DOCSLIB.ORG

Manifolds and Varieties Via Sheaves

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Manifolds and Varieties Via Sheaves Chapter 1 Manifolds and Varieties via Sheaves In rough terms, a manifold is a topological space along with a distinguished collection of functions, which looks locally like Euclidean space. Although it is rarely presented this way in introductory texts (e. g. [Spv, Wa]), sheaf theory is a natural language in which to make such a notion precise. An algebraic variety can be de¯ned similarly as a space which looks locally like the zero set of a collection of polynomials. The sheaf theoretic approach to varieties was introduced by Serre in the early 1950's, this approach was solidi¯ed with the work of Grothendieck shortly thereafter, and algebraic geometry has never been the same since. 1.1 Sheaves of functions In many parts of mathematics, we encounter spaces with distinguished classes of functions on them. When these classes are closed under restriction, as they often are, then they give rise to presheaves. More precisely, let X be a topological space, and T a set. For each open set U X, let MapT (U) be the set of maps from U to T . De¯nition 1.1.1. A collection of subsets P (U) ½ MapT (U), with U ½ X nonempty open, is called a presheaf of ( T -valued) functions on X, if it is closed under restriction, i. e. if f 2 P (U) and V ½ U then fjV 2 P (V ). If the de¯ning conditions for P (U) are local, which means that they can be checked in a neighbourhood of a point, then the presheaf is called sheaf. Or to put it another way: De¯nition 1.1.2. A presheaf of functions P is called a sheaf if f 2 P (U) whenever there is an open cover fUig of U such that fjUi 2 P (Ui). 7 Example 1.1.3. Let PT (U) be the set of constant functions from U to T . This is a presheaf but not a sheaf in general. Example 1.1.4. A function is locally constant if it is constant in a neigh- bourhood of a point. The set of locally constant functions, denoted by T (U) or TX (U), is a sheaf. It is called the constant sheaf. Example 1.1.5. Let T be another topological space, then the set of continuous functions ContX;T (U) from U X to T is a sheaf. When T is discrete, this coincides with the previous example. Example 1.1.6. Let X = Rn, the sets C1(U) of C1 real valued functions form a sheaf. Example 1.1.7. Let X = C (or Cn), the sets O(U) of holomorphic functions on U form a sheaf. Example 1.1.8. Let L be a linear di®erential operator on Rn with C1 coef- 2 2 1 ¯cients (e. g. @ =@xi ). Let S(U) denote the space of C solutions in U. This is a sheaf. P Example 1.1.9. Let X = Rn, the sets L1(U) of L1-functions forms a presheaf which is not a sheaf. We can always force a presheaf to be a sheaf by the following construction. Example 1.1.10. Given a presheaf P of functions to T . De¯ne the s P (U) = ff : U ! T j 8x 2 U; 9 a neighbourhood Ux of x, such that fjUx 2 P (Ux)g This is a sheaf called the shea¯¯cation of P . When P is a presheaf of constant functions, P s is exactly the sheaf of locally constant functions. When this construction is applied to the presheaf L1, we obtain the sheaf of locally L1 functions. Exercise 1.1.11. 1. Check that P s is a sheaf. 2. Let ¼ : B ! X be a surjective continuous map of topological spaces. Prove that the presheaf of sections B(U) = f : U ! B j continuous, 8x 2 U; ¼ ± (x) = xg is a sheaf. 3. Let F : X ! Y be surjective continuous map. Suppose that P is a sheaf of T -valued functions on X. De¯ne f 2 Q(U) ½ MapT (U) if and only if ¤ ¡1 its pullback F f = f ± F jf ¡1U 2 P (F (U)). Show that Q is a sheaf on Y . 8 4. Let Y ½ X be a closed subset of a topological space. Let P be a sheaf of T -valued functions on X. For each open U ½ Y , let PY (U) be the set of functions f : U ! T locally extendible to an element of P , i.e. f 2 PY (U) if and only there for each y 2 U, there exists a neighbourhood V ½ X and an element of P (V ) restricting to fjV \U . Show that PY is a sheaf. 