Celebrating One Hundred Fifty Years of. Topology. ARBEITSTAGUNG Bonn, May 22, 2013
|
|
- Joanna Shaw
- 5 years ago
- Views:
Transcription
1 Celebrating One Hundred Fifty Years of Topology John Milnor Institute for Mathematical Sciences Stony Brook University ( ARBEITSTAGUNG Bonn, May 22, 2013
2 Algebra & Number Theory Geometry
3 3 Algebra & Number Theory 5 4 Geometry Theorema Egregium iπ e = 1 Analysis
4 3 Algebra & Number Theory 5 4 Geometry TOPOLOGY Theorema Egregium iπ e = 1 Analysis
5 When did topology start? V E +F =
6 W C (p) = 1 2πi C dz z p Cauchy, 1825
7 1 L = 4π ZZ x,y (x y ) (dx dy ) kx y k3 1833
8 doubly connected triply connected Riemann, 1857 triply connected
9 August Ferdinand Möbius
10 The Möbius Classification of surfaces in R 3 : 1863 Definition of the class of a surface: On a closed surface of the n-th class [= genus n 1], there exist n 1 closed curves which do not disconnect the surface. Theorem. Any two closed surfaces of the same class are elementarily related. Two geometric figures will be called elementarily related if to any infinitely small element of any dimension in one figure there corresponds an infinitely small element in the other figure, such that two neighboring elements in one figure correspond to two elements in the other which also come together;
11 Camille Jordan, 1877, Jordan curve theorem. Walther von Dyck, 1888: Topology studies properties invariant under continuous functions with continuous inverse. χ(m) = 1 K da. 2π Henri Poincaré, , homology, Betti numbers, duality, homotopy, fundamental group, covering spaces,
12 20 th century Felix Hausdorff L. E. J. Brouwer H. Kneser and H. Hopf
13 Solomon Lefschetz and James Alexander
14 The Alexander Chimney in Colorado
15 Hassler Whitney E Sn 1 X characteristic classes w i H i (X; Z/2).
16 Lev Pontryagin ξ real vector bundle over X p j (ξ) H 4j (X; Z) Shiing-Shen Chern γ complex vector bundle over X c j (γ) H 2j (X; Z)
17 Many people put these into more modern form Wu Wen-Tsün H (B GL(R) ; Z/2) = (Z/2)[w1, w 2, w 3, ] H (B GL(C) ; Z) = Z[c1, c 2, c 3, ] H (B GL(R) ; Z) = Z[p1, p 2, ] (2 torsion)
18 Neue topologische Methoden
19 Todd genus ( arithmetic genus) Francesco Severi David Hilbert Lemma. a unique a'j7" J. A. Todd T = c (c c 2) c 1c 2 + H (B GL(C) ; Q), such that the genus T (V n ) = T(τ V n )[V n] is multiplicative: T (V V ) = T (V ) T (V ), with T ( P n (C) ) = +1. Theorem. T (V n ) = n ( 1) k dim C {holomorphic k forms}. k=0
20 Some notation If γ is a holomorphic vector bundle over V, let (γ) denote the sheaf of germs of local holomorphic sections. Let 1 denote the trivial line bundle. Pierre Dolbeault: Theorem : H k V ; (1) = {holomorphic k forms}. P k k Hence T (V ) = k ( 1) dimc H V ; (1).
21 Classical Riemann-Roch Theorems Gustav Roch Max Noether Andre Weil
22 The Chern character of a complex n-plane bundle over X ch(γ n ) = n + c 1 1! + c 1 2 2c 2 2! + c 3 1 3c 1c 2 + 3c 3 3! + is an element of H (X; Q) characterized by two properties: ch(γ 1 ) = e c 1(γ 1 ). ch(γ m γ n) = ch(γ m ) + ch(γ n), = ch(γ m γ n) = ch(γ m ) ch(γ n). Hirzebruch s Riemann-Roch Theorem: For any holomorphic vector bundle γ over V, ( 1) k dim C H k( V ; (γ) ) ( ) = ch(γ) T(τ V ) [V ]. k
23 Rene Thom s cobordism theory was based on deep geometric intuition, plus hard algebraic topology. Essential ingredients: Jean-Pierre Serre on spectral sequences Norman Steenrod on cohomology operations
24 The L-genus of an oriented 4n-manifold Hirzebruch showed that there is one and only one sum L = 1 + p p 2 p H (B GL(R) ; Q), such that the L-genus, L(M 4n ) = L(τ M 4n)[M 4n ], is multiplicative L(M M ) = L(M) L(M ), with L ( P 2n (C) ) = +1. Theorem (Thom, Hirzebruch). The to the signature of the quadratic form L-genus L(M 4k ) is equal H 2k (M 4k ; Q) Q x (x x)[m 4k ].
