Atiyah-Singer Revisited
|
|
- Andra Ball
- 6 years ago
- Views:
Transcription
1 Atiyah-Singer Revisited Paul Baum Penn State Texas A&M Universty College Station, Texas, USA April 1, 2014
2 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 vector bundle on S n (n even) 3) The spin representation of Spin c (n)
3 W finite dimensional C vector space dim C (W ) < T : W W T Hom C (W, W ) T 2 = I = The eigen-values of T are ± i i = 1 W = W i W i W i = {v W T v = iv} W i = {v W T v = iv}
4 Bott generator vector bundle n even n = 2r S n R n+1 S n = {(a 1, a 2,..., a n+1 ) R n+1 a a a2 n+1 = 1} S n M(2 r, C) (a 1, a 2,..., a n+1 ) a 1 E 1 + a 2 E a n+1 E n+1 (a 1 E 1 + a 2 E a n+1 E n+1 ) 2 = ( a 2 1 a a 2 n+1) I = I = The eigenvalues of a 1 E 1 + a 2 E a n+1 E n+1 are ± i.
5 Bott generator vector bundle β on S n n even n = 2r S n M(2 r, C) (a 1, a 2,..., a n+1 ) a 1 E 1 + a 2 E a n+1 E n+1 β (a1,a 2,...,a n+1 ) = (i eigenspace of a 1 E 1 + a 2 E a n+1 E n+1 )
6 Bott generator vector bundle β on S n n even n = 2r β is determined by: 1 p S n, dim C (β p ) = 2 r 1 2 ch(β)[s n ] = 1
7 Since SO(n) GL(n, R) the usual map Spin c (n) SO(n) can be viewed as mapping Spin c (n) to GL(n, R). Spin c (n) GL(n, R) Let E be an R vector bundle on a paracompact Hausdorff space X. DEFINITION A Spin c datum for E is a homomorphism of principal bundles η : P F(E) whose underlying homomorphism of topological groups is Spin c (n) GL(n, R)
8 A Spin c structure for E is a homotopy class of Spin c data for E. {Spin c structures for E} := {Spin c data for E}/ where = homotopy. A Spin c vector bundle on X is an R vector bundle E on X with a given Spin c structure. A Spin c manifold M is a C manifold (with or without boundary) whose tangent bundle T M is a Spin c vector bundle.
9 Minicourse of five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology? 4. Beyond ellipticity 5. The Riemann-Roch theorem The minicourse is based on joint work with Ron Douglas and is dedicated to him.
10 ATIYAH-SINGER REVISITED This is an expository talk about the Atiyah-Singer index theorem. 1 Dirac operator of R n will be defined. 2 Some low dimensional examples of the theorem will be considered. 3 A special case of the theorem will be proved, with the proof based on Bott periodicity. 4 The proof will be outlined that the special case implies the full theorem.
11 Atiyah-Singer Index theorem M compact C manifold without boundary D an elliptic differential (or pseudo-differential) operator on M E 0, E 1, C C vector bundles on M C (M, E j ) denotes the C vector space of all C sections of E j. D : C (M, E 0 ) C (M, E 1 ) D is a linear transformation of C vector spaces.
12 Atiyah-Singer Index theorem M compact C manifold without boundary D an elliptic differential (or pseudo-differential) operator on M Index(D) := dim C (Kernel D) dim C (Cokernel D) Theorem (M.Atiyah and I.Singer) Index (D) = (a topological formula)
13 Example M = S 1 = {(t 1, t 2 ) R 2 t t2 2 = 1} D f : L 2 (S 1 ) L 2 (S 1 ) is T f 0 0 I where L 2 (S 1 ) = L 2 +(S 1 ) L 2 (S 1 ). L 2 +(S 1 ) has as orthonormal basis e inθ with n = 0, 1, 2,... L 2 (S 1 ) has as orthonormal basis e inθ with n = 1, 2, 3,....