1.2 Manifolds Let k be a ¯eld. De¯nition 1.2.1. Let R be a sheaf of k-valued functions on X. We say that R is a sheaf of algebras if each R(U) Mapk(U) is a subalgebra. We call the pair (X; R) a concrete ringed space over k, or simply a k-space. n n 1 n (R ; CR), (R ; C ) and (C ; O) are examples of R and C-spaces. De¯nition 1.2.2. A morphism of k-spaces (X; R) ! (Y; S) is a continuous map F : X ! Y such that f 2 S(U) implies F ¤f 2 R(F ¡1U): This is good place to introduce, or perhaps remind the reader of, the notion of a category. A category C consists of a set (or class) of objects ObjC and for each pair A; B 2 C, a set HomC(A; B) of morphisms from A to B. There is a composition law ± : HomC(B; C) £ HomC(A; B) ! HomC(A; C); and distinguished elements idA 2 HomC(A; A) which satisfy 1. associativity: f ± (g ± h) = (f ± g) ± h, 2. identity: f ± idA = f and idA ± g = g, whenever these are de¯ned. Categories abound in mathematics. A basic example is the category of Sets consisting of the class of all sets, HomSets(A; B) is just the set of maps from A to B, and composition and idA have the usual meanings. Similarly, we can form the category of groups and group homomorphisms, the category of rings and rings homomorphisms, and the category of topological spaces and continuous maps. We have essentially constructed another example. We can take the objects to be k-spaces, and morphisms as above. These can be seen to constitute a category, once we observe that the identity is a morphism and the composition of morphisms is a morphism. The notion of an isomorphism makes sense in any category, we will spell in the above example. De¯nition 1.2.3. An isomorphism of k-spaces (X; R) =» (Y; S) is a homeo- morphism F : X ! Y such that f 2 S(U) if and only if F ¤f 2 R(F ¡1U). Given a sheaf S on X and open set U ½ X, let SjU denote the sheaf on U de¯ned by V 7! S(V ) for each V U. 9 1 1 De¯nition 1.2.4. An n-dimensional C manifold is an R-space (X; CX ) such that 1. The topology of X is given by a metric1. 1 2. X admits an open covering fUig such that each (Ui; CX jUi ) is isomorphic 1 n to (Bi; C jBi ) for some open ball B ½ R . 1 » 1 The isomorphisms (Ui; C jUi ) = (Bi; C jBi ) correspond to coordinate charts in more conventional treatments. The whole collection of data is called an atlas. There a number of variations on this idea: De¯nition 1.2.5. 1. An n-dimensional topological manifold is de¯ned as n 1 n above but with (R ; C ) replaced by (R ; ContRn;R). 2. An n-dimensional complex manifold can be de¯ned by replacing (Rn; C1) by (Cn; O). One dimensional complex manifolds are usually called Riemann surfaces. De¯nition 1.2.6. A C1 map from one C1 manifold to another is just a morphism of R-spaces. A holomorphic map between complex manifolds is de¯ned as a morphism of C-spaces. C1 (respectively complex) manifolds and maps form a category; an isomor- phism in this category is called a di®eomorphism (respectively biholomorphism). By de¯nition any point of manifold has neighbourhood di®eomorphic or biholo- morphic to a ball. Given a complex manifold (X; OX ), we say that f : X ! R is C1 if and only if f ± g is C1 for each holomorphic map g : B ! X from a ball in Cn. We state for the record: Lemma 1.2.7. An n-dimensional complex manifold together with its sheaf of C1 functions is a 2n-dimensional C1 manifold. Let us consider some examples of manifolds. Certainly any open subset of Rn (Cn) is a (complex) manifold in an obvious fashion. To get less trivial examples, we need one more de¯nition. De¯nition 1.2.8. Given an n-dimensional manifold X, a closed subset Y ½ X is called a closed m-dimensional closed submanifold if for any point x 2 Y , there exists a neighbourhood U of x in X and a di®eomorphism of to a ball B ½ Rn such that Y \ U maps to the intersection of B with an m-dimensional linear space. 1 Given a closed submanifold Y ½ X, de¯ne CY to be the sheaf of functions which are locally extendible to C1 functions on X. For a complex submanifold Y ½ X, we de¯ne OY to be the sheaf of functions which locally extend to holomorphic functions.
