Index Theory with Applications to Mathematics and Physics
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http://dx.doi.org/10.1090/gsm/039 Selected Titles in This Series 39 Larry C. Grove, Classical groups and geometric algebra, 2002 38 Elton P. Hsu, Stochastic analysis on manifolds, 2001 37 Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, 2001 36 Martin Schechter, Principles of functional analysis, second edition, 2001 35 James F. Davis and Paul Kirk, Lecture notes in algebraic topology, 2001 34 Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, 2001 33 Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001 32 Robert G. Bartle, A modern theory of integration, 2001 31 Ralf Korn and Elke Korn, Option pricing and portfolio optimization: Modern methods of financial mathematics, 2001 30 J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, 2001 29 Javier Duoandikoetxea, Fourier analysis, 2001 28 Liviu I. Nicolaescu, Notes on Seiberg-Witten theory, 2000 27 Thierry Aubin, A course in differential geometry, 2001 26 Rolf Berndt, An introduction to symplectic geometry, 2001 25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23 Alberto Candel and Lawrence Conlon, Foliations I, 2000 22 Giinter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000 21 John B. Conway, A course in operator theory, 2000 20 Robert E. Gompf and Andras I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and Martin Weese, Discovering modern set theory. II: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V. -
The Positivity of Local Equivariant Hirzebruch Class for Toric Varieties
THE POSITIVITY OF LOCAL EQUIVARIANT HIRZEBRUCH CLASS FOR TORIC VARIETIES KAMIL RYCHLEWICZ Abstract. The central object of investigation of this paper is the Hirzebruch class, a deformation of the Todd class, given by Hirzebruch (for smooth vari- eties) in his celebrated book [11]. The generalization for singular varieties is due to Brasselet-Sch¨urmann-Yokura. Following the work of Weber, we inves- tigate its equivariant version for (possibly singular) toric varieties. The local decomposition of the Hirzebruch class to the fixed points of the torus action and a formula for the local class in terms of the defining fan are mentioned. After this review part (sections 3–7), we prove the positivity of local Hirze- bruch classes for all toric varieties (Theorem 8.5), thus proving false the alleged counterexample given by Weber. 1. Acknowledgments This paper is an abbreviation of the author’s Master’s Thesis written at Faculty of Mathematics, Informatics and Mechanics of University of Warsaw under the su- pervision of Andrzej Weber. The author owes many thanks to the supervisor, for introducing him to the topic and stating the main problem, for pointing the im- portant parts of the general theory and for the priceless support regarding editorial matters. 2. Introduction The Todd class, which appears in the formulations of the celebrated Hirzebruch- Riemann-Roch theorem, and more general Grothendieck-Riemann-Roch theorem, is originally defined for smooth complete varieties (or schemes) only. In [2] Baum, Fulton and MacPherson gave a generalization for singular varieties, which in general is forced to lie in homology instead of cohomology. -
Notes on the Atiyah-Singer Index Theorem Liviu I. Nicolaescu
Notes on the Atiyah-Singer Index Theorem Liviu I. Nicolaescu Notes for a topics in topology course, University of Notre Dame, Spring 2004, Spring 2013. Last revision: November 15, 2013 i The Atiyah-Singer Index Theorem This is arguably one of the deepest and most beautiful results in modern geometry, and in my view is a must know for any geometer/topologist. It has to do with elliptic partial differential opera- tors on a compact manifold, namely those operators P with the property that dim ker P; dim coker P < 1. In general these integers are very difficult to compute without some very precise information about P . Remarkably, their difference, called the index of P , is a “soft” quantity in the sense that its determination can be carried out relying only on topological tools. You should compare this with the following elementary situation. m n Suppose we are given a linear operator A : C ! C . From this information alone we cannot compute the dimension of its kernel or of its cokernel. We can however compute their difference which, according to the rank-nullity theorem for n×m matrices must be dim ker A−dim coker A = m − n. Michael Atiyah and Isadore Singer have shown in the 1960s that the index of an elliptic operator is determined by certain cohomology classes on the background manifold. These cohomology classes are in turn topological invariants of the vector bundles on which the differential operator acts and the homotopy class of the principal symbol of the operator. Moreover, they proved that in order to understand the index problem for an arbitrary elliptic operator it suffices to understand the index problem for a very special class of first order elliptic operators, namely the Dirac type elliptic operators. -
