TORIC GENERA 3 Or Tangential Form
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Poincar\'E Duality for Spaces with Isolated Singularities
POINCARÉ DUALITY FOR SPACES WITH ISOLATED SINGULARITIES MATHIEU KLIMCZAK Abstract. In this paper we assign, under reasonable hypothesis, to each pseudomanifold with isolated singularities a rational Poincaré duality space. These spaces are constructed with the formalism of intersection spaces defined by Markus Banagl and are indeed related to them in the even dimensional case. 1. Introduction We are concerned with rational Poincaré duality for singular spaces. There is at least two ways to restore it in this context : • As a self-dual sheaf. This is for instance the case with rational intersection homology. • As a spatialization. That is, given a singular space X, trying to associate to it a new topological space XDP that satisfies Poincaré duality. This strategy is at the origin of the concept of intersection spaces. Let us briefly recall this two approaches. While seeking for a theory of characteristic numbers for complex analytic vari- eties and other singular spaces, Mark Goresky and Robert MacPherson discovered (and then defined in [9] for PL pseudomanifolds and in [8] for topological pseudo- p manifolds) a family of groups IH∗ (X) called intersection homology groups of X. These groups depend on a multi-index p called a perversity. Intersection homology is able to restore Poincaré duality on topological stratified pseudomanifolds. If X is a compact oriented pseudomanifold of dimension n and p, q are two complementary perversities, over Q we have an isomorphism p ∼ n−r IHr (X) = IHq (X), n−r q With IHq (X) := hom(IHn−r(X), Q). Intersection spaces were defined by Markus Banagl in [2] as an attempt to spa- tialize Poincaré duality for singular spaces. -
Homotopy Properties of Thom Complexes (English Translation with the Author’S Comments) S.P.Novikov1
Homotopy Properties of Thom Complexes (English translation with the author’s comments) S.P.Novikov1 Contents Introduction 2 1. Thom Spaces 3 1.1. G-framed submanifolds. Classes of L-equivalent submanifolds 3 1.2. Thom spaces. The classifying properties of Thom spaces 4 1.3. The cohomologies of Thom spaces modulo p for p > 2 6 1.4. Cohomologies of Thom spaces modulo 2 8 1.5. Diagonal Homomorphisms 11 2. Inner Homology Rings 13 2.1. Modules with One Generator 13 2.2. Modules over the Steenrod Algebra. The Case of a Prime p > 2 16 2.3. Modules over the Steenrod Algebra. The Case of p = 2 17 1The author’s comments: As it is well-known, calculation of the multiplicative structure of the orientable cobordism ring modulo 2-torsion was announced in the works of J.Milnor (see [18]) and of the present author (see [19]) in 1960. In the same works the ideas of cobordisms were extended. In particular, very important unitary (”complex’) cobordism ring was invented and calculated; many results were obtained also by the present author studying special unitary and symplectic cobordisms. Some western topologists (in particular, F.Adams) claimed on the basis of private communication that J.Milnor in fact knew the above mentioned results on the orientable and unitary cobordism rings earlier but nothing was written. F.Hirzebruch announced some Milnors results in the volume of Edinburgh Congress lectures published in 1960. Anyway, no written information about that was available till 1960; nothing was known in the Soviet Union, so the results published in 1960 were obtained completely independently. -
Characteristic Classes and K-Theory Oscar Randal-Williams
Characteristic classes and K-theory Oscar Randal-Williams https://www.dpmms.cam.ac.uk/∼or257/teaching/notes/Kthy.pdf 1 Vector bundles 1 1.1 Vector bundles . 1 1.2 Inner products . 5 1.3 Embedding into trivial bundles . 6 1.4 Classification and concordance . 7 1.5 Clutching . 8 2 Characteristic classes 10 2.1 Recollections on Thom and Euler classes . 10 2.2 The projective bundle formula . 12 2.3 Chern classes . 14 2.4 Stiefel–Whitney classes . 16 2.5 Pontrjagin classes . 17 2.6 The splitting principle . 17 2.7 The Euler class revisited . 18 2.8 Examples . 18 2.9 Some tangent bundles . 20 2.10 Nonimmersions . 21 3 K-theory 23 3.1 The functor K ................................. 23 3.2 The fundamental product theorem . 26 3.3 Bott periodicity and the cohomological structure of K-theory . 28 3.4 The Mayer–Vietoris sequence . 36 3.5 The Fundamental Product Theorem for K−1 . 36 3.6 K-theory and degree . 38 4 Further structure of K-theory 39 4.1 The yoga of symmetric polynomials . 39 4.2 The Chern character . 41 n 4.3 K-theory of CP and the projective bundle formula . 44 4.4 K-theory Chern classes and exterior powers . 46 4.5 The K-theory Thom isomorphism, Euler class, and Gysin sequence . 47 n 4.6 K-theory of RP ................................ 49 4.7 Adams operations . 51 4.8 The Hopf invariant . 53 4.9 Correction classes . 55 4.10 Gysin maps and topological Grothendieck–Riemann–Roch . 58 Last updated May 22, 2018. -