14 Example f : S 1 R 2 {0} is a C map. S 1 R 2 {(0, 0)} f T f : L 2 +(S 1 ) L 2 +(S 1 ) is the composition L 2 +(S 1 ) M f L 2 (S 1 ) L 2 +(S 1 ) T f : L 2 +(S 1 ) L 2 +(S 1 ) is the Toeplitz operator associated to f
15 Example Thus T f is composition T f : L 2 +(S 1 ) M f L 2 (S 1 P ) L 2 +(S 1 ) where L 2 +(S 1 ) M f L 2 (S 1 ) is v fv fv(t 1, t 2 ) := f(t 1, t 2 )v(t 1, t 2 ) (t 1, t 2 ) S 1 R 2 = C and L 2 (S 1 ) P L 2 +(S 1 ) is the Hilbert space projection. D f (v + w) := T f (v) + w v L 2 +(S 1 ), w L 2 (S 1 ) Index(D f ) = -winding number (f).
16 RIEMANN - ROCH M compact connected Riemann surface genus of M = # of holes = 1 2 [rankh 1(M; Z)]
17 D a divisor of M D consists of a finite set of points of M p 1, p 2,..., p l and an integer assigned to each point n 1, n 2,..., n l Equivalently D is a function D : M Z with finite support Support(D) = {p M D(p) 0} Support(D) is a finite subset of M
18 D a divisor on M deg(d) := p M D(p) Remark D 1, D 2 two divisors D 1 D 2 iff p M, D 1 (p) D 2 (p) Remark D a divisor, D is ( D)(p) = D(p)
19 Example Let f : M C { } be a meromorphic function. Define a divisor δ(f) by: 0 if p is neither a zero nor a pole of f δ(f)(p) = order of the zero if f(p) = 0 (order of the pole) if p is a pole of f
20 Example Let w be a meromorphic 1-form on M. Locally w is f(z)dz where f is a (locally defined) meromorphic function. Define a divisor δ(w) by: 0 if p is neither a zero nor a pole of w δ(w)(p) = order of the zero if w(p) = 0 (order of the pole) if p is a pole of w
21 D a divisor on M { } meromorphic functions H 0 (M, D) := δ(f) D f : M C { } { } meromorphic 1-forms H 1 (M, D) := w on M δ(w) D Lemma H 0 (M, D) and H 1 (M, D) are finite dimensional C vector spaces dim C H 0 (M, D) < dim C H 1 (M, D) <
22 Theorem (R. R.) Let M be a compact connected Riemann surface and let D be a divisor on M. Then: dim C H 0 (M, D) dim C H 1 (M, D) = d g + 1 d = degree (D) g = genus (M)
23 HIRZEBRUCH-RIEMANN-ROCH M non-singular projective algebraic variety / C E an algebraic vector bundle on M E = sheaf of germs of algebraic sections of E H j (M, E) := j-th cohomology of M using E, j = 0, 1, 2, 3,...
24 LEMMA For all j = 0, 1, 2,... dim C H j (M, E) <. For all j > dim C (M), H j (M, E) = 0. χ(m, E) := n = dim C (M) n ( 1) j dim C H j (M, E) j=0 THEOREM[HRR] Let M be a non-singular projective algebraic variety / C and let E be an algebraic vector bundle on M. Then χ(m, E) = (ch(e) T d(m))[m]
25 Hirzebruch-Riemann-Roch Theorem (HRR) Let M be a non-singular projective algebraic variety / C and let E be an algebraic vector bundle on M. Then χ(m, E) = (ch(e) T d(m))[m]
26 Various well-known structures on a C manifold M make M into a Spin c manifold (complex-analytic) (symplectic) (almost complex) (contact) (stably almost complex) Spin Spin c (oriented) A Spin c manifold can be thought of as an oriented manifold with a slight extra bit of structure. Most of the oriented manifolds which occur in practice are Spin c manifolds.
27 Two Out Of Three Lemma Lemma Let 0 E E E 0 be a short exact sequence of R-vector bundles on X. If two out of three are Spin c vector bundles, then so is the third.
28 Definition Let M be a C manifold (with or without boundary). M is a Spin c manifold iff the tangent bundle T M of M is a Spin c vector bundle on M. The Two Out Of Three Lemma implies that the boundary M of a Spin c manifold M with boundary is again a Spin c manifold.
