# Dirac Operator. Göttingen Mathematical Institute. Paul Baum Penn State 6 February, 2017

Size: px
Start display at page:

Download "Dirac Operator. Göttingen Mathematical Institute. Paul Baum Penn State 6 February, 2017"

Transcription

1 Dirac Operator Göttingen Mathematical Institute Paul Baum Penn State 6 February, 2017

2 Five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology? 4. The Riemann-Roch theorem 5. K-theory for group C algebras (BC conjecture)

3 DIRAC OPERATOR The Dirac operator of R n will be defined. This is a first order elliptic differential operator with constant coefficients. Next, the class of differentiable manifolds which come equipped with an order one differential operator which at the symbol level is locally isomorphic to the Dirac operator of R n will be considered. These are the Spin c manifolds. Spin c is slightly stronger than oriented, so Spin c can be viewed as oriented plus epsilon. Most of the oriented manifolds that occur in practice are Spin c. The Dirac operator of a closed Spin c manifold is the basic example for the Hirzebruch-Riemann-Roch theorem and the Atiyah-Singer index theorem.

4 What is the Dirac operator of R n? To answer this, shall construct matrices E 1, E 2,..., E n with the following properties :

5 Properties of E 1, E 2,..., E n Each E j is a 2 r 2 r matrix of complex numbers, where r is the largest integer n/2. Each E j is skew adjoint, i.e. E j = E j (* = conjugate transpose) Ej 2 = I j = 1, 2,..., n (I is the 2 r 2 r identity matrix.) E j E k + E k E j = 0 whenever j k. For n odd, (n = 2r + 1) i r+1 E 1 E 2 E n = I For n even, (n = 2r) each E j is of the form E j = [ 0 0 ] and ir E 1 E 2 E n = [ ] I 0 0 I i = 1

6 These matrices are constructed by a simple inductive procedure. n = 1, E 1 = [ i] n n + 1 with n odd (r r + 1) The new matrices Ẽ1, Ẽ2,..., Ẽn+1 are ] Ẽ j = [ 0 Ej E j 0 for j = 1,..., n and Ẽ n+1 = [ ] 0 I I 0 where E 1, E 2,..., E n are the old matrices. n n + 1 with n even (r does not change) The new matrices Ẽ1, Ẽ2,..., Ẽn+1 are Ẽ j = E j for j = 1,..., n and Ẽ n+1 = [ ] ii 0 0 ii where E 1, E 2,..., E n are the old matrices.

7 Example n = 1: E 1 = [ i] n = 2: E 1 = [ 0 i i 0 n = 3: E 1 = [ 0 i i 0 ], E2 = [ ] ], E2 = [ ], E3 = [ ] i 0 0 i

8 Example n = 4: E 1 = [ 0 ] 0 0 i 0 0 i 0 0 i 0 0 i E 2 = [ ] E 3 = [ 0 0 i 0 ] i i i 0 0 E 4 = [ ]

9 D = Dirac operator of R n { n = 2r n even n = 2r + 1 n odd D = n j=1 E j x j D is an unbounded symmetric operator on the Hilbert space L 2 (R n ) L 2 (R n )... L 2 (R n ) (2 r times) To begin, the domain of D is C c (R n ) C c (R n )... C c (R n ) (2 r times) D is essentially self-adjoint (i.e. D has a unique self-adjoint extension) so it is natural to view D as an unbounded self-adjoint operator on the Hilbert space L 2 (R n ) L 2 (R n )... L 2 (R n ) (2 r times)

10 QUESTION : Let M be a C manifold of dimension n. Does M admit a differential operator which (at the symbol level) is locally isomorphic to the Dirac operator of R n? To answer this question, will define Spin c vector bundle.

11 What is a Spin c vector bundle? Let X be a paracompact Hausdorff topological space. On X let E be an R vector bundle which has been oriented. i.e. the structure group of E has been reduced from GL(n, R) to GL + (n, R) GL + (n, R) = {[a ij ] GL(n, R) det[a ij ] > 0} n= fiber dimension (E) Assume n 3 and recall that for n 3 H 2 (GL + (n, R); Z) = Z/2Z Denote by F + (E) the principal GL + (n, R) bundle on X consisting of all positively oriented frames.