Arxiv:1402.0409V1 [Math.HO]
THE PICARD SCHEME STEVEN L. KLEIMAN Abstract. This article introduces, informally, the substance and the spirit of Grothendieck’s theory of the Picard scheme, highlighting its elegant simplicity, natural generality, and ingenious originality against the larger historical record. 1. Introduction A scientific biography should be written in which we indicate the “flow” of mathematics ... discussing a certain aspect of Grothendieck’s work, indicating possible roots, then describing the leap Grothendieck made from those roots to general ideas, and finally setting forth the impact of those ideas. Frans Oort [60, p. 2] Alexander Grothendieck sketched his proof of the existence of the Picard scheme in his February 1962 Bourbaki talk. Then, in his May 1962 Bourbaki talk, he sketched his proofs of various general properties of the scheme. Shortly afterwards, these two talks were reprinted in [31], commonly known as FGA, along with his commentaries, which included statements of nine finiteness theorems that refine the single finiteness theorem in his May talk and answer several related questions. However, Grothendieck had already defined the Picard scheme, via the functor it represents, on pp.195-15,16 of his February 1960 Bourbaki talk. Furthermore, on p.212-01 of his February 1961 Bourbaki talk, he had announced that the scheme can be constructed by combining results on quotients sketched in that talk along with results on the Hilbert scheme to be sketched in his forthcoming May 1961 Bourbaki talk. Those three talks plus three earlier talks, which prepare the way, were also reprinted in [31]. arXiv:1402.0409v1 [math.HO] 3 Feb 2014 Moreover, Grothendieck noted in [31, p. -
Characteristic Classes and Smooth Structures on Manifolds Edited by S
7102 tp.fh11(path) 9/14/09 4:35 PM Page 1 SERIES ON KNOTS AND EVERYTHING Editor-in-charge: Louis H. Kauffman (Univ. of Illinois, Chicago) The Series on Knots and Everything: is a book series polarized around the theory of knots. Volume 1 in the series is Louis H Kauffman’s Knots and Physics. One purpose of this series is to continue the exploration of many of the themes indicated in Volume 1. These themes reach out beyond knot theory into physics, mathematics, logic, linguistics, philosophy, biology and practical experience. All of these outreaches have relations with knot theory when knot theory is regarded as a pivot or meeting place for apparently separate ideas. Knots act as such a pivotal place. We do not fully understand why this is so. The series represents stages in the exploration of this nexus. Details of the titles in this series to date give a picture of the enterprise. Published*: Vol. 1: Knots and Physics (3rd Edition) by L. H. Kauffman Vol. 2: How Surfaces Intersect in Space — An Introduction to Topology (2nd Edition) by J. S. Carter Vol. 3: Quantum Topology edited by L. H. Kauffman & R. A. Baadhio Vol. 4: Gauge Fields, Knots and Gravity by J. Baez & J. P. Muniain Vol. 5: Gems, Computers and Attractors for 3-Manifolds by S. Lins Vol. 6: Knots and Applications edited by L. H. Kauffman Vol. 7: Random Knotting and Linking edited by K. C. Millett & D. W. Sumners Vol. 8: Symmetric Bends: How to Join Two Lengths of Cord by R. -
THOM COBORDISM THEOREM 1. Introduction in This Short Notes We
THOM COBORDISM THEOREM MIGUEL MOREIRA 1. Introduction In this short notes we will explain the remarkable theorem of Thom that classifies cobordisms. This theorem was originally proven by Ren´eThom in [] but we will follow a more modern exposition. We want to give a precise idea of the everything involved but we won't give complete details at some points. Let's define the concept of cobordism. Definition 1. Let M; N be two smooth closed n-manifolds. We say that M; N are cobordant if there exists a compact (n + 1)-manifold W such that @W = M t N. This is an equivalence relation. Given a space X and f : M ! X we call the pair (M; f) a singular manifold. We say that (M; f) and (N; g) are cobordant if there is a cobordism W between M and N and f t g extends to F : W ! X. We denote by Nn(X) the set of cobordism classes of singular n-manifolds (M; f). In particular Nn ≡ Nn(∗) is the set of cobordism sets. We'll already state the two amazing theorems by Thom. The first gives a homotopy-theoretical interpretation of Nn and the second actually computes it. Theorem 1. N∗ is a generalized homology theory associated to the Thom spectrum MO, i.e. Nn(X) = colim πn+k(MO(k) ^ X+): k!1 Theorem 2. We have an isomorphism of graded Z=2-algebras ∼ ` π∗MO = Z=2[xi : 0 < i 6= 2 − 1] = Z=2[x2; x4; x5; x6;:::] where xi has grading i. -
All That Math Portraits of Mathematicians As Young Researchers
