘‘fibration’’ means that, locally, the total space is the product of the base and the fiber, hence the bigger spheres are filed with smaller ones. For instance, through every point of S3 there passes one circle, S1, and different circles do not intersect.
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Heinz Hopf, Selected Chapters of Geometry
Selected Chapters of Geometry ETH Zurich,¨ summer semester 1940 Heinz Hopf (This is the text of a course that Heinz Hopf gave in the summer of 1940 at the ETH in Zurich.¨ It has been reconstructed and translated by Hans Samelson from the notes that he took as student during the course and that resurfaced in 2002.) Contents Chapter I. Euler’s Formula I.1. The Euclidean and the Spherical Triangle .............1 I.2. The n Dimensional Simplex, Euclidean and Spherical .......2 I.3. Euler’s Theorem on Polyhedra, other Proofs and Consequences .4 I.4. Convex Polyhedra ..........................6 I.5. More Applications ..........................8 I.6. Extension of Legendre’s Proof to n Dimensions ..........9 I.7. Steiner’s Proof of Euler’s Theorem ................ 10 I.8. The Euler Characteristic of a (Bounded Convex) 3 Cell .... 11 Chapter II. Graphs II.1. The Euler Formula for the Characteristic ........... 15 II.2. Graphs in the Plane (and on the Sphere) ............. 16 II.3. Comments and Applications .................... 18 II.4. A Result of Cauchy’s ........................ 20 Chapter III. The Four Vertex Theorem and Related Matters III.1. The Theorem of Friedrich Schur (Erhard Schmidt’s Proof) .. 22 III.2. The Theorem of W. Vogt ...................... 25 III.3. Mukhopadaya’s Four Vertex Theorem .............. 26 III.4. Cauchy’s Congruence Theorem for Convex Polyhedra ..... 27 Chapter IV. The Isoperimetric Inequality IV.1. Proofs of H.A. Schwarz (1884), A. Hurwitz, Erhard Schmidt .. 30 IV.2. The Isoperimetric Inequality in Rn ............... 33 selected chapters of geometry 1 Chapter I. Euler’s Formula I.1. The Euclidean and the Spherical Triangle For the Euclidean triangle we have the well known P fact “Sum of the angles = π ”or α−π=0.
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Saunders Mac Lane, the Knight of Mathematics S. S
Scientiae Mathematicae Japonicae Online, e-2006, 73–77 73 SAUNDERS MAC LANE, THE KNIGHT OF MATHEMATICS S. S. Kutateladze Received June 12, 2005 Abstract. This is a short obituary of Saunders Mac Lane (1909–2005). San Francisco and April 14, 2005 form the terminal place and date of the marvellous almost centennial life of the prominent American mathematician Saunders Mac Lane who shared with Samuel Eilenberg (1913–1998) the honor of creation of category theory which ranks among the most brilliant, controversial, ambitious, and heroic mathematical achieve- ments of the 20th century. Category theory, alongside set theory, serves as a universal language of modern math- ematics. Categories, functors, and natural transformations are widely used in all areas of mathematics, allowing us to look uniformly and consistently on various constructions and formulate the general properties of diverse structures. The impact of category theory is irreducible to the narrow frameworks of its great expressive conveniences. This theory has drastically changed our general outlook on the foundations of mathematics and widened the room of free thinking in mathematics. Set theory, a great and ingenious creation of Georg Cantor, occupies in the common opin- ion of the 20th century the place of the sole solid base of modern mathematics. Mathematics becomes sinking into a section of the Cantorian set theory. Most active mathematicians, teachers, and philosophers consider as obvious and undisputable the thesis that mathemat- ics cannot be grounded on anything but set theory. The set-theoretic stance transforms paradoxically into an ironclad dogma, a clear-cut forbiddance of thinking (as L. Feuerbach once put it wittily).