12 A point of F + (E) is a pair ( x, (v 1, v 2,..., v n ) ) where x X and (v 1, v 2,..., v n ) is a positively oriented basis of E x. The projection F + (E) X is ( x, (v1, v 2,..., v n ) ) x For x X, denote by ι x : F x + (E) F + (E) the inclusion of the fiber at x into F + (E). Note that (with n 3) H 2 (F x + (E); Z) = Z/2Z

13 A Spin c vector bundle on X is an R vector bundle E on X (fiber dimension E 3) with 1 E has been oriented. 2 α H 2 (F + (E); Z) has been chosen such that x X ι x(α) H 2 (F + x (E); Z) is non-zero.

14 Remarks 1.For n = 1, 2 E is a Spin c vector bundle = E has been oriented and an element α H 2 (X; Z) has been chosen. (α can be zero.) 2. For all values of n = fiber dimension(e), E is a Spin c vector bundle iff the structure group of E has been changed from GL(n, R) to Spin c (n). i.e. Such a change of structure group is equivalent to the above definition of Spin c vector bundle.

15 Topological obstruction to Spin c -able E an R vector bundle on X. w 1 (E), w 2 (E),..., w n (E) Stiefel-Whitney classes of E w j (E) H j (X; Z/2Z) E is Spin c -able iff: (i) w 1 (E) = 0 (i.e. E is orientable). and (ii) w 2 (E) is in the image of the mod 2 reduction map H 2 (X; Z) H 2 (X; Z/2Z)

16 By forgetting some structure a complex vector bundle or a Spin vector bundle canonically becomes a Spin c vector bundle complex Spin Spin c oriented A Spin c structure for an R vector bundle E can be thought of as an orientation for E plus a slight extra bit of structure. Spin c structures behave very much like orientations. For example, an orientation on two out of three R vector bundles in a short exact sequence determines an orientation on the third vector bundle. An analogous assertion is true for Spin c structures.

17 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.

18 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.

19 Various well-known structures on a manifold M make M into a Spin c manifold. (complex-analytic) (symplectic) (almost complex) (contact) (stably almost complex) Spin Spin c (oriented)

20 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. A Spin c manifold comes equipped with a first-order elliptic differential operator known as its Dirac operator. This operator is locally isomorphic (at the symbol level) to the Dirac operator of R n.

21 EXAMPLE. Let M be a compact complex-analytic manifold. Set Ω p,q = C (M, Λ p,q T C 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.

22 TWO POINTS OF VIEW ON SPIN c MANIFOLDS 1. Spin c is a slight strengthening of oriented. Most of 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)

23 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 )

24 WARNING!!! The Todd class of a Spin c manifold is not obtained by complexifying the tangent bundle T M of M and then applying the Todd polynomial to the Chern classes of T C M. T d(t C M) = Â(M)2 = Â(M) Â(M) Correct formula for the Todd class of a Spin c manifold M is: T d(m) := exp c 1(M)/2 Â(M) T d(m) H (M; Q)

25 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].

26 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. THEOREM Index(D E ) = (ch(e) T d(m))[m]. This theorem will be proved in the next lecture as a corollary of Bott periodicity. In particular, this will prove the Hirzebruch-Riemann-Roch theorem. Also, this will prove (for closed even-dimensional Spin c manifolds) the Hirzebruch signature theorem.

27 From E 1, E 2,..., E n obtain : 1) The Dirac operator of R n 2) The Bott generator vector bundle on S n (n even) 3) The spin representation of Spin c (n)

28 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 ± 1 W = W 1 W 1 W 1 = {v W T v = v} W 1 = {v W T v = v}

29 Bott generator vector bundle n even n = 2r S n R n+1 S n M(2 r, C) S n = {(a 1, a 2,..., a n+1 ) R n+1 a a a2 n+1 = 1} (a 1, a 2,..., a n+1 ) i(a 1 E 1 + a 2 E a n+1 E n+1 ) i = 1 (i) 2 (a 1 E 1 + a 2 E a n+1 E n+1 ) 2 = ( 1)( a 2 1 a a 2 n+1) I = I = The eigenvalues of i(a 1 E 1 + a 2 E a n+1 E n+1 ) are ± 1.