29 A Spin c manifold comes equipped with a first-order elliptic differential operator known as its Dirac operator. If M is a Spin c manifold, then T d(m) is T d(m) := exp c 1(M)/2 Â(M) T d(m) H (M; Q) If M is a complex-analyic manifold, then M has Chern classes c 1, c 2,..., c n and exp c 1(M)/2 Â(M) = P T odd (c 1, c 2,..., c n )
30 EXAMPLE. Let M be a compact complex-analytic manifold. Set Ω p,q = C (M, Λ p,q T M) Ω p,q is the C vector space of all C differential forms of type (p, q) Dolbeault complex 0 Ω 0,0 Ω 0,1 Ω 0,2 Ω 0,n 0 The Dirac operator (of the underlying Spin c manifold) is the assembled Dolbeault complex + : j Ω 0, 2j j 0, 2j+1 Ω The index of this operator is the arithmetic genus of M i.e. is the Euler number of the Dolbeault complex.
31 TWO POINTS OF VIEW ON SPIN c MANIFOLDS 1. Spin c is a slight strengthening of oriented. The oriented manifolds that occur in practice are Spin c. 2. Spin c is much weaker than complex-analytic. BUT the assempled Dolbeault complex survives (as the Dirac operator). AND the Todd class survives. M Spin c = T d(m) H (M; Q)
32 SPECIAL CASE OF ATIYAH-SINGER Let M be a compact even-dimensional Spin c manifold without boundary. Let E be a C vector bundle on M. D E denotes the Dirac operator of M tensored with E. D E : C (M, S + E) C (M, S E) S +, S are the positive (negative) spinor bundles on M. THEOREM Index(D E ) = (ch(e) T d(m))[m].
33 K 0 ( ) Definition Define an abelian group denoted K 0 ( ) by considering pairs (M, E) such that: 1 M is a compact even-dimensional Spin c manifold without boundary. 2 E is a C vector bundle on M.
34 Set K 0 ( ) = {(M, E)}/ where the the equivalence relation is generated by the three elementary steps Bordism Direct sum - disjoint union Vector bundle modification Addition in K 0 ( ) is disjoint union. (M, E) + (M, E ) = (M M, E E ) In K 0 ( ) the additive inverse of (M, E) is ( M, E) where M denotes M with the Spin c structure reversed. (M, E) = ( M, E)
35 Isomorphism (M, E) is isomorphic to (M, E ) iff a diffeomorphism ψ : M M preserving the Spin c -structures on M, M and with ψ (E ) = E.
36 Bordism (M 0, E 0 ) is bordant to (M 1, E 1 ) iff (Ω, E) such that: 1 Ω is a compact odd-dimensional Spin c manifold with boundary. 2 E is a C vector bundle on Ω. 3 ( Ω, E Ω ) = (M 0, E 0 ) ( M 1, E 1, ) M 1 is M 1 with the Spin c structure reversed.
37 (M 0, E 0 ) ( M 1, E 1 )
38 Direct sum - disjoint union Let E, E be two C vector bundles on M (M, E) (M, E ) (M, E E )
39 Vector bundle modification (M, E) Let F be a Spin c vector bundle on M Assume that dim R (F p ) 0 mod 2 p M for every fiber F p of F 1 R = M R S(F 1 R ) := unit sphere bundle of F 1 R (M, E) (S(F 1 R ), β π E)
40 S(F 1 R ) M π This is a fibration with even-dimensional spheres as fibers. F 1 R is a Spin c vector bundle on M with odd-dimensional fibers. The Spin c structure for F causes there to appear on S(F 1 R ) a C-vector bundle β whose restriction to each fiber of π is the Bott generator vector bundle of that even-dimensional sphere. (M, E) (S(F 1 R ), β π E)
41 Addition in K 0 ( ) is disjoint union. (M, E) + (M, E ) = (M M, E E ) In K 0 ( ) the additive inverse of (M, E) is ( M, E) where M denotes M with the Spin c structure reversed. (M, E) = ( M, E)
42 DEFINITION. (M, E) bounds (Ω, Ẽ) with : 1 Ω is a compact odd-dimensional Spin c manifold with boundary. 2 Ẽ is a C vector bundle on Ω. 3 ( Ω, Ẽ Ω) = (M, E) REMARK. (M, E) = 0 in K 0 ( ) (M, E) (M, E ) where (M, E ) bounds.
43 Consider the homomorphism of abelian groups K 0 ( ) Z (M, E) Index(D E ) Notation D E is the Dirac operator of M tensored with E.