25 Hirzebruch also defined the Â-genus Â(M4k ), where  = 1 p p 1 2 4p If M 4k is a spin manifold (w 2 = 0), then Michael Atiyah Is Singer proved that Â(M 4k ) is equal to the index of the associated Dirac operator, and hence is an integer.
26 Almost parallelizable manifolds Suppose that M 4k (point) is parallelizable, so that p j (τ M 4k ) is zero for j < k. Hirzebruch s formulas then take the form L(M 4k ) = and since w 2 = 0, ( 2 2k (2 2k 1 1) B ) k p k [M 4k ], (2k)! Â(M 4k ) = B k 2 (2k)! p k[m 4k ].
27 Raoul Bott at the Arbeitstagung in 1969 Bott showed that π 4k 1 (SO) = Z. Furthermore, a generator gives rise to a vector bundle ξ over S 4k with { p k (ξ)[s 4k (2k 1)! for k even ] = 2(2k 1)! for k odd. = the Pontrjagin number p k [M 4k ] of an almost parallelizable manifold is always divisible by (2k 1)!.
28 The Pontrjagin-Thom construction. S p+q f S q S p (or M p ) = f -1 (x 0 ) x 0 Any M p S p+q with framed normal bundle determines a homotopy class in π p+q (S q ). Taking M p = S p we obtain the J-homomorphism J : π p (SO q ) π p+q (S q ). In the stable case q >> p, I will write J : π p (SO) Π p.
29 The Adams Conjecture Frank Adams J ( π 4k 1 (SO) ) ( ) Bk = denominator 4k.
30 The E 8 manifold-with-boundary W 4k The boundary W 4k is a homotopy (4k 1)-sphere if k > 1.
31 Michel Kervaire Theorem The group of homotopy spheres which bound parallelizable manifolds is cyclic of order ( ) 2 2k 2 (2 2k 1 4Bk 1) numerator, k with generator W 4k 1.
32 The last 50 years Amazing progress in low dimensional topology: Freedman, Donaldson, Thurston Geometrization, Perelman Ever deeper connections with mathematical physics: gauge theory, Seiberg-Witten theory, symplectic topology, TO BE CONTINUED!
Spheres John Milnor Institute for Mathematical Sciences Stony Brook University (
Spheres John Milnor Institute for Mathematical Sciences Stony Brook University (www.math.sunysb.edu) ABEL LECTURE Oslo, May 25, 2011 Examples of Spheres: 2. The standard sphere S n R n+1 is the locus x
More informationBERNOULLI NUMBERS, HOMOTOPY GROUPS, AND A THEOREM OF ROHLIN
454 BERNOULLI NUMBERS, HOMOTOPY GROUPS, AND A THEOREM OF ROHLIN By JOHN W. MILNOR AND MICHEL A. KERVAIRE A homomorphism J: 7T k _ 1 (SO w ) -> n m+k _ 1 (S m ) from the homotopy groups of rotation groups
More informationAtiyah-Singer Revisited
Atiyah-Singer Revisited Paul Baum Penn State Texas A&M Universty College Station, Texas, USA April 1, 2014 From E 1, E 2,..., E n obtain : 1) The Dirac operator of R n D = n j=1 E j x j 2) The Bott generator
More informationDirac Operator. Göttingen Mathematical Institute. Paul Baum Penn State 6 February, 2017
Dirac Operator Göttingen Mathematical Institute Paul Baum Penn State 6 February, 2017 Five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology? 4. The Riemann-Roch theorem 5. K-theory
More informationThe Riemann-Roch Theorem
The Riemann-Roch Theorem TIFR Mumbai, India Paul Baum Penn State 7 August, 2015 Five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology? 4. Beyond ellipticity 5. The Riemann-Roch
More informationInvariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem
PETER B. GILKEY Department of Mathematics, University of Oregon Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem Second Edition CRC PRESS Boca Raton Ann Arbor London Tokyo Contents
More informationRemarks on the Milnor number
José 1 1 Instituto de Matemáticas, Universidad Nacional Autónoma de México. Liverpool, U. K. March, 2016 In honour of Victor!! 1 The Milnor number Consider a holomorphic map-germ f : (C n+1, 0) (C, 0)
More informationClassification of (n 1)-connected 2n-dimensional manifolds and the discovery of exotic spheres
Classification of (n 1)-connected 2n-dimensional manifolds and the discovery of exotic spheres John Milnor At Princeton in the fifties I was very much interested in the fundamental problem of understanding
More informationDirac Operator. Texas A&M University College Station, Texas, USA. Paul Baum Penn State. March 31, 2014
Dirac Operator Paul Baum Penn State Texas A&M University College Station, Texas, USA March 31, 2014 Miniseries of five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology? 4. Beyond
More informationThe Riemann-Roch Theorem
The Riemann-Roch Theorem Paul Baum Penn State Texas A&M University College Station, Texas, USA April 4, 2014 Minicourse of five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology?