30 Bott generator vector bundle β on S n n even n = 2r β (a1,a 2,...,a n+1 ) = (+1 eigenspace of i(a 1 E 1 + a 2 E a n+1 E n+1 )) = Hom C ({v C 2r i(a 1 E 1 + a 2 E a n+1 E n+1 ) v = v}, C) K 0 (S n ) = Z Z 1 β 1 = S n C

31 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

32 n even n = 2r S n R n+1 With the Spin (or Spin c ) structure S n has as the boundary of the unit ball B n+1 of R n+1, the Spinor bundle S of S n is: S = S n C 2r The positive (negative) Spinor bundles S + (S ) are defined by : S + (a 1,a 2,...,a n+1 ) = +1 eigenspace of i(a 1E 1 +a 2 E 2 + +a n+1 E n+1 ) S (a 1,a 2,...,a n+1 ) = 1 eigenspace of i(a 1E 1 +a 2 E 2 + +a n+1 E n+1 ) S = S n C 2r = S + S β = (S + )

33 M Spin c manifold M might be non-empty T M = the tangent bundle of M Dirac operator D : C c (M, S) C c (M, S) S is the Spinor bundle C c (M, S) = {C sections with compact support of S}

34 such that D : C c (M, S) C c (M, S) (1) D is C-linear D(s 1 + s 2 ) = Ds 1 + Ds 2 s j Cc (M, S) D(λx) = λds λ C (2) If f : M C is a C function, then D(fs) = (df)s + f(ds) (3) If s j Cc (M, S) then (Ds 1 x, s 2 x) = (s 1 x, Ds 2 x)dx M (4) If dim M is even, then D is off-diagonal S = S + S M D = 0 D D + 0

35 D : C c (M, S) C c (M, S) is an elliptic first-order differential operator. D can be viewed as an unbounded operator on the Hilbert space L 2 (M, S) (s 1, s 2 ) = M (s 1 x, s 2 x)dx D : C c (M, S) C c (M, S) is a symmetric operator

36 EXAMPLE. Let M be a compact complex-analytic manifold. The positive (negative) Spinor bundles of the underlying Spin c manifold are : S + = Λ 0, 2j TC M j S = j Λ 0, 2j+1 T C M + : C (M, j D + : C (M, S + ) C (M, S ) is Λ 0, 2j T C M) C (M, j Λ 0, 2j+1 T C M) The index of this operator is the arithmetic genus of M i.e. is the Euler number of the Dolbeault complex.

37 EXAMPLE. Let M be a compact even-dimensional Spin c manifold without boundary. D + S : C (M, S + S ) C (M, S S ) is the Hirzebruch signature operator of M. If the dimension of M is divisible by 4, the index of this operator is the signature of the quadratic form H r (M; R) R H r (M, R) R n = 2r r even a b (a b)[m]

38 Example. n even n = 2r S n R n+1 D = Dirac operator of S n S = Spinor bundle of S n = S n C 2r S = S + S D : C (S n, S) C (S n, S) 0 D D = D + 0 D + : C (S n, S + ) C (S n, S ) Index (D + ) := dim C (Kernel D + ) dim C (Cokernel D + ) = 0 Theorem. Index (D + ) = 0

39 Tensor D + with the Bott generator vector bundle β D + β : C (S n, S + β) C (S n, S β) Theorem. On S n, with n even, Index(D + ) = 0 and Index (D + β ) = 1.

40 BOTT PERIODICITY Z j odd π j GL(n, C) = 0 j even j = 0, 1, 2,..., 2n 1

41 Why???? does Bott periodicity imply 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. THEOREM Index(D E ) = (ch(e) T d(m))[m]. This will be explained in the next lecture tomorrow.