44 It is a corollary of Bott periodicity that this homomorphism of abelian groups is an isomorphism. Equivalently, Index(D E ) is a complete invariant for the equivalence relation generated by the three elementary steps; i.e. (M, E) (M, E ) if and only if Index(D E ) = Index(D E ).
45 BOTT PERIODICITY Z j odd π j GL(n, C) = 0 j even j = 0, 1, 2,..., 2n 1
46 Why does Bott periodicity imply that K 0 ( ) Z (M, E) Index(D E ) is an isomorphism?
47 To prove surjectivity must find an (M, E) with Index(D E ) = 1. e.g. Let M = CP n, and let E be the trivial (complex) line bundle on CP n E=1 C = CP n C Index(CP n, 1 C ) = 1 Thus Bott periodicity is not used in the proof of surjectivity.
48 Lemma used in the Proof of Injectivity Given any (M, E) there exists an even-dimensional sphere S 2n and a C-vector bundle F on S 2n with (M, E) (S 2n, F ). Bott periodicity is not used in the proof of this lemma. The lemma is proved by a direct argument using the definition of the equivalence relation on the pairs (M, E).
49 Let r be a positive integer, and let Vect C (S 2n, r) be the set of isomorphism classes of C vector bundles on S 2n of rank r, i.e. of fiber dimension r. Vect C (S 2n, r) π 2n 1 GL(r, C)
50 PROOF OF INJECTIVITY Let (M, E) have Index(M, E) = 0. By the above lemma, we may assume that (M, E) = (S 2n, F ). Using Bott periodicity plus the bijection Vect C (S 2n, r) π 2n 1 GL(r, C) we may assume that F is of the form F = θ p qβ θ p = S 2n C p and β is the Bott generator vector bundle on S 2n. Convention. If q < 0, then qβ = q β.
51 Index(S 2n, β) = 1 Index(S 2n, θ p ) = 0 Therefore Index(S 2n, F ) = 0 = q = 0 Hence (S 2n, F ) = (S 2n, θ p ). This bounds and so is zero in K 0 ( ). QED (S 2n, θ p ) = (B 2n+1, B 2n+1 C p )
52 Define a homomorphism of abelian groups K 0 ( ) Q (M, E) ( ch(e) Td(M) ) [M] where ch(e) is the Chern character of E and Td(M) is the Todd class of M. ch(e) H (M, Q) and Td(M) H (M, Q). [M] is the orientation cycle of M. [M] H (M, Z).
53 Granted that K 0 ( ) Z (M, E) Index(D E ) is an isomorphism, to prove that these two homomorphisms are equal, it suffices to check one nonzero example.
Dirac 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 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 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 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 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 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 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 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 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 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 informationTOEPLITZ OPERATORS. Toeplitz studied infinite matrices with NW-SE diagonals constant. f e C :
TOEPLITZ OPERATORS EFTON PARK 1. Introduction to Toeplitz Operators Otto Toeplitz lived from 1881-1940 in Goettingen, and it was pretty rough there, so he eventually went to Palestine and eventually contracted
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 informationCelebrating One Hundred Fifty Years of. Topology. ARBEITSTAGUNG Bonn, May 22, 2013
Celebrating One Hundred Fifty Years of Topology John Milnor Institute for Mathematical Sciences Stony Brook University (www.math.sunysb.edu) ARBEITSTAGUNG Bonn, May 22, 2013 Algebra & Number Theory 3 4
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 information1 Structures 2. 2 Framework of Riemann surfaces Basic configuration Holomorphic functions... 3
Compact course notes Riemann surfaces Fall 2011 Professor: S. Lvovski transcribed by: J. Lazovskis Independent University of Moscow December 23, 2011 Contents 1 Structures 2 2 Framework of Riemann surfaces
More informationMath 231b Lecture 16. G. Quick
Math 231b Lecture 16 G. Quick 16. Lecture 16: Chern classes for complex vector bundles 16.1. Orientations. From now on we will shift our focus to complex vector bundles. Much of the theory for real vector