Load more

Recommended publications
  • EXERCISES 1 Exercise 1.1. Let X = (|X|,O X) Be a Scheme and I Is A
    EXERCISES 1 Exercise 1.1. Let X = (|X|, OX ) be a scheme and I is a quasi-coherent OX -module. Show that the ringed space X[I] := (|X|, OX [I]) is also a scheme. Exercise 1.2. Let C be a category and h : C → Func(C◦, Set) the Yoneda embedding. Show ∼ that for any arrows X → Y and Z → Y in C, there is a natural isomorphism hX×Z Y → hX ×hY hZ . Exercise 1.3. Let A → R be a ring homomorphism. Verify that under the identification 2 of DerA(R, ΩR/A) with the sections of the diagonal map R ⊗A R/J → R given in lecture, 1 the universal derivation d : R → ΩR/A corresponds to the section given by sending x ∈ R to 1 ⊗ x. Exercise 1.4. Let AffZ be the category of affine schemes. The Yoneda functor h gives rise to a related functor hAff : Sch → Func(Aff◦ , Set) Z Z which only considers the functor of points for affine schemes. Is this functor still fully faithful? Cultural note: since Aff◦ is equivalent to the category of rings ( -algebras!), we can also Z Z view h as defining a (covariant) functor on the category of rings. When X is an affine scheme Aff of the form Z[{xi}]/({fj}], the value of hX on A is just the set of solutions of the fj with coordinates in A. Exercise 1.5. Let R be a ring and H : ModR → Set a functor which commutes with finite products. Verify the claim in lecture that H(I) has an R-module structure.
    [Show full text]
  • Bézout's Theorem
    Bézout's theorem Toni Annala Contents 1 Introduction 2 2 Bézout's theorem 4 2.1 Ane plane curves . .4 2.2 Projective plane curves . .6 2.3 An elementary proof for the upper bound . 10 2.4 Intersection numbers . 12 2.5 Proof of Bézout's theorem . 16 3 Intersections in Pn 20 3.1 The Hilbert series and the Hilbert polynomial . 20 3.2 A brief overview of the modern theory . 24 3.3 Intersections of hypersurfaces in Pn ...................... 26 3.4 Intersections of subvarieties in Pn ....................... 32 3.5 Serre's Tor-formula . 36 4 Appendix 42 4.1 Homogeneous Noether normalization . 42 4.2 Primes associated to a graded module . 43 1 Chapter 1 Introduction The topic of this thesis is Bézout's theorem. The classical version considers the number of points in the intersection of two algebraic plane curves. It states that if we have two algebraic plane curves, dened over an algebraically closed eld and cut out by multi- variate polynomials of degrees n and m, then the number of points where these curves intersect is exactly nm if we count multiple intersections and intersections at innity. In Chapter 2 we dene the necessary terminology to state this theorem rigorously, give an elementary proof for the upper bound using resultants, and later prove the full Bézout's theorem. A very modest background should suce for understanding this chapter, for example the course Algebra II given at the University of Helsinki should cover most, if not all, of the prerequisites. In Chapter 3, we generalize Bézout's theorem to higher dimensional projective spaces.