Downloaded from orbit.dtu.dk on: Oct 06, 2021 All that Math Portraits of mathematicians as young researchers Hansen, Vagn Lundsgaard Published in: EMS Newsletter Publication date: 2012 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Hansen, V. L. (2012). All that Math: Portraits of mathematicians as young researchers. EMS Newsletter, (85), 61-62. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Editorial Obituary Feature Interview 6ecm Marco Brunella Alan Turing’s Centenary Endre Szemerédi p. 4 p. 29 p. 32 p. 39 September 2012 Issue 85 ISSN 1027-488X S E European M M Mathematical E S Society Applied Mathematics Journals from Cambridge journals.cambridge.org/pem journals.cambridge.org/ejm journals.cambridge.org/psp journals.cambridge.org/flm journals.cambridge.org/anz journals.cambridge.org/pes journals.cambridge.org/prm journals.cambridge.org/anu journals.cambridge.org/mtk Receive a free trial to the latest issue of each of our mathematics journals at journals.cambridge.org/maths Cambridge Press Applied Maths Advert_AW.indd 1 30/07/2012 12:11 Contents Editorial Team Editors-in-Chief Jorge Buescu (2009–2012) European (Book Reviews) Vicente Muñoz (2005–2012) Dep. -
Analysis in Complex Geometry
1946-4 School and Conference on Differential Geometry 2 - 20 June 2008 Analysis in Complex Geometry Shing-Tung Yau Harvard University, Dept. of Mathematics Cambridge, MA 02138 United States of America Analysis in Complex Geometry Shing-Tung Yau Harvard University ICTP, June 18, 2008 1 An important question in complex geometry is to char- acterize those topological manifolds that admit a com- plex structure. Once a complex structure is found, one wants to search for the existence of algebraic or geo- metric structures that are compatible with the complex structure. 2 Most geometric structures are given by Hermitian met- rics or connections that are compatible with the com- plex structure. In most cases, we look for connections with special holonomy group. A connection may have torsion. The torsion tensor is not well-understood. Much more research need to be done especially for those Hermitian connections with special holonomy group. 3 Karen Uhlenbeck Simon Donaldson By using the theorem of Donaldson-Uhlenbeck-Yau, it is possible to construct special holonomy connections with torsion. Their significance in relation to complex or algebraic structure need to be explored. 4 K¨ahler metrics have no torsion and their geometry is very close to that of algebraic geometry. Yet, as was demonstrated by Voisin, there are K¨ahler manifolds that are not homotopy equivalent to any algebraic manifolds. The distinction between K¨ahler and algebraic geometry is therefore rather delicate. Erich K¨ahler 5 Algebraic geometry is a classical subject and there are several natural equivalences of algebraic manifolds: bi- rational equivalence, biregular equivalence, and arith- metic equivalence. -
Floer Homotopy Theory, Realizing Chain Complexes by Module Spectra, And
Floer homotopy theory, realizing chain complexes by module spectra, and manifolds with corners Ralph L. Cohen ∗ Department of Mathematics Stanford University Stanford, CA 94305 February 13, 2013 Abstract In this paper we describe and continue the study begun in [5] of the homotopy theory that underlies Floer theory. In that paper the authors addressed the question of realizing a Floer complex as the celluar chain complex of a CW -spectrum or pro-spectrum, where the attaching maps are determined by the compactified moduli spaces of connecting orbits. The basic obstructions to the existence of this realization are the smoothness of these moduli spaces, and the existence of compatible collections of framings of their stable tangent bundles. In this note we describe a generalization of this, to show that when these moduli spaces are smooth, ∗ and are oriented with respect to a generalized cohomology theory E , then a Floer E∗-homology theory can be defined. In doing this we describe a functorial viewpoint on how chain complexes can be realized by E-module spectra, generalizing the stable homotopy realization criteria given in [5]. Since these moduli spaces, if smooth, will be manifolds with corners, we give a discussion about the appropriate notion of orientations of manifolds with corners. Contents 1 Floer homotopy theory 5 arXiv:0802.2752v1 [math.AT] 20 Feb 2008 1.1 PreliminariesfromMorsetheory . ......... 5 1.2 SmoothFloertheories ............................. ...... 9 2 Realizing chain complexes by E-module spectra 10 ∗ 3 Manifolds with corners, E -orientations of flow categories, and Floer E∗ - homol- ogy 14 ∗The author was partially supported by a grant from the NSF 1 Introduction In [5], the authors began the study of the homotopy theoretic aspects of Floer theory. -
Programme & Information Brochure