More informationThe Riemann-Roch Theorem
The Riemann-Roch Theorem Paul Baum Penn State TIFR Mumbai, India 20 February, 2013 THE RIEMANN-ROCH THEOREM Topics in this talk : 1. Classical Riemann-Roch 2. Hirzebruch-Riemann-Roch (HRR) 3. Grothendieck-Riemann-Roch
More informationOn the non-existence of elements of Kervaire invariant one
On the non-existence of elements of Kervaire invariant one Michael University of Virginia ICM Topology Session, Seoul, Korea, 2014 Poincaré Conjecture & Milnor s Question Milnor s Questions How many smooth
More informationExotic spheres. Overview and lecture-by-lecture summary. Martin Palmer / 22 July 2017
Exotic spheres Overview and lecture-by-lecture summary Martin Palmer / 22 July 2017 Abstract This is a brief overview and a slightly less brief lecture-by-lecture summary of the topics covered in the course
More informationOxford 13 March Surgery on manifolds: the early days, Or: What excited me in the 1960s. C.T.C.Wall
Oxford 13 March 2017 Surgery on manifolds: the early days, Or: What excited me in the 1960s. C.T.C.Wall In 1956 Milnor amazed the world by giving examples of smooth manifolds homeomorphic but not diffeomorphic
More informationhttp://dx.doi.org/10.1090/pspum/003 DIFFERENTIAL GEOMETRY PROCEEDINGS OF THE THIRD SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY Held at the University of Arizona Tucson, Arizona February
More informationCHARACTERISTIC CLASSES
1 CHARACTERISTIC CLASSES Andrew Ranicki Index theory seminar 14th February, 2011 2 The Index Theorem identifies Introduction analytic index = topological index for a differential operator on a compact
More informationRecall for an n n matrix A = (a ij ), its trace is defined by. a jj. It has properties: In particular, if B is non-singular n n matrix,
Chern characters Recall for an n n matrix A = (a ij ), its trace is defined by tr(a) = n a jj. j=1 It has properties: tr(a + B) = tr(a) + tr(b), tr(ab) = tr(ba). In particular, if B is non-singular n n
More informationExercises on characteristic classes
Exercises on characteristic classes April 24, 2016 1. a) Compute the Stiefel-Whitney classes of the tangent bundle of RP n. (Use the method from class for the tangent Chern classes of complex projectives
More informationk=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula
20 VASILY PESTUN 3. Lecture: Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer Index theorems 3.. Index for a holomorphic vector bundle. For a holomorphic vector bundle E over a complex variety of dim
More informationField theories and algebraic topology
Field theories and algebraic topology Tel Aviv, November 2011 Peter Teichner Max-Planck Institut für Mathematik, Bonn University of California, Berkeley Mathematics as a language for physical theories
More information1. Introduction SOME REMARKS ON ALMOST AND STABLE ALMOST COMPLEX MANIFOLDS
Math. Nachr. ( ), SOME REMARKS ON ALMOST AND STABLE ALMOST COMPLEX MANIFOLDS By Anand Dessai (Received ) Abstract. In the first part we give necessary and sufficient conditions for the existence of a stable
More informationSummary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture)
Building Geometric Structures: Summary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture) A geometric structure on a manifold is a cover
More informationarxiv: v2 [math.gt] 26 Sep 2013
HOMOLOGY CLASSES OF NEGATIVE SQUARE AND EMBEDDED SURFACES IN 4-MANIFOLDS arxiv:1301.3733v2 [math.gt] 26 Sep 2013 M. J. D. HAMILTON ABSTRACT. Let X be a simply-connected closed oriented 4-manifold and A
More informationBackground and history
lecture[1]browder's work on the Arf-Kervaire invariant problemlecture-text Browder s work on Arf-Kervaire invariant problem Panorama of Topology A Conference in Honor of William Browder May 10, 2012 Mike
More informationFormal groups laws and genera*
Bulletin of the Manifold Atlas (2011) Formal groups laws and genera* TARAS PANOV Abstract. The article reviews role of formal group laws in bordism theory. 55N22, 57R77 1. Introduction The theory of formal
More informationSPECTRAL ASYMMETRY AND RIEMANNIAN GEOMETRY