    [Show full text]
  • Arxiv:0704.2891V3 [Math.AG] 5 Dec 2007 Schwartz Functions on Nash Manifolds
    Schwartz functions on Nash manifolds Avraham Aizenbud and Dmitry Gourevitch ∗ July 11, 2011 Abstract In this paper we extend the notions of Schwartz functions, tempered func- tions and generalized Schwartz functions to Nash (i.e. smooth semi-algebraic) manifolds. We reprove for this case classically known properties of Schwartz functions on Rn and build some additional tools which are important in rep- resentation theory. Contents 1 Introduction 2 1.1 Mainresults................................ 3 1.2 Schwartz sections of Nash bundles . 4 1.3 Restricted topologyand sheaf properties . .... 4 1.4 Possibleapplications ........................... 5 1.5 Summary ................................. 6 1.6 Remarks.................................. 6 2 Semi-algebraic geometry 8 2.1 Basicnotions ............................... 8 arXiv:0704.2891v3 [math.AG] 5 Dec 2007 2.2 Tarski-Seidenberg principle of quantifier elimination anditsapplications............................ 8 2.3 Additional preliminary results . .. 10 ∗Avraham Aizenbud and Dmitry Gourevitch, Faculty of Mathematics and Computer Science, The Weizmann Institute of Science POB 26, Rehovot 76100, ISRAEL. E-mails: [email protected], [email protected]. Keywords: Schwartz functions, tempered functions, generalized functions, distributions, Nash man- ifolds. 1 3 Nash manifolds 11 3.1 Nash submanifolds of Rn ......................... 11 3.2 Restricted topological spaces and sheaf theory over them........ 12 3.3 AbstractNashmanifolds . 14 3.3.1 ExamplesandRemarks. 14 3.4 Nashvectorbundles ........................... 15 3.5 Nashdifferentialoperators . 16 3.5.1 Algebraic differential operators on a Nash manifold . .... 17 3.6 Nashtubularneighborhood . 18 4 Schwartz and tempered functions on affine Nash manifolds 19 4.1 Schwartzfunctions ............................ 19 4.2 Temperedfunctions. .. .. .. 20 4.3 Extension by zero of Schwartz functions . .. 20 4.4 Partitionofunity ............................. 21 4.5 Restriction and sheaf property of tempered functions .
    [Show full text]
  • ALGEBRAIC GEOMETRY (PART III) EXAMPLE SHEET 3 (1) Let X Be a Scheme and Z a Closed Subset of X. Show That There Is a Closed Imme
    ALGEBRAIC GEOMETRY (PART III) EXAMPLE SHEET 3 CAUCHER BIRKAR (1) Let X be a scheme and Z a closed subset of X. Show that there is a closed immersion f : Y ! X which is a homeomorphism of Y onto Z. (2) Show that a scheme X is affine iff there are b1; : : : ; bn 2 OX (X) such that each D(bi) is affine and the ideal generated by all the bi is OX (X) (see [Hartshorne, II, exercise 2.17]). (3) Let X be a topological space and let OX to be the constant sheaf associated to Z. Show that (X; OX ) is a ringed space and that every sheaf on X is in a natural way an OX -module. (4) Let (X; OX ) be a ringed space. Show that the kernel and image of a morphism of OX -modules are also OX -modules. Show that the direct sum, direct product, direct limit and inverse limit of OX -modules are OX -modules. If F and G are O H F G O L X -modules, show that omOX ( ; ) is also an X -module. Finally, if is a O F F O subsheaf of an X -module , show that the quotient sheaf L is also an X - module. O F O O F ' (5) Let (X; X ) be a ringed space and an X -module. Prove that HomOX ( X ; ) F(X). (6) Let f :(X; OX ) ! (Y; OY ) be a morphism of ringed spaces. Show that for any O F O G ∗G F ' G F X -module and Y -module we have HomOX (f ; ) HomOY ( ; f∗ ).