### 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

### Atiyah-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

### WHAT 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

### The 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

### The 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?

### BEYOND 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

### Invariance 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

### k=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

### The 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

### CHARACTERISTIC 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

### Introduction 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

### SPECTRAL 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

### K-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.

### Celebrating 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

### Remarks 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)

### Math 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

### TRANSVERSAL DIRAC OPERATORS ON DISTRIBUTIONS, FOLIATIONS, AND G-MANIFOLDS LECTURE NOTES

TRANSVERSAL DIRAC OPERATORS ON DISTRIBUTIONS, FOLIATIONS, AND G-MANIFOLDS LECTURE NOTES KEN RICHARDSON Abstract. In these lectures, we investigate generalizations of the ordinary Dirac operator to manifolds

### LECTURE 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

### Lecture 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

### Chern 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

### On 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

### Introduction 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

### KR-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

### arxiv: 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

### Index 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

### NOTES 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

### LECTURE 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

### The Spinor Representation

The Spinor Representation Math G4344, Spring 2012 As we have seen, the groups Spin(n) have a representation on R n given by identifying v R n as an element of the Clifford algebra C(n) and having g Spin(n)

### A 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

### The topology of positive scalar curvature ICM Section Topology Seoul, August 2014

The topology of positive scalar curvature ICM Section Topology Seoul, August 2014 Thomas Schick Georg-August-Universität Göttingen ICM Seoul, August 2014 All pictures from wikimedia. Scalar curvature My

### Geometric 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

### The 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

### Universität 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

### Introduction (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

### A SHORT OVERVIEW OF SPIN REPRESENTATIONS. Contents

A SHORT OVERVIEW OF SPIN REPRESENTATIONS YUNFENG ZHANG Abstract. In this note, we make a short overview of spin representations. First, we introduce Clifford algebras and spin groups, and classify their

### TOEPLITZ 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

### On algebraic index theorems. Ryszard Nest. Introduction. The index theorem. Deformation quantization and Gelfand Fuks. Lie algebra theorem

s The s s The The term s is usually used to describe the equality of, on one hand, analytic invariants of certain operators on smooth manifolds and, on the other hand, topological/geometric invariants

### Clifford Algebras and Spin Groups

Clifford Algebras and Spin Groups Math G4344, Spring 2012 We ll now turn from the general theory to examine a specific class class of groups: the orthogonal groups. Recall that O(n, R) is the group of

### Complex 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

### MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES

MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES NILAY KUMAR In these lectures I want to introduce the Chern-Weil approach to characteristic classes on manifolds, and in particular, the Chern classes.

### Summary 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

### The 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

### LECTURE 3 MATH 261A. Office hours are now settled to be after class on Thursdays from 12 : 30 2 in Evans 815, or still by appointment.

LECTURE 3 MATH 261A LECTURES BY: PROFESSOR DAVID NADLER PROFESSOR NOTES BY: JACKSON VAN DYKE Office hours are now settled to be after class on Thursdays from 12 : 30 2 in Evans 815, or still by appointment.

### Exotic 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

### arxiv: 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

### 15 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

### FROM 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.

### Overview 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

### Recall 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

### 30 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

### The Standard Model in Noncommutative Geometry: fermions as internal Dirac spinors

1/21 The Standard Model in Noncommutative Geometry: fermions as internal Dirac spinors Ludwik Dabrowski SISSA, Trieste (I) (based on JNCG in print with F. D Andrea) Corfu, 22 September 2015 Goal Establish

### Modern index Theory lectures held at CIRM rencontré Theorie d indice, Mar 2006

Modern index Theory lectures held at CIRM rencontré Theorie d indice, Mar 2006 Thomas Schick, Göttingen Abstract Every elliptic (pseudo)-differential operator D on a closed manifold gives rise to a Fredholm

### Fractional Index Theory

Fractional Index Theory Index a ( + ) = Z Â(Z ) Q Workshop on Geometry and Lie Groups The University of Hong Kong Institute of Mathematical Research 26 March 2011 Mathai Varghese School of Mathematical

### Stable complex and Spin c -structures

APPENDIX D Stable complex and Spin c -structures In this book, G-manifolds are often equipped with a stable complex structure or a Spin c structure. Specifically, we use these structures to define quantization.