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 informationA users guide to K-theory
A users guide to K-theory K-theory Alexander Kahle alexander.kahle@rub.de Mathematics Department, Ruhr-Universtät Bochum Bonn-Cologne Intensive Week: Tools of Topology for Quantum Matter, July 2014 Outline
More informationOn the Equivalence of Geometric and Analytic K-Homology
On the Equivalence of Geometric and Analytic K-Homology Paul Baum, Nigel Higson, and Thomas Schick Abstract We give a proof that the geometric K-homology theory for finite CWcomplexes defined by Baum and
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 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 informationIntroduction to the Baum-Connes conjecture
Introduction to the Baum-Connes conjecture Nigel Higson, John Roe PSU NCGOA07 Nigel Higson, John Roe (PSU) Introduction to the Baum-Connes conjecture NCGOA07 1 / 15 History of the BC conjecture Lecture
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 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 informationCW-complexes. Stephen A. Mitchell. November 1997
CW-complexes Stephen A. Mitchell November 1997 A CW-complex is first of all a Hausdorff space X equipped with a collection of characteristic maps φ n α : D n X. Here n ranges over the nonnegative integers,
More informationChern Classes and the Chern Character
Chern Classes and the Chern Character German Stefanich Chern Classes In this talk, all our topological spaces will be paracompact Hausdorff, and our vector bundles will be complex. Let Bun GLn(C) be the
More informationThe kernel of the Dirac operator
The kernel of the Dirac operator B. Ammann 1 M. Dahl 2 E. Humbert 3 1 Universität Regensburg Germany 2 Institutionen för Matematik Kungliga Tekniska Högskolan, Stockholm Sweden 3 Laboratoire de Mathématiques
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 informationDISTINGUISHING EMBEDDED CURVES IN RATIONAL COMPLEX SURFACES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 16, Number 1, January 1998, Pages 305 310 S 000-9939(98)04001-5 DISTINGUISHING EMBEDDED CURVES IN RATIONAL COMPLEX SURFACES TERRY FULLER (Communicated
More informationNOTES ON DIVISORS AND RIEMANN-ROCH
NOTES ON DIVISORS AND RIEMANN-ROCH NILAY KUMAR Recall that due to the maximum principle, there are no nonconstant holomorphic functions on a compact complex manifold. The next best objects to study, as
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 informationLECTURE 26: THE CHERN-WEIL THEORY
LECTURE 26: THE CHERN-WEIL THEORY 1. Invariant Polynomials We start with some necessary backgrounds on invariant polynomials. Let V be a vector space. Recall that a k-tensor T k V is called symmetric if
More informationMultiplicity of singularities is not a bi-lipschitz invariant
Multiplicity of singularities is not a bi-lipschitz invariant Misha Verbitsky Joint work with L. Birbrair, A. Fernandes, J. E. Sampaio Geometry and Dynamics Seminar Tel-Aviv University, 12.12.2018 1 Zariski
More informationThe Dirac-Ramond operator and vertex algebras
The Dirac-Ramond operator and vertex algebras Westfälische Wilhelms-Universität Münster cvoigt@math.uni-muenster.de http://wwwmath.uni-muenster.de/reine/u/cvoigt/ Vanderbilt May 11, 2011 Kasparov theory
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 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 informationUniversität Regensburg Mathematik
Universität Regensburg Mathematik Harmonic spinors and local deformations of the metric Bernd Ammann, Mattias Dahl, and Emmanuel Humbert Preprint Nr. 03/2010 HARMONIC SPINORS AND LOCAL DEFORMATIONS OF
More informationEach is equal to CP 1 minus one point, which is the origin of the other: (C =) U 1 = CP 1 the line λ (1, 0) U 0
Algebraic Curves/Fall 2015 Aaron Bertram 1. Introduction. What is a complex curve? (Geometry) It s a Riemann surface, that is, a compact oriented twodimensional real manifold Σ with a complex structure.