    [Show full text]
  • SHEAVES of MODULES 01AC Contents 1. Introduction 1 2
    SHEAVES OF MODULES 01AC Contents 1. Introduction 1 2. Pathology 2 3. The abelian category of sheaves of modules 2 4. Sections of sheaves of modules 4 5. Supports of modules and sections 6 6. Closed immersions and abelian sheaves 6 7. A canonical exact sequence 7 8. Modules locally generated by sections 8 9. Modules of finite type 9 10. Quasi-coherent modules 10 11. Modules of finite presentation 13 12. Coherent modules 15 13. Closed immersions of ringed spaces 18 14. Locally free sheaves 20 15. Bilinear maps 21 16. Tensor product 22 17. Flat modules 24 18. Duals 26 19. Constructible sheaves of sets 27 20. Flat morphisms of ringed spaces 29 21. Symmetric and exterior powers 29 22. Internal Hom 31 23. Koszul complexes 33 24. Invertible modules 33 25. Rank and determinant 36 26. Localizing sheaves of rings 38 27. Modules of differentials 39 28. Finite order differential operators 43 29. The de Rham complex 46 30. The naive cotangent complex 47 31. Other chapters 50 References 52 1. Introduction 01AD This is a chapter of the Stacks Project, version 77243390, compiled on Sep 28, 2021. 1 SHEAVES OF MODULES 2 In this chapter we work out basic notions of sheaves of modules. This in particular includes the case of abelian sheaves, since these may be viewed as sheaves of Z- modules. Basic references are [Ser55], [DG67] and [AGV71]. We work out what happens for sheaves of modules on ringed topoi in another chap- ter (see Modules on Sites, Section 1), although there we will mostly just duplicate the discussion from this chapter.
    [Show full text]
  • Arxiv:1307.5568V2 [Math.AG]
    PARTIAL POSITIVITY: GEOMETRY AND COHOMOLOGY OF q-AMPLE LINE BUNDLES DANIEL GREB AND ALEX KURONYA¨ To Rob Lazarsfeld on the occasion of his 60th birthday Abstract. We give an overview of partial positivity conditions for line bundles, mostly from a cohomological point of view. Although the current work is to a large extent of expository nature, we present some minor improvements over the existing literature and a new result: a Kodaira-type vanishing theorem for effective q-ample Du Bois divisors and log canonical pairs. Contents 1. Introduction 1 2. Overview of the theory of q-ample line bundles 4 2.1. Vanishing of cohomology groups and partial ampleness 4 2.2. Basic properties of q-ampleness 7 2.3. Sommese’s geometric q-ampleness 15 2.4. Ample subschemes, and a Lefschetz hyperplane theorem for q-ample divisors 17 3. q-Kodaira vanishing for Du Bois divisors and log canonical pairs 19 References 23 1. Introduction Ampleness is one of the central notions of algebraic geometry, possessing the extremely useful feature that it has geometric, numerical, and cohomological characterizations. Here we will concentrate on its cohomological side. The fundamental result in this direction is the theorem of Cartan–Serre–Grothendieck (see [Laz04, Theorem 1.2.6]): for a complete arXiv:1307.5568v2 [math.AG] 23 Jan 2014 projective scheme X, and a line bundle L on X, the following are equivalent to L being ample: ⊗m (1) There exists a positive integer m0 = m0(X, L) such that L is very ample for all m ≥ m0. (2) For every coherent sheaf F on X, there exists a positive integer m1 = m1(X, F, L) ⊗m for which F ⊗ L is globally generated for all m ≥ m1.
    [Show full text]
  • 4. Coherent Sheaves Definition 4.1. If (X,O X) Is a Locally Ringed Space
    4. Coherent Sheaves Definition 4.1. If (X; OX ) is a locally ringed space, then we say that an OX -module F is locally free if there is an open affine cover fUig of X such that FjUi is isomorphic to a direct sum of copies of OUi . If the number of copies r is finite and constant, then F is called locally free of rank r (aka a vector bundle). If F is locally free of rank one then we way say that F is invertible (aka a line bundle). The group of all invertible sheaves under tensor product, denoted Pic(X), is called the Picard group of X. A sheaf of ideals I is any OX -submodule of OX . Definition 4.2. Let X = Spec A be an affine scheme and let M be an A-module. M~ is the sheaf which assigns to every open subset U ⊂ X, the set of functions a s: U −! Mp; p2U which can be locally represented at p as a=g, a 2 M, g 2 R, p 2= Ug ⊂ U. Lemma 4.3. Let A be a ring and let M be an A-module. Let X = Spec A. ~ (1) M is a OX -module. ~ (2) If p 2 X then Mp is isomorphic to Mp. ~ (3) If f 2 A then M(Uf ) is isomorphic to Mf . Proof. (1) is clear and the rest is proved mutatis mutandis as for the structure sheaf. Definition 4.4. An OX -module F on a scheme X is called quasi- coherent if there is an open cover fUi = Spec Aig by affines and ~ isomorphisms FjUi ' Mi, where Mi is an Ai-module.