SPECTRAL ASYMMETRY AND RIEMANNIAN GEOMETRY M. F. ATIYAH, V. K. PATODI AND I. M. SINGER 1 Main Theorems If A is a positive self-adjoint elliptic (linear) differential operator on a compact manifold then
More informationThe Theorem of Gauß-Bonnet in Complex Analysis 1
The Theorem of Gauß-Bonnet in Complex Analysis 1 Otto Forster Abstract. The theorem of Gauß-Bonnet is interpreted within the framework of Complex Analysis of one and several variables. Geodesic triangles
More informationarxiv: v2 [math.dg] 3 Nov 2016
THE NON-EXISTENT COMPLEX 6-SPHERE arxiv:1610.09366v2 [math.dg] 3 Nov 2016 MICHAEL ATIYAH Dedicated to S.S.Chern, Jim Simons and Nigel Hitchin Abstract. The possible existence of a complex structure on
More informationThe Classification of (n 1)-connected 2n-manifolds
The Classification of (n 1)-connected 2n-manifolds Kyler Siegel December 18, 2014 1 Prologue Our goal (following [Wal]): Question 1.1 For 2n 6, what is the diffeomorphic classification of (n 1)-connected
More informationIntroduction to surgery theory
Introduction to surgery theory Wolfgang Lück Bonn Germany email wolfgang.lueck@him.uni-bonn.de http://131.220.77.52/lueck/ Bonn, 17. & 19. April 2018 Wolfgang Lück (MI, Bonn) Introduction to surgery theory
More informationChern numbers and Hilbert Modular Varieties
Chern numbers and Hilbert Modular Varieties Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 9, 2011 A Topological Point
More informationPossible Advanced Topics Course
Preprint typeset in JHEP style - HYPER VERSION Possible Advanced Topics Course Gregory W. Moore Abstract: Potential List of Topics for an Advanced Topics version of Physics 695, Fall 2013 September 2,
More informationTopological rigidity for non-aspherical manifolds by M. Kreck and W. Lück
Topological rigidity for non-aspherical manifolds by M. Kreck and W. Lück July 11, 2006 Abstract The Borel Conjecture predicts that closed aspherical manifolds are topological rigid. We want to investigate
More informationTOPOLOGICALLY INVARIANT CHERN NUMBERS OF PROJECTIVE VARIETIES
TOPOLOGICALLY INVARIANT CHERN NUMBERS OF PROJECTIVE VARIETIES D. KOTSCHICK ABSTRACT. We prove that a rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex
More informationE 0 0 F [E] + [F ] = 3. Chern-Weil Theory How can you tell if idempotents over X are similar?
. Characteristic Classes from the viewpoint of Operator Theory. Introduction Overarching Question: How can you tell if two vector bundles over a manifold are isomorphic? Let X be a compact Hausdorff space.
More informationIntroduction (Lecture 1)
Introduction (Lecture 1) February 2, 2011 In this course, we will be concerned with variations on the following: Question 1. Let X be a CW complex. When does there exist a homotopy equivalence X M, where
More informationWEDNESDAY SEMINAR SUMMER 2017 INDEX THEORY
WEDNESDAY SEMINAR SUMMER 2017 INDEX THEORY The Atiyah-Singer index theorem is certainly one of the most influential collection of results from the last century. The theorem itself relates an analytic invariant,
More informationClassifying complex surfaces and symplectic 4-manifolds
Classifying complex surfaces and symplectic 4-manifolds UT Austin, September 18, 2012 First Cut Seminar Basics Symplectic 4-manifolds Definition A symplectic 4-manifold (X, ω) is an oriented, smooth, 4-dimensional
More informationRECURRENCE RELATIONS IN THOM SPECTRA. The squaring operations themselves appear as coefficients in the resulting polynomial:
RECURRENCE RELATIONS IN THOM SPECTRA ERIC PETERSON (Throughout, H will default to mod- cohomology.). A HIGHLY INTERESTING SPACE We begin with a love letter to P. Its first appearance in the theory of algebraic
More informationExotic spheres and topological modular forms. Mark Behrens (MIT) (joint with Mike Hill, Mike Hopkins, and Mark Mahowald)
Exotic spheres and topological modular forms Mark Behrens (MIT) (joint with Mike Hill, Mike Hopkins, and Mark Mahowald) Fantastic survey of the subject: Milnor, Differential topology: 46 years later (Notices
More informationBackground and history. Classifying exotic spheres. A historical introduction to the Kervaire invariant problem. ESHT boot camp.