Arxiv:2009.09697V1 [Math.AG] 21 Sep 2020 Btuto Hoy Uhcasswr Endfrcmlxan Complex for [ Defined in Mo Were Tian a Classes Given Such Fundamental Classes
VIRTUAL EQUIVARIANT GROTHENDIECK-RIEMANN-ROCH FORMULA CHARANYA RAVI AND BHAMIDI SREEDHAR Abstract. For a G-scheme X with a given equivariant perfect obstruction theory, we prove a virtual equivariant Grothendieck-Riemann-Roch formula, this is an extension of a result of Fantechi-G¨ottsche ([FG10]) to the equivariant context. We also prove a virtual non-abelian localization theorem for schemes over C with proper actions. Contents 1. Introduction 1 2. Preliminaries 5 2.1. Notations and conventions 5 2.2. Equivariant Chow groups 7 2.3. Equivariant Riemann-Roch theorem 7 2.4. Perfect obstruction theories and virtual classes 8 3. Virtual classes and the equivariant Riemann-Roch map 11 3.1. Induced perfect obstruction theory 11 3.2. Virtual equivariant Riemann-Roch formula 15 4. Virtual structure sheaf and the Atiyah-Segal map 18 4.1. Atiyah-Segal isomorphism 18 4.2. Morita equivalence 21 4.3. Virtual structure sheaf on the inertia scheme 23 References 26 1. Introduction As several moduli spaces that one encounters in algebraic geometry have a well- arXiv:2009.09697v1 [math.AG] 21 Sep 2020 defined expected dimension, one is interested in constructing a fundamental class of the expected dimension in its Chow group. Interesting numerical invariants like Gromov-Witten invariants and Donaldson-Thomas invariants are obtained by inte- grating certain cohomology classes over such a fundamental class. This motivated the construction of virtual fundamental classes. Given a moduli space with a perfect obstruction theory, such classes were defined for complex analytic spaces by Li and Tian in [LT98] and in the algebraic sense for Deligne-Mumford stacks by Behrend and Fantechi in [BF97]. -
An Introduction to Pseudo-Differential Operators
An introduction to pseudo-differential operators Jean-Marc Bouclet1 Universit´ede Toulouse 3 Institut de Math´ematiquesde Toulouse [email protected] 2 Contents 1 Background on analysis on manifolds 7 2 The Weyl law: statement of the problem 13 3 Pseudodifferential calculus 19 3.1 The Fourier transform . 19 3.2 Definition of pseudo-differential operators . 21 3.3 Symbolic calculus . 24 3.4 Proofs . 27 4 Some tools of spectral theory 41 4.1 Hilbert-Schmidt operators . 41 4.2 Trace class operators . 44 4.3 Functional calculus via the Helffer-Sj¨ostrandformula . 50 5 L2 bounds for pseudo-differential operators 55 5.1 L2 estimates . 55 5.2 Hilbert-Schmidt estimates . 60 5.3 Trace class estimates . 61 6 Elliptic parametrix and applications 65 n 6.1 Parametrix on R ................................ 65 6.2 Localization of the parametrix . 71 7 Proof of the Weyl law 75 7.1 The resolvent of the Laplacian on a compact manifold . 75 7.2 Diagonalization of ∆g .............................. 78 7.3 Proof of the Weyl law . 81 A Proof of the Peetre Theorem 85 3 4 CONTENTS Introduction The spirit of these notes is to use the famous Weyl law (on the asymptotic distribution of eigenvalues of the Laplace operator on a compact manifold) as a case study to introduce and illustrate one of the many applications of the pseudo-differential calculus. The material presented here corresponds to a 24 hours course taught in Toulouse in 2012 and 2013. We introduce all tools required to give a complete proof of the Weyl law, mainly the semiclassical pseudo-differential calculus, and then of course prove it! The price to pay is that we avoid presenting many classical concepts or results which are not necessary for our purpose (such as Borel summations, principal symbols, invariance by diffeomorphism or the G˚ardinginequality). -
Dual Pairs in the Pin-Group and Duality for the Corresponding Spinorial