    [Show full text]
  • The Historical Development of Algebraic Geometry∗
    The Historical Development of Algebraic Geometry∗ Jean Dieudonn´e March 3, 1972y 1 The lecture This is an enriched transcription of footage posted by the University of Wis- consin{Milwaukee Department of Mathematical Sciences [1]. The images are composites of screenshots from the footage manipulated with Python and the Python OpenCV library [2]. These materials were prepared by Ryan C. Schwiebert with the goal of capturing and restoring some of the material. The lecture presents the content of Dieudonn´e'sarticle of the same title [3] as well as the content of earlier lecture notes [4], but the live lecture is natu- rally embellished with different details. The text presented here is, for the most part, written as it was spoken. This is why there are some ungrammatical con- structions of spoken word, and some linguistic quirks of a French speaker. The transcriber has taken some liberties by abbreviating places where Dieudonn´e self-corrected. That is, short phrases which turned out to be false starts have been elided, and now only the subsequent word-choices that replaced them ap- pear. Unfortunately, time has taken its toll on the footage, and it appears that a brief portion at the beginning, as well as the last enumerated period (VII. Sheaves and schemes), have been lost. (Fortunately, we can still read about them in [3]!) The audio and video quality has contributed to several transcription errors. Finally, a lack of thorough knowledge of the history of algebraic geometry has also contributed some errors. Contact with suggestions for improvement of arXiv:1803.00163v1 [math.HO] 1 Mar 2018 the materials is welcome.
    [Show full text]
  • University of Alberta
    University of Alberta Extensions of Skorohod’s almost sure representation theorem by Nancy Hernandez Ceron A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Master of Science in Mathematics Department of Mathematical and Statistical Sciences ©Nancy Hernandez Ceron Fall 2010 Edmonton, Alberta Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission. Library and Archives Bibliothèque et Canada Archives Canada Published Heritage Direction du Branch Patrimoine de l’édition 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre référence ISBN:978-0-494-62500-2 Our file Notre référence ISBN: 978-0-494-62500-2 NOTICE: AVIS: The author has granted a non- L’auteur a accordé une licence non exclusive exclusive license allowing Library and permettant à la Bibliothèque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par télécommunication ou par l’Internet, prêter, telecommunication or on the Internet, distribuer et vendre des thèses partout dans le loan, distribute and sell theses monde, à des fins commerciales ou autres, sur worldwide, for commercial or non- support microforme, papier, électronique et/ou commercial purposes, in microform, autres formats.
    [Show full text]
  • Compact Ringed Spaces
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 52, 41 l-436 (1978) Compact Ringed Spaces CHRISTOPHER J. MULVEY * Department of Mathematics, Columbia University, New York, New York 10027 and Mathematics Division, University of Sussex, Falmer, Brighton, BNl9QH, England Communicated by Saunders MacLane Received April 16, 1976 The properties of complete regularity, paracompactness, and compactness of a topological space X all may be described in terms of the ring R(X) of continuous real functions on X. In this paper we are concerned with extending these concepts from topological to ringed spaces, replacing the ring of continuous real functions on the space by the ring of sections of the ringed space. Our interest in the problem arose from attempting to construct the tangent module 7M of a smooth manifold IM directly from the sheaf QM of smooth real functions. For each open set U of M, the module rM( U)is the s2,( U)-module of derivations of In,(U). Yet, since taking derivations is not functorial, the evident restriction maps are not canonically forthcoming. For paracompact manifolds the construction needed was known to Boardman [5]. In general, it can be carried out using the complete regularity of the ringed space (M, 62,) [18, 321. Although spaces satisfying these conditions appear naturally in analytical contexts, there emerges a fundamental connection with sectional representations of rings [19, 21, 231. In turn, this has provided a technique for using intuitionistic mathematics in applying these representations [22]. Extending these properties to ringed spaces, we prove a compactness theorem for completely regular ringed spaces generalizing the Gelfand-Kolmogoroff criterion concerning maximal ideals in the ring R(X) of continuous real functions on a completely regular space X.