A historical introduction to the Kervaire invariant problem ESHT boot camp April 4, 2016 Mike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester 1.1 Mike Hill,
More informationRohlin s Invariant and 4-dimensional Gauge Theory. Daniel Ruberman Nikolai Saveliev
Rohlin s Invariant and 4-dimensional Gauge Theory Daniel Ruberman Nikolai Saveliev 1 Overall theme: relation between Rohlin-type invariants and gauge theory, especially in dimension 4. Background: Recall
More informationThe Kervaire invariant in homotopy theory
The Kervaire invariant in homotopy theory Mark Mahowald and Paul Goerss June 20, 2011 Abstract In this note we discuss how the first author came upon the Kervaire invariant question while analyzing the
More informationCobordant differentiable manifolds
Variétés différentiables cobordant, Colloque Int. du C. N. R. S., v. LII, Géométrie différentielle, Strasbourg (1953), pp. 143-149. Cobordant differentiable manifolds By R. THOM (Strasbourg) Translated
More informationFrom Algebraic Geometry to Homological Algebra
From Algebraic Geometry to Homological Algebra Sepehr Jafari Università Degli Studi di Genova Dipartimento Di Matematica November 8, 2016 November 8, 2016 1 / 24 Outline 1 Historical Events Algebraic Geometry
More informationDEFINITE MANIFOLDS BOUNDED BY RATIONAL HOMOLOGY THREE SPHERES
DEFINITE MANIFOLDS BOUNDED BY RATIONAL HOMOLOGY THREE SPHERES BRENDAN OWENS AND SAŠO STRLE Abstract. This paper is based on a talk given at the McMaster University Geometry and Topology conference, May
More informationTopology in the 20th century: a view from the inside
Russian Math. Surveys 59:5 803 829 Uspekhi Mat. Nauk 59:5 3 28 c 2004 RAS(DoM) and LMS DOI 10.1070/RM2004v059n05ABEH000770 Topology in the 20th century: a view from the inside S. P. Novikov Foreword The
More informationINTEGRALS ON SPIN MANIFOLDS AND THE K-THEORY OF
INTEGRALS ON SPIN MANIFOLDS AND THE K-THEORY OF K(Z, 4) JOHN FRANCIS Abstract. We prove that certain integrals M P (x)â take only integer values on spin manifolds. Our method of proof is to calculate the
More informationOsaka Journal of Mathematics. 37(2) P.1-P.4
Title Katsuo Kawakubo (1942 1999) Author(s) Citation Osaka Journal of Mathematics. 37(2) P.1-P.4 Issue Date 2000 Text Version publisher URL https://doi.org/10.18910/4128 DOI 10.18910/4128 rights KATSUO
More informationThe Hopf invariant one problem
The Hopf invariant one problem Ishan Banerjee September 21, 2016 Abstract This paper will discuss the Adams-Atiyah solution to the Hopf invariant problem. We will first define and prove some identities
More informationELLIPTIC COHOMOLOGY: A HISTORICAL OVERVIEW
ELLIPTIC COHOMOLOGY: A HISTORICAL OVERVIEW CORBETT REDDEN The goal of this overview is to introduce concepts which underlie elliptic cohomology and reappear in the construction of tmf. We begin by defining
More informationComplex Bordism and Cobordism Applications
Complex Bordism and Cobordism Applications V. M. Buchstaber Mini-course in Fudan University, April-May 2017 Main goals: --- To describe the main notions and constructions of bordism and cobordism; ---
More informationMonodromy of the Dwork family, following Shepherd-Barron X n+1. P 1 λ. ζ i = 1}/ (µ n+1 ) H.
Monodromy of the Dwork family, following Shepherd-Barron 1. The Dwork family. Consider the equation (f λ ) f λ (X 0, X 1,..., X n ) = λ(x n+1 0 + + X n+1 n ) (n + 1)X 0... X n = 0, where λ is a free parameter.
More informationLECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS
LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS WEIMIN CHEN, UMASS, SPRING 07 1. Basic elements of J-holomorphic curve theory Let (M, ω) be a symplectic manifold of dimension 2n, and let J J (M, ω) be
More informationRemarks on Chern-Simons Theory. Dan Freed University of Texas at Austin
Remarks on Chern-Simons Theory Dan Freed University of Texas at Austin 1 MSRI: 1982 2 Classical Chern-Simons 3 Quantum Chern-Simons Witten (1989): Integrate over space of connections obtain a topological
More informationCollected Paper III, Differential Topology
Collected Paper III, Differential Topology Introduction to Part 1: Exotic Spheres This section will consist of the following eight papers: On manifolds homeomorphic to the 7-sphere, Annals of Mathematics
More informationHODGE NUMBERS OF COMPLETE INTERSECTIONS
HODGE NUMBERS OF COMPLETE INTERSECTIONS LIVIU I. NICOLAESCU 1. Holomorphic Euler characteristics Suppose X is a compact Kähler manifold of dimension n and E is a holomorphic vector bundle. For every p
More informationThe Hopf Bracket. Claude LeBrun SUNY Stony Brook and Michael Taylor UNC Chapel Hill. August 11, 2013
The Hopf Bracket Claude LeBrun SUY Stony Brook and ichael Taylor UC Chapel Hill August 11, 2013 Abstract Given a smooth map f : between smooth manifolds, we construct a hierarchy of bilinear forms on suitable
More informationNONNEGATIVE CURVATURE AND COBORDISM TYPE. 1. Introduction
NONNEGATIVE CURVATURE AND COBORDISM TYPE ANAND DESSAI AND WILDERICH TUSCHMANN Abstract. We show that in each dimension n = 4k, k 2, there exist infinite sequences of closed simply connected Riemannian
More informationTraces and Determinants of
Traces and Determinants of Pseudodifferential Operators Simon Scott King's College London OXFORD UNIVERSITY PRESS CONTENTS INTRODUCTION 1 1 Traces 7 1.1 Definition and uniqueness of a trace 7 1.1.1 Traces
More informationA geometric solution of the Kervaire Invariant One problem
A geometric solution of the Kervaire Invariant One problem Petr M. Akhmet ev 19 May 2009 Let f : M n 1 R n, n = 4k + 2, n 2 be a smooth generic immersion of a closed manifold of codimension 1. Let g :
More informationLecture 11: Hirzebruch s signature theorem
Lecture 11: Hirzebruch s signature theorem In this lecture we define the signature of a closed oriented n-manifold for n divisible by four. It is a bordism invariant Sign: Ω SO n Z. (Recall that we defined
More informationMike Hill University of Virginia Mike Hopkins Harvard University Doug Ravenel University of Rochester
A solution to the Arf-Kervaire invariant problem Second Abel Conference: A Mathematical Celebration of John Milnor February 1, 2012 Mike Hill University of Virginia Mike Hopkins Harvard University Doug
More informationLecture 8: More characteristic classes and the Thom isomorphism
Lecture 8: More characteristic classes and the Thom isomorphism We begin this lecture by carrying out a few of the exercises in Lecture 1. We take advantage of the fact that the Chern classes are stable
More informationKO -theory of complex Stiefel manifolds
KO -theory of complex Stiefel manifolds Daisuke KISHIMOTO, Akira KONO and Akihiro OHSHITA 1 Introduction The purpose of this paper is to determine the KO -groups of complex Stiefel manifolds V n,q which
More informationIntroduction to Index Theory. Elmar Schrohe Institut für Analysis
Introduction to Index Theory Elmar Schrohe Institut für Analysis Basics Background In analysis and pde, you want to solve equations. In good cases: Linearize, end up with Au = f, where A L(E, F ) is a
More informationEulerian polynomials
Münster J. of Math. 1 (2008), 9 14 Münster Journal of Mathematics urn:nbn:de:hbz:6-43529469890 c by the author 2008 Eulerian polynomials Friedrich Hirzebruch (Communicated by Linus Kramer) Abstract. A
More informationWHAT IS K-HOMOLOGY? Paul Baum Penn State. Texas A&M University College Station, Texas, USA. April 2, 2014
WHAT IS K-HOMOLOGY? Paul Baum Penn State Texas A&M University College Station, Texas, USA April 2, 2014 Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, 2014 1 / 56 Let X be a compact C manifold without
More informationAn Introduction to Complex K-Theory
An Introduction to Complex K-Theory May 23, 2010 Jesse Wolfson Abstract Complex K-Theory is an extraordinary cohomology theory defined from the complex vector bundles on a space. This essay aims to provide
More informationMorse theory and stable pairs
Richard A. SCGAS 2010 Joint with Introduction Georgios Daskalopoulos (Brown University) Jonathan Weitsman (Northeastern University) Graeme Wilkin (University of Colorado) Outline Introduction 1 Introduction
More informationFriedrich Hirzebruch Thomas Berger Rainer jung. Manifolds and Modular Forms
Friedrich Hirzebruch Thomas Berger Rainer jung Manifolds and Modular Forms Aspect~f tv\athematic~ Edited by Klas Diederich Vol. E 2: Vol. E 3: Vol. E 5: Vol. E 6: Vol. E 7: Vol. E 9: M. Knebusch/M. Kolster:
More informationTHE METHODS OF ALGEBRAIC TOPOLOGY FROM THE VIEWPOINT OF COBORDISM THEORY
THE METHODS OF ALGEBRAIC TOPOLOGY FROM THE VIEWPOINT OF COBORDISM THEORY S. P. NOVIKOV Abstract. The goal of this wor is the construction of the analogue to the Adams spectral sequence in cobordism theory,
More informationCharacteristic classes and Invariants of Spin Geometry
Characteristic classes and Invariants of Spin Geometry Haibao Duan Institue of Mathematics, CAS 2018 Workshop on Algebraic and Geometric Topology, Southwest Jiaotong University July 29, 2018 Haibao Duan
More informationSOME ASPECTS OF STABLE HOMOTOPY THEORY
SOME ASPECTS OF STABLE HOMOTOPY THEORY By GEORGE W. WHITEHEAD 1. The suspension category Many of the phenomena of homotopy theory become simpler in the "suspension range". This fact led Spanier and J.
