DOCSLIB.ORG

Infinite Dimensional Lie Theory from the Point of View of Functional

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Infinite dimensional Lie Theory from the point of view of Functional Analysis Josef Teichmann F¨urmeine Eltern, Michael, Michaela und Hannah Zsofia Hazel. Abstract. Convenient analysis is enlarged by a powerful theory of Hille-Yosida type. More precisely asymptotic spectral properties of bounded operators on a convenient vector space are related to the existence of smooth semigroups in a necessary and sufficient way. An approximation theorem of Trotter-type is proved, too. This approximation theorem is in fact an existence theorem for smooth right evolutions of non-autonomous differential equations on convenient locally convex spaces and crucial for the following applications. To enlighten the generically "unsolved" (even though H. Omori et al. gave interesting and concise conditions for regularity) question of the existence of product integrals on convenient Lie groups, we provide by the given approximation formula some simple criteria. On the one hand linearization is used, on the other hand remarkable families of right invariant distance functions, which exist on all up to now known Lie groups, are the ingredients: Assuming some natural global conditions regularity can be proved on convenient Lie groups. The existence of product integrals is an essential basis for Lie theory in the convenient setting, since generically differential equations cannot be solved on non-normable locally convex spaces. The relationship between infinite dimensional Lie algebras and Lie groups, which is well under- stood in the regular case, is also reviewed from the point of view of local Lie groups: Namely the question under which conditions the existence of a local Lie group for a given convenient Lie algebra implies the existence of a global Lie group is treated by cohomological methods. It is shown that the considerations do not depend on the convergence of the Campbell-Baker-Hausdorff-Formula as in the original paper of W.T. van Est and Th.J. Korthagen. Contents Danksagung v Introduction vii Chapter 1. Convenient Analysis 1 1. Smooth Calculus 1 2. Analytic Calculus 9 3. Convenient Manifolds 11 Chapter 2. Convenient Hille-Yosida-Theory 17 1. Classical concepts and modern points of view 17 2. Smooth semigroups 25 3. ACP and Zoology of locally convex Spaces 30 4. The Trotter approximation theorem for product integrals 38 5. Analytic Semigroups 42 6. Refinements, Applications and Examples 44 Chapter 3. Product Integrals on infinite dimensional groups 51 1. Convenient Lie groups 51 2. Tempered Lie groups 55 3. Lipschitz-metrizable smooth groups 61 4. Product integration via Metrization 66 5. Product integration via Linearization 71 Chapter 4. The inverse of S. Lie's third theorem 73 1. Local Lie groups and the CBH-Formula 73 2. Local Lie groups without the CBH-Formula 79 3. Cohomological conditions for enlargibility 79 4. Enlargibility of Lie Algebras 90 Appendix A. Smooth bump functions on convenient vector spaces 95 Appendix B. Homological algebra and spectral sequences 97 Bibliography 99 iii iv CONTENTS Danksagung Ein Mensch setzt sich zur Aufgabe, die Welt abzuzeichnen. Im Laufe der Jahre bev¨olkerter einen Raum mit Bildern von Provinzen, K¨onigreichen, Gebirgen, Buchten, Schiffen, Inseln, Fischen, Behausungen, Werkzeugen, Gestirnen, Pferden und Personen. Kurz bevor er stirbt, entdeckt er, daß dieses geduldige Labyrinth das Bild seines Gesichts wiedergibt. (Jorge Luis Borges) Im wesentlichen glaube ich ja, daß ich niemals sterben werde, aber trotzdem, daß in all dem, was man so tut und schafft mehr von einem steckt als auch auf dem zweiten Blick erkennbar ist. Damit schimmern auch die Personen, die durch Unterst¨utzungjedweder Art und durch Freundschaft in meinem Leben wichtig gewesen sind und sein werden zwischen der hochabstrakten Mathematik durch. Meinen Eltern Erna und Josef Teichmann geb¨uhrtder erste und gr¨oßteDank, unter vielen Dingen ist f¨urmich besonders zu erw¨ahnen,daß Diskussion ¨uber alles m¨ogliche, um den Sachen auf den Grund zu gehen, bei uns zu Hause eine große Rolle spielten. Im speziellen bin ich meinen Eltern daf¨urdankbar, diesen Stil mitbekommen zu haben f¨urmein Leben. Meinem Bruder, der die ersten Worte dieses Textes vor Jahren zu nachtschlafener Stunde in mein Hermes-Baby tippte, m¨ochte ich danken, nat¨urlich f¨urdie vielen interessanten Diskussionen, f¨urdie neuen Zug¨angezum Leben, f¨urseine fortw¨ahrendeUnterst¨utzungund die Liebe zur K-Theorie. Meiner Tante Hilde Darani m¨ochte ich f¨urdie