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Snapshots of Topology in the 50'S
Snapshots of Topology in the 50’s John Milnor Stony Brook University ETH, APRIL 12, 2007 Zum 90. Geburtstag von Beno Eckmann Beno Eckmann and Heinz Hopf in 1953 1 (from the Oberwolfach collection) Paul Alexandroff and Heinz Hopf in 1931 2 in Zürich (from the Oberwolfach collection) A Big Family 3 Hopf had many students. According to the Mathematics Genealogy Project, he has 2212 mathematical descendants More than one third of these are descended from Hopf via Beno Eckmann. Genealogy 4 Euler Lagrange Poisson Fourier Gauss Dirichlet Gerling Lipschitz Pl"ucker Klein Lindemann Hilbert Bieberbach Schmidt Hopf Eckmann The J-homomorphism of George Whitehead 5 q J : πn SO(q) → πn+q(S ) n+q S q f D n -1 0 S = f (0) Consider a tubular neighborhood N of a great n-sphere in Sn+q. A framing of its normal bundle, described by an element q of πn SO(q) , gives rise to a map from N to the unit disk D . The J-homomorphism (continued) 6 n+q S q S f n n S (or M ) p -1 0 = f (p ) 0 q Now collapse the boundary of D to a point p1, yielding a sphere Sq, so that f q q q (N, ∂N) → (D , ∂D ) → (S , p1) , n+q and map all of S r N to p1. The resulting map n+q q q g : S → S represents the required element of πn+q(S ). Some References 7 George Whitehead, 1950: A generalization of the Hopf invariant, Ann. of Math. 51.
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A History of Duality in Algebraic Topology James C
A History of Duality in Algebraic Topology James C. Becker and Daniel Henry Gottlieb 1. Introduction. § Duality in the general course of human affairs seems to be a juxtaposition of comple- mentary or opposite concepts. This frequently leads to poetical sounding uses of language, both in the common language and in the precision of mathematical theorems. Thus the duality of Projective Geometry: Two points determine a line; two lines determine a point. Gergonne first introduced the word duality in mathematics in 1826. He defined it for Pro- jective Geometry. By the time of Poincaré’s note in the Comptes Rendus of 1893, duality was very much in vogue. There are many dualities in algebraic topology. An informal survey of some topologists has revealed the following names of duality in current use. There are Poincaré, Alexan- der, Lefschetz, Pontrjagin, Spanier–Whitehead, Hodge, Vogell, Ranici, Whitney, Serre, Eckmann–Hilton, Atiyah, Brown–Comenetz: These are a few whose names reflect those of their discoverers. There are the dual categories, the duality between homology and cohomology and that between homotopy and cohomotopy. There is a duality between cup products and cap products, and between suspension and looping. The remarkable things are: First, a great many of these seemingly separated dualities are intimately related; and Second, those workers who tried to extend or generalize various of these dualities were led to invent widely important notions such as infinite simplicial complexes or spectra; or they discovered remarkable new relationships among important classical concepts. Such a story demands a point of view. Fortunately, one has been provided by A.
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Topology Becomes Algebraic with Vietoris and Noether
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 39 (1986) 305-307 305 North-Holland TOPOLOGY BECOMES ALGEBRAIC WITH VIETORIS AND NOETHER Saunders MAC LANE Department of Mathematics, The University of Chicago, Chicago, IL 60637, USA Communicated by P. Freyd and A. Heller Received 26 February 1985 An inspiring article by Dieudonne recently published in this journal [1] describes how Emmy Noether persuaded Heinz Hopf that some of his arguments in combi- natorial topology could be more perpicuous if the traditional use of Betti numbers and torsion coefficients were replaced by homology groups. Since these groups were finitely generated abelian groups, they are direct products of cyclic groups; in this decomposition; the number of infinite cyclic groups gives the corresponding Betti numbers, and the torsion coefficients are similarly determined (DieudonnC disguises the abelian groups by calling them Z-modules; the historical term is abelian group [7]). Heinz Hopf, with Paul Alexandroff, had visited Princeton University in 1927-1928. Then (and now) fixed-point theorems were a central interest. The famous Brouwer fixed-point theorem for the n-cell had been extended by Lefschetz [5] to a fixed-point formula for mappings of a manifold into itself. Hopf found a further extension from n-manifolds to n-complexes, and published an announce- ment [2], submitted January 9,1928, of his result in the Proceedings of the National Academy of Sciences, U.S.A. (At that time, and during the whole 50 year period when the mathematician Edwin B.
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