DUAL PAIRS IN THE PIN-GROUP AND DUALITY FOR THE CORRESPONDING SPINORIAL REPRESENTATION CLÉMENT GUÉRIN, GANG LIU, AND ALLAN MERINO Abstract. In this paper, we give a complete picture of Howe correspondence for the setting (O(E, b), Pin(E, b), Π), where O(E, b) is an orthogonal group (real or complex), Pin(E, b) is the two-fold Pin-covering of O(E, b), and Π is the spinorial representation of Pin(E, b). More pre- cisely, for a dual pair (G, G′) in O(E, b), we determine explicitly the nature of its preimages (G, G′) in Pin(E, b), and prove that apart from some exceptions, (G, G′) is always a dual pair in e f e f Pin(E, b); then we establish the Howe correspondence for Π with respect to (G, G′). e f Contents 1. Introduction 1 2. Preliminaries 3 3. Dual pairs in the Pin-group 6 3.1. Pull-back of dual pairs 6 3.2. Identifyingtheisomorphismclassofpull-backs 8 4. Duality for the spinorial representation 14 References 21 1. Introduction The first duality phenomenon has been discovered by H. Weyl who pointed out a correspon- dence between some irreducible finite dimensional representations of the general linear group GL(V) and the symmetric group Sk where V is a finite dimensional vector space over C. In- d deed, considering the joint action of GL(V) and Sd on the space V⊗ , we get the following decomposition: arXiv:1907.09093v1 [math.RT] 22 Jul 2019 d λ V⊗ = M λ σ , [ ⊗ (Vλ,λ) GL(V) ∈ π [ where GL(V)π is the set of equivalence classes of irreducible finite dimensional representations d of GL(V) such that Hom (V , V⊗ ) , 0 and σ is an irreducible representation of S . -
The Riemann-Roch Theorem Is a Special Case of the Atiyah-Singer Index Formula
S.C. Raynor The Riemann-Roch theorem is a special case of the Atiyah-Singer index formula Master thesis defended on 5 March, 2010 Thesis supervisor: dr. M. L¨ubke Mathematisch Instituut, Universiteit Leiden Contents Introduction 5 Chapter 1. Review of Basic Material 9 1. Vector bundles 9 2. Sheaves 18 Chapter 2. The Analytic Index of an Elliptic Complex 27 1. Elliptic differential operators 27 2. Elliptic complexes 30 Chapter 3. The Riemann-Roch Theorem 35 1. Divisors 35 2. The Riemann-Roch Theorem and the analytic index of a divisor 40 3. The Euler characteristic and Hirzebruch-Riemann-Roch 42 Chapter 4. The Topological Index of a Divisor 45 1. De Rham Cohomology 45 2. The genus of a Riemann surface 46 3. The degree of a divisor 48 Chapter 5. Some aspects of algebraic topology and the T-characteristic 57 1. Chern classes 57 2. Multiplicative sequences and the Todd polynomials 62 3. The Todd class and the Chern Character 63 4. The T-characteristic 65 Chapter 6. The Topological Index of the Dolbeault operator 67 1. Elements of topological K-theory 67 2. The difference bundle associated to an elliptic operator 68 3. The Thom Isomorphism 71 4. The Todd genus is a special case of the topological index 76 Appendix: Elliptic complexes and the topological index 81 Bibliography 85 3 Introduction The Atiyah-Singer index formula equates a purely analytical property of an elliptic differential operator P (resp. elliptic complex E) on a compact manifold called the analytic index inda(P ) (resp. inda(E)) with a purely topological prop- erty, the topological index indt(P )(resp. -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
Sylow P -Subgroups of the Classical Groups Over Finite Fields with Characteristic Prime to P
SYLOW /--SUBGROUPS OF THE CLASSICAL GROUPS OVER FINITE FIELDS WITH CHARACTERISTIC PRIME TO p A. J. WEIR If Sn is a Sylow /--subgroup of the symmetric group of degree pn, then any group of order pn may be imbedded in 5„. We may express Sn as the complete product1 C o C o • • • o C of « cyclic groups of order p and the purpose of this paper is to show that any Sylow p- subgroup of a classical group (see §1) over the finite field GF(q) with q elements, where (q, p) = 1, is expressible as a direct product of basic subgroups 5„= C o C o • • • o C (n factors), where C is cyclic of order pT. (We assume always that p7£2.) Since C may be imbedded in ST, we see that Sn is imbedded in Sn+r-i in a particularly simple way. The above r is defined by the equation q*— \ =pT * where g* is the first power of q which is congruent to 1 mod p and * denotes some unspecified number prime to p. The case r = 1 is therefore of frequent occurrence, and then clearly Sn=Sn. Professor Philip Hall was my research supervisor in Cambridge (England) during the years 1949-1952 and it is a pleasure to acknowl- edge here my indebtedness to him for his generous encouragement. 1. We shall refer to the following groups as the Classical Groups:* I. The general linear group GL(n, q) is the group of all nonsingular (nXn) matrices with coefficients in GF(q). The order of GL(n, q) is qn(n-l)l2fq _ 1)(?2 _ J) . -
The Resolvent Parametrix of the General Elliptic Linear Differential Operator: a Closed Form for the Intrinsic Symbol