More informationCohomology of the classifying spaces of gauge groups over 3-manifolds in low dimensions
Cohomology of the classifying spaces of gauge groups over 3-manifolds in low dimensions by Shizuo Kaji Department of Mathematics Kyoto University Kyoto 606-8502, JAPAN e-mail: kaji@math.kyoto-u.ac.jp Abstract
More informationarxiv: v1 [math.ag] 13 Mar 2019
THE CONSTRUCTION PROBLEM FOR HODGE NUMBERS MODULO AN INTEGER MATTHIAS PAULSEN AND STEFAN SCHREIEDER arxiv:1903.05430v1 [math.ag] 13 Mar 2019 Abstract. For any integer m 2 and any dimension n 1, we show
More informationScalar curvature and the Thurston norm
Scalar curvature and the Thurston norm P. B. Kronheimer 1 andt.s.mrowka 2 Harvard University, CAMBRIDGE MA 02138 Massachusetts Institute of Technology, CAMBRIDGE MA 02139 1. Introduction Let Y be a closed,
More informationKunihiko Kodaira: Mathematician, Friend, and Teacher
Kunihiko Kodaira: Mathematician, Friend, and Teacher F. Hirzebruch Kunihiko Kodaira died July 26, 1997, and a memorial article appeared in the March 1998 Notices, pp. 388 389. Kunihiko Kodaira was friend
More informationThe Yang-Mills equations over Klein surfaces
The Yang-Mills equations over Klein surfaces Chiu-Chu Melissa Liu & Florent Schaffhauser Columbia University (New York City) & Universidad de Los Andes (Bogotá) Seoul ICM 2014 Outline 1 Moduli of real
More informationPeriods of meromorphic quadratic differentials and Goldman bracket
Periods of meromorphic quadratic differentials and Goldman bracket Dmitry Korotkin Concordia University, Montreal Geometric and Algebraic aspects of integrability, August 05, 2016 References D.Korotkin,
More informationHOMOTOPICALLY EQUIVALENT SMOOTH MANIFOLDS. I
HOMOTOPICALLY EQUIVALET SMOOTH MAIFOLDS. I S. P. OVIKOV In this paper we introduce a method for the investigation of smooth simply connected manifolds of dimension n 5 that permits a classification of
More informationKR-theory. Jean-Louis Tu. Lyon, septembre Université de Lorraine France. IECL, UMR 7502 du CNRS
Jean-Louis Tu Université de Lorraine France Lyon, 11-13 septembre 2013 Complex K -theory Basic definition Definition Let M be a compact manifold. K (M) = {[E] [F] E, F vector bundles } [E] [F] [E ] [F
More informationBEYOND ELLIPTICITY. Paul Baum Penn State. Fields Institute Toronto, Canada. June 20, 2013
BEYOND ELLIPTICITY Paul Baum Penn State Fields Institute Toronto, Canada June 20, 2013 Paul Baum (Penn State) Beyond Ellipticity June 20, 2013 1 / 47 Minicourse of five lectures: 1. Dirac operator 2. Atiyah-Singer
More informationCLOSED (J-I)-CONNECTED (2J+1)-MANIFOLDS, s = 3, 7.
CLOSED (J-I)-CONNECTED (2J+1)-MANIFOLDS, s = 3, 7. DAVID L. WILKENS 1. Introduction This paper announces certain results concerning closed (s l)-connected (2s +1)- manifolds P, where s = 3 or 7. (Here
More informationFROM HOLOMORPHIC FUNCTIONS TO HOLOMORPHIC SECTIONS
FROM HOLOMORPHIC FUNCTIONS TO HOLOMORPHIC SECTIONS ZHIQIN LU. Introduction It is a pleasure to have the opportunity in the graduate colloquium to introduce my research field. I am a differential geometer.