### An 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

### Bott Periodicity. Anthony Bosman Senior Honors Thesis Department of Mathematics, Stanford University Adviser: Eleny Ionel

Bott Periodicity Anthony Bosman Senior Honors Thesis Department of Mathematics, Stanford University Adviser: Eleny Ionel Acknowledgements This paper is being written as a Senior honors thesis. I m indebted

### Hyperkähler geometry lecture 3

Hyperkähler geometry lecture 3 Misha Verbitsky Cohomology in Mathematics and Physics Euler Institute, September 25, 2013, St. Petersburg 1 Broom Bridge Here as he walked by on the 16th of October 1843

### KO -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

### Spectral Theorem for Self-adjoint Linear Operators

Notes for the undergraduate lecture by David Adams. (These are the notes I would write if I was teaching a course on this topic. I have included more material than I will cover in the 45 minute lecture;

### THE GAUSS-BONNET THEOREM FOR VECTOR BUNDLES

THE GAUSS-BONNET THEOREM FOR VECTOR BUNDLES Denis Bell 1 Department of Mathematics, University of North Florida 4567 St. Johns Bluff Road South,Jacksonville, FL 32224, U. S. A. email: dbell@unf.edu This

### Invariants from noncommutative index theory for homotopy equivalences

Invariants from noncommutative index theory for homotopy equivalences Charlotte Wahl ECOAS 2010 Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS 2010 1 / 12 Basics in noncommutative

### Elliptic Regularity. Throughout we assume all vector bundles are smooth bundles with metrics over a Riemannian manifold X n.

Elliptic Regularity Throughout we assume all vector bundles are smooth bundles with metrics over a Riemannian manifold X n. 1 Review of Hodge Theory In this note I outline the proof of the following Fundamental

### Lecture 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

### Donaldson and Seiberg-Witten theory and their relation to N = 2 SYM

Donaldson and Seiberg-Witten theory and their relation to N = SYM Brian Williams April 3, 013 We ve began to see what it means to twist a supersymmetric field theory. I will review Donaldson theory and

### HODGE 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

### The 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

### Symplectic critical surfaces in Kähler surfaces

Symplectic critical surfaces in Kähler surfaces Jiayu Li ( Joint work with X. Han) ICTP-UNESCO and AMSS-CAS November, 2008 Symplectic surfaces Let M be a compact Kähler surface, let ω be the Kähler form.

### REAL INSTANTONS, DIRAC OPERATORS AND QUATERNIONIC CLASSIFYING SPACES PAUL NORBURY AND MARC SANDERS Abstract. Let M(k; SO(n)) be the moduli space of ba

REAL INSTANTONS, DIRAC OPERATORS AND QUATERNIONIC CLASSIFYING SPACES PAUL NORBURY AND MARC SANDERS Abstract. Let M(k; SO(n)) be the moduli space of based gauge equivalence classes of SO(n) instantons on

### A PRIMER ON SESQUILINEAR FORMS

A PRIMER ON SESQUILINEAR FORMS BRIAN OSSERMAN This is an alternative presentation of most of the material from 8., 8.2, 8.3, 8.4, 8.5 and 8.8 of Artin s book. Any terminology (such as sesquilinear form

### Index theory on manifolds with corners: Generalized Gauss-Bonnet formulas

Index theory on singular manifolds I p. 1/4 Index theory on singular manifolds I Index theory on manifolds with corners: Generalized Gauss-Bonnet formulas Paul Loya Index theory on singular manifolds I

### Representation Theory

Representation Theory Representations Let G be a group and V a vector space over a field k. A representation of G on V is a group homomorphism ρ : G Aut(V ). The degree (or dimension) of ρ is just dim

### LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori

LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.