    [Show full text]
  • A Radial Basis Function Partition of Unity Method for Transport on the Sphere
    A RADIAL BASIS FUNCTION PARTITION OF UNITY METHOD FOR TRANSPORT ON THE SPHERE by Kevin Aiton A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics Boise State University May 2014 c 2014 Kevin Aiton ALL RIGHTS RESERVED BOISE STATE UNIVERSITY GRADUATE COLLEGE DEFENSE COMMITTEE AND FINAL READING APPROVALS of the thesis submitted by Kevin Aiton Thesis Title: A Radial Basis Function Partition of Unity Method for Transport on the Sphere Date of Final Oral Examination: 06 December 2013 The following individuals read and discussed the thesis submitted by student Kevin Aiton, and they evaluated his presentation and response to questions during the final oral examination. They found that the student passed the final oral examination. Grady Wright, Ph.D. Chair, Supervisory Committee Donna Calhoun, Ph.D. Member, Supervisory Committee Inanc Senocak, Ph.D. Member, Supervisory Committee The final reading approval of the thesis was granted by Grady Wright, Ph.D., Chair of the Supervisory Committee. The thesis was approved for the Graduate College by John R. Pelton, Ph.D., Dean of the Graduate College. ACKNOWLEDGMENTS This work was supported, in part, by the National Science Foundation (NSF) under grant DMS-0934581 and also by the Boise State Mathematics Department under the 2013 summer graduate fellowship. I would like to express gratitude to Professor Grady Wright. He has been both kind and patient. I would also like to thank the MOOSE development team at Idaho National Laboratory for letting me work remotely so I could work on my thesis. Finally, I would like to give gratitude to my family.
    [Show full text]
  • INTERSECTION NUMBERS, TRANSFERS, and GROUP ACTIONS by Daniel H. Gottlieb and Murad Ozaydin Purdue University University of Oklah
    INTERSECTION NUMBERS, TRANSFERS, AND GROUP ACTIONS by Daniel H. Gottlieb and Murad Ozaydin Purdue University University of Oklahoma Abstract We introduce a new definition of intersection number by means of umkehr maps. This definition agrees with an obvious definition up to sign. It allows us to extend the concept parametrically to fibre bundles. For fibre bundles we can then define intersection number transfers. Fibre bundles arise naturally in equivariant situations, so the trace of the action divides intersection numbers. We apply this to equivariant projective varieties and other examples. Also we study actions of non-connected groups and their traces. AMS subject classification 1991: Primary: 55R12, 57S99, 55M25 Secondary: 14L30 Keywords: Poincare Duality, trace of action, Sylow subgroups, Euler class, projective variety. 1 1. Introduction When two manifolds with complementary dimensions inside a third manifold intersect transversally, their intersection number is defined. This classical topological invariant has played an important role in topology and its applications. In this paper we introduce a new definition of intersection number. The definition naturally extends to the case of parametrized manifolds over a base B; that is to fibre bundles over B . Then we can define transfers for the fibre bundles related to the intersection number. This is our main objective , and it is done in Theorem 6. The method of producing transfers depends upon the Key Lemma which we prove here. Special cases of this lemma played similar roles in the establishment of the Euler–Poincare transfer, the Lefschetz number transfer, and the Nakaoka transfer. Two other features of our definition are: 1) We replace the idea of submanifolds with the idea of maps from manifolds, and in effect we are defining the intersection numbers of the maps; 2) We are not restricted to only two maps, we can define the intersection number for several maps, or submanifolds.
    [Show full text]