More informationRATIONAL PONTRJAGIN CLASSES. HOMEOMORPHISM AND HOMOTOPY TYPE OF CLOSED MANIFOLDS. I
RATIONAL PONTRJAGIN CLASSES. HOMEOMORPHISM AND HOMOTOPY TYPE OF CLOSED MANIFOLDS. I S. P. NOVIKOV In a number of special cases it is proved that the rational Pontrjagin Hirzebruch classes may be computed
More informationIntroductory Lectures on Manifold Topology: Signposts
Surveys of Modern Mathematics Volume VII Introductory Lectures on Manifold Topology: Signposts Thomas Farrell Department of Mathematical Sciences Binghamton University Yang Su Academy of Mathematics and
More informationA TASTE OF TWO-DIMENSIONAL COMPLEX ALGEBRAIC GEOMETRY. We also have an isomorphism of holomorphic vector bundles
A TASTE OF TWO-DIMENSIONAL COMPLEX ALGEBRAIC GEOMETRY LIVIU I. NICOLAESCU ABSTRACT. These are notes for a talk at a topology seminar at ND.. GENERAL FACTS In the sequel, for simplicity we denote the complex
More informationClassics in Mathematics
Classics in Mathematics Friedrich Hirzebruch Topological Methods in Algebraic Geometry Friedrich Hirzebruch Topological Methods in Algebraic Geometry Reprint of the 1978 Edition Springer Friedrich Hirzebruch
More informationL6: Almost complex structures
L6: Almost complex structures To study general symplectic manifolds, rather than Kähler manifolds, it is helpful to extract the homotopy-theoretic essence of having a complex structure. An almost complex
More informationDiagonal Subschemes and Vector Bundles
Pure and Applied Mathematics Quarterly Volume 4, Number 4 (Special Issue: In honor of Jean-Pierre Serre, Part 1 of 2 ) 1233 1278, 2008 Diagonal Subschemes and Vector Bundles Piotr Pragacz, Vasudevan Srinivas
More informationCobordism of fibered knots and related topics
Advanced Studies in Pure Mathematics 46, 2007 Singularities in Geometry and Topology 2004 pp. 1 47 Cobordism of fibered knots and related topics Vincent Blanlœil and Osamu Saeki Abstract. This is a survey
More informationSymplectic Lefschetz fibrations and the geography of symplectic 4-manifolds
Symplectic Lefschetz fibrations and the geography of symplectic 4-manifolds Matt Harvey January 17, 2003 This paper is a survey of results which have brought techniques from the theory of complex surfaces
More informationModuli spaces of graphs and homology operations on loop spaces of manifolds
Moduli spaces of graphs and homology operations on loop spaces of manifolds Ralph L. Cohen Stanford University July 2, 2005 String topology:= intersection theory in loop spaces (and spaces of paths) of
More informationFROM CLASSICAL THETA FUNCTIONS TO TOPOLOGICAL QUANTUM FIELD THEORY
FROM CLASSICAL THETA FUNCTIONS TO TOPOLOGICAL QUANTUM FIELD THEORY Răzvan Gelca Texas Tech University Alejandro Uribe University of Michigan WE WILL CONSTRUCT THE ABELIAN CHERN-SIMONS TOPOLOGICAL QUANTUM
More informationSOME EXERCISES. By popular demand, I m putting up some fun problems to solve. These are meant to give intuition for messing around with spectra.
SOME EXERCISES By popular demand, I m putting up some fun problems to solve. These are meant to give intuition for messing around with spectra. 1. The algebraic thick subcategory theorem In Lecture 2,
More information30 Surfaces and nondegenerate symmetric bilinear forms
80 CHAPTER 3. COHOMOLOGY AND DUALITY This calculation is useful! Corollary 29.4. Let p, q > 0. Any map S p+q S p S q induces the zero map in H p+q ( ). Proof. Let f : S p+q S p S q be such a map. It induces
More informationK-Homology, Assembly and Rigidity Theorems for Relative Eta Invariants
K-Homology, Assembly and Rigidity Theorems for Relative Eta Invariants Department of Mathematics Pennsylvania State University Potsdam, May 16, 2008 Outline K-homology, elliptic operators and C*-algebras.
More informationAlgebraic Curves and Riemann Surfaces
Algebraic Curves and Riemann Surfaces Rick Miranda Graduate Studies in Mathematics Volume 5 If American Mathematical Society Contents Preface xix Chapter I. Riemann Surfaces: Basic Definitions 1 1. Complex
More informationThe relationship between framed bordism and skew-framed bordism
The relationship between framed bordism and sew-framed bordism Pyotr M. Ahmet ev and Peter J. Eccles Abstract A sew-framing of an immersion is an isomorphism between the normal bundle of the immersion
More information