transactions of the american mathematical society Volume 310, Number 2, December 1988 THE RESOLVENT PARAMETRIX OF THE GENERAL ELLIPTIC LINEAR DIFFERENTIAL OPERATOR: A CLOSED FORM FOR THE INTRINSIC SYMBOL S. A. FULLING AND G. KENNEDY ABSTRACT. Nonrecursive, explicit expressions are obtained for the term of arbitrary order in the asymptotic expansion of the intrinsic symbol of a resol- vent parametrix of an elliptic linear differential operator, of arbitrary order and algebraic structure, which acts on sections of a vector bundle over a manifold. Results for the conventional symbol are included as a special case. 1. Introduction. As is well known, the resolvent operator, (A - A)-1, plays a central role in the functional analysis associated with an elliptic linear differential operator A. In particular, from it one can easily obtain the corresponding heat operator, e~tA, for t G R+ and semibounded A. Furthermore, detailed knowledge of the terms in the asymptotic expansions of the integral kernels of the resolvent and heat operators is of great value in calculating the asymptotics of eigenvalues and spectral functions [12, 25, 2, 3, 33]; partial solutions of inverse problems [26, 15]; indices of Fredholm operators [1, 19, 20]; and various physical quantities, including specific heats [4], partition functions [49], renormalized effective actions [38, 14, 34, 45, 8], and renormalized energy-momentum tensors [9, 43, 44]. (The references given here are merely representative.) It is important, therefore, to have available an efficient and general method for calculating such terms. The present paper addresses this topic for the resolvent operator, or, more precisely, a resolvent parametrix; later papers will treat the heat operator. -
Equivariant Grothendieck-Riemann-Roch, Theorem 6.2.13)
Equivariant Grothendieck-Riemann-Roch theorem via formal deformation theory Grigory Kondyrev, Artem Prikhodko Abstract We use the formalism of traces in higher categories to prove a common generalization of the holomorphic Atiyah-Bott fixed point formula and the Grothendieck-Riemann-Roch theorem. The proof is quite different from the original one proposed by Grothendieck et al.: it relies on the interplay between self dualities of quasi- and ind- coherent sheaves on X and formal deformation theory of Gaitsgory-Rozenblyum. In particular, we give a description of the Todd class in terms of the difference of two formal group structures on the derived loop scheme LX. The equivariant case is reduced to the non-equivariant one by a variant of the Atiyah-Bott localization theorem. Contents 0 Introduction 2 1 Categorical Chern character 7 1.1 Self-duality of quasi-coherent sheaves and Chern character . ....................... 8 1.2 Canonical LX-equivariantstructure ................................ .... 9 1.3 CategoricalCherncharacterasexponential . ................... 12 1.4 Comparison with the classical Chern character . ................. 13 2 Trace of pushforward functor via ind-coherent sheaves 17 2.1 ReminderonInd-coherentsheaves . ................ 18 2.2 Computingthetraceofpushforward . ................ 20 3 Orientations and traces 22 3.1 Serreorientation ................................... ............ 23 3.2 Canonicalorientation................................ ............. 24 3.3 Grouporientations .................................. ........... -
Representations and Invariants of the Classical Groups, by Roe Goodman and Nolan R
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 36, Number 4, Pages 533{538 S 0273-0979(99)00795-8 Article electronically published on July 28, 1999 Representations and invariants of the classical groups, by Roe Goodman and Nolan R. Wallach, Cambridge Univ. Press, Cambridge, 1998, xvi + 685 pp., $100.00, ISBN 0-521-58273-3, paperback, $39.95, ISBN 0-521-66348-2 This 685-page text by Goodman and Wallach constitutes a carefully organized exposition, not indeed of the entire finite-dimensional representation-theory over R or C of the classical groups (which would be hard to imagine in a single text three times the size of this one), but rather of a selection of some key topics therein. This selection of topics, especially as concerns invariant-theory, seems (as Goodman and Wallach indicate in the preface to their text) to have been in many ways inspired by the contents of H. Weyl’s The Classical Groups, Their Invariants and Representations [31]. Approximately half of the text under review is purely introductory; the remaining half then utilizes this introductory material to develop the main themes. The introductory material, covered in a total of approximately 280 pages in the first three chapters and in four appendices, introduces the reader to some necessary topics in elementary algebraic and differential geometry and to such basic concepts of representation theory as: linear algebraic groups and Lie groups, Lie algebras associated to such, Jordan decomposition, maximal tori, roots and weights, real forms of classical groups, the PBW theorem, invariant integration over compact Lie groups, etc.