More information15 Elliptic curves and Fermat s last theorem
15 Elliptic curves and Fermat s last theorem Let q > 3 be a prime (and later p will be a prime which has no relation which q). Suppose that there exists a non-trivial integral solution to the Diophantine
More informationCharacteristic classes in the Chow ring
arxiv:alg-geom/9412008v1 10 Dec 1994 Characteristic classes in the Chow ring Dan Edidin and William Graham Department of Mathematics University of Chicago Chicago IL 60637 Let G be a reductive algebraic
More informationHandlebody Decomposition of a Manifold
Handlebody Decomposition of a Manifold Mahuya Datta Statistics and Mathematics Unit Indian Statistical Institute, Kolkata mahuya@isical.ac.in January 12, 2012 contents Introduction What is a handlebody
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 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 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 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 informationHodge Structures. October 8, A few examples of symmetric spaces
Hodge Structures October 8, 2013 1 A few examples of symmetric spaces The upper half-plane H is the quotient of SL 2 (R) by its maximal compact subgroup SO(2). More generally, Siegel upper-half space H
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 informationIndex Theory and Spin Geometry
Index Theory and Spin Geometry Fabian Lenhardt and Lennart Meier March 20, 2010 Many of the familiar (and not-so-familiar) invariants in the algebraic topology of manifolds may be phrased as an index of
More informationA Problem of Hsiang-Palais-Terng on Isoparametric Submanifolds
arxiv:math/0312251v1 [math.dg] 12 Dec 2003 A Problem of Hsiang-Palais-Terng on Isoparametric Submanifolds Haibao Duan Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, dhb@math.ac.cn
More informationFAKE PROJECTIVE SPACES AND FAKE TORI
FAKE PROJECTIVE SPACES AND FAKE TORI OLIVIER DEBARRE Abstract. Hirzebruch and Kodaira proved in 1957 that when n is odd, any compact Kähler manifold X which is homeomorphic to P n is isomorphic to P n.
More informationALGEBRAIC TOPOLOGY: MATH 231BR NOTES
ALGEBRAIC TOPOLOGY: MATH 231BR NOTES AARON LANDESMAN CONTENTS 1. Introduction 4 2. 1/25/16 5 2.1. Overview 5 2.2. Vector Bundles 5 2.3. Tautological bundles on projective spaces and Grassmannians 7 2.4.
More informationRIEMANN SURFACES. max(0, deg x f)x.
RIEMANN SURFACES 10. Weeks 11 12: Riemann-Roch theorem and applications 10.1. Divisors. The notion of a divisor looks very simple. Let X be a compact Riemann surface. A divisor is an expression a x x x
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 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 informationLECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE
LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE JOHANNES EBERT 1.1. October 11th. 1. Recapitulation from differential topology Definition 1.1. Let M m, N n, be two smooth manifolds
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 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 informationLecture 15: Duality. Next we spell out the answer to Exercise It is part of the definition of a TQFT.
Lecture 15: Duality We ended the last lecture by introducing one of the main characters in the remainder of the course, a topological quantum field theory (TQFT). At this point we should, of course, elaborate
More informationComplex manifolds, Kahler metrics, differential and harmonic forms
Complex manifolds, Kahler metrics, differential and harmonic forms Cattani June 16, 2010 1 Lecture 1 Definition 1.1 (Complex Manifold). A complex manifold is a manifold with coordinates holomorphic on
More informationSYMPLECTIC GEOMETRY: LECTURE 5
SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The
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 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 informationTopological K-theory
Topological K-theory Robert Hines December 15, 2016 The idea of topological K-theory is that spaces can be distinguished by the vector bundles they support. Below we present the basic ideas and definitions
More informationConformal field theory in the sense of Segal, modified for a supersymmetric context
Conformal field theory in the sense of Segal, modified for a supersymmetric context Paul S Green January 27, 2014 1 Introduction In these notes, we will review and propose some revisions to the definition
More informationNodal symplectic spheres in CP 2 with positive self intersection
Nodal symplectic spheres in CP 2 with positive self intersection Jean-François BARRAUD barraud@picard.ups-tlse.fr Abstract 1 : Let ω be the canonical Kähler structure on CP 2 We prove that any ω-symplectic
More informationPullbacks of hyperplane sections for Lagrangian fibrations are primitive
Pullbacks of hyperplane sections for Lagrangian fibrations are primitive Ljudmila Kamenova, Misha Verbitsky 1 Dedicated to Professor Claire Voisin Abstract. Let p : M B be a Lagrangian fibration on a hyperkähler
More informationMath 797W Homework 4
Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition
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 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 informationALGEBRAIC GEOMETRY I, FALL 2016.