Programme & Information 6th European Congress of Mathematics Kraków 2012 6ECM Programme Coordinator Witold Majdak Editors Agnieszka Bojanowska Wojciech Słomczyński Anna Valette Typestetting Leszek Pieniążek Cover Design Podpunkt Contents Welcome to the 6ECM! 5 Scientific Programme 7 Plenary and Invited Lectures 7 Special Lectures and Session 10 Friedrich Hirzebruch Memorial Session 10 Mini-symposia 11 Satellite Thematic Sessions 12 Panel Discussions 13 Poster Sessions 14 Schedule 15 Social events 21 Exhibitions 23 Books and Software Exhibition 23 Old Mathematical Manuscripts and Books 23 Art inspired by mathematics 23 Films 25 6ECM Specials 27 Wiadomości Matematyczne and Delta 27 Maths busking – Mathematics in the streets of Kraków 27 6ECM Medal 27 Coins commemorating Stefan Banach 28 6ECM T-shirt 28 Where to eat 29 Practical Information 31 6ECM Tourist Programme 33 Tours in Kraków 33 Excursions in Kraków’s vicinity 39 More Tourist Attractions 43 Old City 43 Museums 43 Parks and Mounds 45 6ECM Organisers 47 Maps & Plans 51 Honorary Patron President of Poland Bronisław Komorowski Honorary Committee Minister of Science and Higher Education Barbara Kudrycka Voivode of Małopolska Voivodship Jerzy Miller Marshal of Małopolska Voivodship Marek Sowa Mayor of Kraków Jacek Majchrowski WELCOME to the 6ECM! We feel very proud to host you in Poland’s oldest medieval university, in Kraków. It was in this city that the Polish Mathematical Society was estab- lished ninety-three years ago. And it was in this country, Poland, that the European Mathematical Society was established in 1991. Thank you very much for coming to Kraków. The European Congresses of Mathematics are quite different from spe- cialized scientific conferences or workshops. -
EUROPEAN MATHEMATICAL SOCIETY EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics the Open University Milton Keynes MK7 6AA, UK E-Mail: [email protected]
CONTENTS EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics The Open University Milton Keynes MK7 6AA, UK e-mail: [email protected] ASSOCIATE EDITORS VASILE BERINDE Department of Mathematics, University of Baia Mare, Romania e-mail: [email protected] NEWSLETTER No. 47 KRZYSZTOF CIESIELSKI Mathematics Institute March 2003 Jagiellonian University Reymonta 4 EMS Agenda ................................................................................................. 2 30-059 Kraków, Poland e-mail: [email protected] Editorial by Sir John Kingman .................................................................... 3 STEEN MARKVORSEN Department of Mathematics Executive Committee Meeting ....................................................................... 4 Technical University of Denmark Building 303 Introducing the Committee ............................................................................ 7 DK-2800 Kgs. Lyngby, Denmark e-mail: [email protected] An Answer to the Growth of Mathematical Knowledge? ............................... 9 SPECIALIST EDITORS Interview with Vagn Lundsgaard Hansen .................................................. 15 INTERVIEWS Steen Markvorsen [address as above] Interview with D V Anosov .......................................................................... 20 SOCIETIES Krzysztof Ciesielski [address as above] Israel Mathematical Union ......................................................................... 25 EDUCATION Tony Gardiner -
Life and Work of Friedrich Hirzebruch
Jahresber Dtsch Math-Ver (2015) 117:93–132 DOI 10.1365/s13291-015-0114-1 HISTORICAL ARTICLE Life and Work of Friedrich Hirzebruch Don Zagier1 Published online: 27 May 2015 © Deutsche Mathematiker-Vereinigung and Springer-Verlag Berlin Heidelberg 2015 Abstract Friedrich Hirzebruch, who died in 2012 at the age of 84, was one of the most important German mathematicians of the twentieth century. In this article we try to give a fairly detailed picture of his life and of his many mathematical achievements, as well as of his role in reshaping German mathematics after the Second World War. Mathematics Subject Classification (2010) 01A70 · 01A60 · 11-03 · 14-03 · 19-03 · 33-03 · 55-03 · 57-03 Friedrich Hirzebruch, who passed away on May 27, 2012, at the age of 84, was the outstanding German mathematician of the second half of the twentieth century, not only because of his beautiful and influential discoveries within mathematics itself, but also, and perhaps even more importantly, for his role in reshaping German math- ematics and restoring the country’s image after the devastations of the Nazi years. The field of his scientific work can best be summed up as “Topological methods in algebraic geometry,” this being both the title of his now classic book and the aptest de- scription of an activity that ranged from the signature and Hirzebruch-Riemann-Roch theorems to the creation of the modern theory of Hilbert modular varieties. Highlights of his activity as a leader and shaper of mathematics inside and outside Germany in- clude his creation of the Arbeitstagung,