### RIEMANN SURFACES: TALK V: DOLBEAULT COHOMOLOGY

RIEMANN SURFACES: TALK V: DOLBEAULT COHOMOLOGY NICK MCCLEEREY 0. Complex Differential Forms Consider a complex manifold X n (of complex dimension n) 1, and consider its complexified tangent bundle T C

### Lecture on Equivariant Cohomology

Lecture on Equivariant Cohomology Sébastien Racanière February 20, 2004 I wrote these notes for a hours lecture at Imperial College during January and February. Of course, I tried to track down and remove

### Characteristic 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

### ERRATA FOR INTRODUCTION TO SYMPLECTIC TOPOLOGY

ERRATA FOR INTRODUCTION TO SYMPLECTIC TOPOLOGY DUSA MCDUFF AND DIETMAR A. SALAMON Abstract. These notes correct a few typos and errors in Introduction to Symplectic Topology (2nd edition, OUP 1998, reprinted

### GEOMETRIC QUANTIZATION

GEOMETRIC QUANTIZATION 1. The basic idea The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical

### Cobordant 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

### SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction

SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS CRAIG JACKSON 1. Introduction Generally speaking, geometric quantization is a scheme for associating Hilbert spaces

### Topological 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

### LECTURE 5: COMPLEX AND KÄHLER MANIFOLDS

LECTURE 5: COMPLEX AND KÄHLER MANIFOLDS Contents 1. Almost complex manifolds 1. Complex manifolds 5 3. Kähler manifolds 9 4. Dolbeault cohomology 11 1. Almost complex manifolds Almost complex structures.

### RIMS-1897 On multiframings of 3-manifolds By Tatsuro SHIMIZU December 2018 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan

RIMS-1897 On multiframings of 3-manifolds By Tatsuro SHIMIZU December 2018 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan On multiframings of 3 manifolds Tatsuro Shimizu 1

### LECTURE 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

### THE HODGE DECOMPOSITION

THE HODGE DECOMPOSITION KELLER VANDEBOGERT 1. The Musical Isomorphisms and induced metrics Given a smooth Riemannian manifold (X, g), T X will denote the tangent bundle; T X the cotangent bundle. The additional

### The Riemann-Roch Theorem

The Riemann-Roch Theorem In this lecture F/K is an algebraic function field of genus g. Definition For A D F, is called the index of specialty of A. i(a) = dim A deg A + g 1 Definition An adele of F/K

### Operator algebras and topology

Operator algebras and topology Thomas Schick 1 Last compiled November 29, 2001; last edited November 29, 2001 or later 1 e-mail: schick@uni-math.gwdg.de www: http://uni-math.gwdg.de/schick Fax: ++49-251/83

### Chern 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

### Notes by Maksim Maydanskiy.

SPECTRAL FLOW IN MORSE THEORY. 1 Introduction Notes by Maksim Maydanskiy. Spectral flow is a general formula or computing the Fredholm index of an operator d ds +A(s) : L1,2 (R, H) L 2 (R, H) for a family

### HYPERKÄHLER MANIFOLDS

HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly

### LECTURE 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

### HASSE-MINKOWSKI THEOREM

HASSE-MINKOWSKI THEOREM KIM, SUNGJIN 1. Introduction In rough terms, a local-global principle is a statement that asserts that a certain property is true globally if and only if it is true everywhere locally.

### LIMITING CASES OF BOARDMAN S FIVE HALVES THEOREM

Proceedings of the Edinburgh Mathematical Society Submitted Paper Paper 14 June 2011 LIMITING CASES OF BOARDMAN S FIVE HALVES THEOREM MICHAEL C. CRABB AND PEDRO L. Q. PERGHER Institute of Mathematics,

### The 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

### Characteristic 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

### THE STRONG NOVIKOV CONJECTURE FOR LOW DEGREE COHOMOLOGY

THE STRONG NOVIKOV CONJECTURE FOR LOW DEGREE COHOMOLOGY BERNHARD HANKE AND THOMAS SCHICK ABSTRACT. We show that for each discrete group Γ, the rational assembly map K (BΓ Q K (C maxγ Q is injective on

### L6: 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

### Cohomology of the Mumford Quotient

Cohomology of the Mumford Quotient Maxim Braverman Abstract. Let X be a smooth projective variety acted on by a reductive group G. Let L be a positive G-equivariant line bundle over X. We use a Witten

### Dirac operators with torsion

Dirac operators with torsion Prof.Dr. habil. Ilka Agricola Philipps-Universität Marburg Golden Sands / Bulgaria, September 2011 1 Relations between different objects on a Riemannian manifold (M n, g):