ALGEBRAIC GEOMETRY I, FALL 2016. DIVISORS. 1. Weil and Cartier divisors Let X be an algebraic variety. Define a Weil divisor on X as a formal (finite) linear combination of irreducible subvarieties of
More informationBordism and the Pontryagin-Thom Theorem
Bordism and the Pontryagin-Thom Theorem Richard Wong Differential Topology Term Paper December 2, 2016 1 Introduction Given the classification of low dimensional manifolds up to equivalence relations such
More informationINERTIA GROUPS AND SMOOTH STRUCTURES OF (n - 1)- CONNECTED 2n-MANIFOLDS. Osaka Journal of Mathematics. 53(2) P.309-P.319
Title Author(s) INERTIA GROUPS AND SMOOTH STRUCTURES OF (n - 1)- CONNECTED 2n-MANIFOLDS Ramesh, Kaslingam Citation Osaka Journal of Mathematics. 53(2) P.309-P.319 Issue Date 2016-04 Text Version publisher
More informationLECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM
LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM Contents 1. The Atiyah-Guillemin-Sternberg Convexity Theorem 1 2. Proof of the Atiyah-Guillemin-Sternberg Convexity theorem 3 3. Morse theory
More informationLECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS. 1. Lie groups
LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups A Lie group is a special smooth manifold on which there is a group structure, and moreover, the two structures are compatible. Lie groups are
More informationGeometric Structures in Mathematical Physics Non-existence of almost complex structures on quaternion-kähler manifolds of positive type
Geometric Structures in Mathematical Physics Non-existence of almost complex structures on quaternion-kähler manifolds of positive type Paul Gauduchon Golden Sands, Bulgaria September, 19 26, 2011 1 Joint
More informationOverview of Atiyah-Singer Index Theory
Overview of Atiyah-Singer Index Theory Nikolai Nowaczyk December 4, 2014 Abstract. The aim of this text is to give an overview of the Index Theorems by Atiyah and Singer. Our primary motivation is to understand
More informationIntroduction to. Riemann Surfaces. Lecture Notes. Armin Rainer. dim H 0 (X, L D ) dim H 0 (X, L 1 D ) = 1 g deg D
Introduction to Riemann Surfaces Lecture Notes Armin Rainer dim H 0 (X, L D ) dim H 0 (X, L 1 D ) = 1 g deg D June 17, 2018 PREFACE i Preface These are lecture notes for the course Riemann surfaces held
More informationTHE H-PRINCIPLE, LECTURE 14: HAEFLIGER S THEOREM CLASSIFYING FOLIATIONS ON OPEN MANIFOLDS
THE H-PRINCIPLE, LECTURE 14: HAELIGER S THEOREM CLASSIYING OLIATIONS ON OPEN MANIOLDS J. RANCIS, NOTES BY M. HOYOIS In this lecture we prove the following theorem: Theorem 0.1 (Haefliger). If M is an open
More informationNOTES ON FIBER BUNDLES
NOTES ON FIBER BUNDLES DANNY CALEGARI Abstract. These are notes on fiber bundles and principal bundles, especially over CW complexes and spaces homotopy equivalent to them. They are meant to supplement
More informationMath 215B: Solutions 3
Math 215B: Solutions 3 (1) For this problem you may assume the classification of smooth one-dimensional manifolds: Any compact smooth one-dimensional manifold is diffeomorphic to a finite disjoint union
More informationTopics in Algebraic Geometry
Topics in Algebraic Geometry Nikitas Nikandros, 3928675, Utrecht University n.nikandros@students.uu.nl March 2, 2016 1 Introduction and motivation In this talk i will give an incomplete and at sometimes
More informationarxiv:math/ v1 [math.ag] 18 Oct 2003
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 113, No. 2, May 2003, pp. 139 152. Printed in India The Jacobian of a nonorientable Klein surface arxiv:math/0310288v1 [math.ag] 18 Oct 2003 PABLO ARÉS-GASTESI
More informationRIEMANN S INEQUALITY AND RIEMANN-ROCH
RIEMANN S INEQUALITY AND RIEMANN-ROCH DONU ARAPURA Fix a compact connected Riemann surface X of genus g. Riemann s inequality gives a sufficient condition to construct meromorphic functions with prescribed
More informationTopological K-theory, Lecture 3
Topological K-theory, Lecture 3 Matan Prasma March 2, 2015 1 Applications of the classification theorem continued Let us see how the classification theorem can further be used. Example 1. The bundle γ
More informationarxiv: v4 [math.dg] 26 Jun 2018
POSITIVE SCALAR CURVATURE METRICS VIA END-PERIODIC MANIFOLDS MICHAEL HALLAM AND VARGHESE MATHAI arxiv:1706.09354v4 [math.dg] 26 Jun 2018 Abstract. We obtain two types of results on positive scalar curvature
More informationRemarks on Fully Extended 3-Dimensional Topological Field Theories
Remarks on Fully Extended 3-Dimensional Topological Field Theories Dan Freed University of Texas at Austin June 6, 2011 Work in progress with Constantin Teleman Manifolds and Algebra: Abelian Groups Pontrjagin
More informationGeometric Aspects of Quantum Condensed Matter
Geometric Aspects of Quantum Condensed Matter January 15, 2014 Lecture XI y Classification of Vector Bundles over Spheres Giuseppe De Nittis Department Mathematik room 02.317 +49 09131 85 67071 @ denittis.giuseppe@gmail.com
More informationThe eta invariant and the equivariant spin. bordism of spherical space form 2 groups. Peter B Gilkey and Boris Botvinnik
The eta invariant and the equivariant spin bordism of spherical space form 2 groups Peter B Gilkey and Boris Botvinnik Mathematics Department, University of Oregon Eugene Oregon 97403 USA Abstract We use
More informationSpin(10,1)-metrics with a parallel null spinor and maximal holonomy
Spin(10,1)-metrics with a parallel null spinor and maximal holonomy 0. Introduction. The purpose of this addendum to the earlier notes on spinors is to outline the construction of Lorentzian metrics in
More informationGeometry Qualifying Exam Notes
Geometry Qualifying Exam Notes F 1 F 1 x 1 x n Definition: The Jacobian matrix of a map f : N M is.. F m F m x 1 x n square matrix, its determinant is called the Jacobian determinant.. When this is a Definition:
More informationNotes 10: Consequences of Eli Cartan s theorem.
Notes 10: Consequences of Eli Cartan s theorem. Version 0.00 with misprints, The are a few obvious, but important consequences of the theorem of Eli Cartan on the maximal tori. The first one is the observation
More informationTHE POINCARE-HOPF THEOREM
THE POINCARE-HOPF THEOREM ALEX WRIGHT AND KAEL DIXON Abstract. Mapping degree, intersection number, and the index of a zero of a vector field are defined. The Poincare-Hopf theorem, which states that under
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 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 informationThe Atiyah bundle and connections on a principal bundle
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 120, No. 3, June 2010, pp. 299 316. Indian Academy of Sciences The Atiyah bundle and connections on a principal bundle INDRANIL BISWAS School of Mathematics, Tata
More informationRIEMANN SURFACES. LECTURE NOTES. WINTER SEMESTER 2015/2016.
RIEMANN SURFACES. LECTURE NOTES. WINTER SEMESTER 2015/2016. A PRELIMINARY AND PROBABLY VERY RAW VERSION. OLEKSANDR IENA Contents Some prerequisites for the whole lecture course. 5 1. Lecture 1 5 1.1. Definition
More information1. Algebraic vector bundles. Affine Varieties
0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.
More informationThe Strominger Yau Zaslow conjecture
The Strominger Yau Zaslow conjecture Paul Hacking 10/16/09 1 Background 1.1 Kähler metrics Let X be a complex manifold of dimension n, and M the underlying smooth manifold with (integrable) almost complex
More informationOn Eilenberg-MacLanes Spaces (Term paper for Math 272a)
On Eilenberg-MacLanes Spaces (Term paper for Math 272a) Xi Yin Physics Department Harvard University Abstract This paper discusses basic properties of Eilenberg-MacLane spaces K(G, n), their cohomology
More informationTwo-sided multiplications and phantom line bundles
Two-sided multiplications and phantom line bundles Ilja Gogić Department of Mathematics University of Zagreb 19th Geometrical Seminar Zlatibor, Serbia August 28 September 4, 2016 joint work with Richard
More information