sch¨onenGespr¨ache ¨uber Religion und das Leben im allgemeinen danken, f¨urdie fortw¨ahrendeUnterst¨utzungund ihren Charme. Michaela und ich sind zu Beginn unserer Zeit in Wien in das jeweilig andere Leben getreten, Monate sp¨aterbegannen wir dann uns auf Hannah Zsofia Hazel zu freuen. Michaela m¨ochte ich f¨urdie ungarischen T¨anzeim Santo Spirito danken, nat¨urlich die vielen Diskussionen, den Widerstandsgeist und die Liebe. H.Z. Hazel, obwohl noch ein sehr junger Mensch, hat den gr¨oßtenEinfluß auf uns alle. Bei ihr sind Eigenschaften wie Lust, Freude, Widerstandskraft und Wille wunderbar ausgepr¨agt. An dieser Stelle m¨ochte ich auch Sabine und Ingo Franz, Herbert Oberhuber und Sigrid Winkler f¨urdie Unterst¨utzungunserer kleinen Familie danken. Mit Stefan Haller hatte ich die Freude den Raum zu teilen, den wir regelm¨aßigin eine Spelunke mit tiefh¨angendenRauchschwaden verwandelten, vielen Dank f¨urdie sch¨onenDiskussionen. Dank geb¨uhrtauch den TeilnehmerInnen an den Wissenschaftlichen Arbeiten, insbesondere Andreas Cap, f¨urdie anregende und profunde Kritik und das Zuh¨oren. Mein mathematischer Weg wurde von zwei Pers¨onlichkeiten an der Universit¨atGraz gepr¨agt: Peter Sch¨opfhabe ich gleich zu Beginn meines Studiums bei einer Erl¨auterunglogischer Probleme der Analysis 1 kennengelernt, Franz Halter-Koch etwas sp¨aterbei seinen Algebra-Vorlesungen und auch als sein Sekret¨ar. Beiden verdanke ich mathematische Bildung, viele Diskussionen und Un- terst¨utzungen.Besonders erw¨ahnenm¨ochte ich dabei, daß Franz Halter-Koch mich auch pers¨onlich finanziell unterst¨utzte,um mir die M¨oglichkeit zu bieten, eine andere Universit¨atkennenzulernen und dort weiter zu studieren. Danken m¨ochte ich außerdem Alfred Geroldinger, Ludwig Pittner, Ernst Seidel und Bernd Thaller. Wolgang Arendt habe ich in Besan¸conim Studienjahr 1994/95 kennengelernt: Ihm m¨ochte ich f¨urdie beiden Einladungen nach Ulm, wo ich meine Arbeiten pr¨asentieren konnte, danken, auch f¨ur die sch¨onenDiskussionen ¨uber Osterreich¨ donauaufw¨arts. Mein Weg nach Wien war m¨oglicherweise mathematisch etwas unerwartet, da meine Diplomarbeit im funktionalanalytischen Bereich angesiedelt gewesen war, aber die Offenheit Peter Michors ließ alle Zweifel schwinden. Die liberale und intellektuelle Atmosph¨areum Peter Michor war f¨urmeine pers¨onliche Entwicklung und f¨urdas Gedeihen der Dissertation wesentlich. Außerdem hat er dadurch, v vi DANKSAGUNG daß er an meine Arbeit glaubte und sie aus Mitteln des FWF (Projekt P-10037-PHY) unterst¨utzte, nicht nur mein Leben als reiner Mathematiker erm¨oglicht, sondern auch mein Leben als Hausmann. F¨urdiese liberale und seltene Haltung, n¨amlich bez¨uglich der Zeiteinteilung mir freie Hand zu lassen, m¨ochte ich besonders danken. Andreas Kriegl m¨ochte ich f¨urdie vielen detaillierten Hinweise und die Redlichkeit beim Disku- tieren danken, auch f¨urdie Signale der Bereitschaft dazu. Anregungen zur Dissertation und zu meinem mathematischen Denken kamen auch von nicht- mathematischer Seite, einerseits durch die Aufforderung zu erkl¨aren,was ich da so mache, andererseits durch kluge und pr¨aziseStatements nach meinen Erl¨auterungen.So war Martin Feldbacher vor zwei Jahren fest davon ¨uberzeugt, daß das Konzept mit den Lipschitz-Metriken aufgehen wird. Sebastian Markt legte mir nahe, einige Aussagen zur mathematischen Logik noch einmal zu ¨uberlegen. Roman Urbaner sah, daß die Erstfassung meiner dilletierenden Ausf¨uhrungenzu Robert Musil doch etwas im Dunkeln liegen, wo sie vielleicht noch immer sind. Niko Wahl und Gregor Weiss m¨ochte ich f¨urdie Tatsache danken, mir die Bibel und das Kapital n¨ahergebracht zu haben. Josef Teichmann, Wien im Juli 1999 Introduction Er war weniger wissenschaftlich als menschlich verliebt in die Wissenschaft. Er sah, daß sie in allen Fragen, wo sie sich f¨urzust¨andigh¨alt,anders denkt als gew¨ohnlicheMenschen. Wenn man statt wissenschaftlicher Anschauungen Lebensanschauung setzen w¨urde, statt Hypothese Versuch und statt Wahrheit Tat, so g¨abe es kein Lebenswerk eines ansehnlichen Naturforschers oder Mathematikers, das an Mut und Umsturzkraft nicht die gr¨oßtenTaten der Geschichte weit ¨ubertreffen w¨urde. Der Mann war noch nicht auf der Welt, der zu seinen Gl¨aubigenh¨attesagen k¨onnen: Stehlt, mordet, treibt Unzucht - unsere Lehre ist so stark, daß sie aus der Jauche eurer S¨undensch¨aumendhelle Bergw¨assermacht; aber in der Wissenschaft kommt es alle paar Jahre vor, daß etwas was bis dahin als Fehler galt, pl¨otzlichalle Anschauungen umkehrt oder daß ein unscheinbarer und verachteter Gedanke zum Herrscher ¨uber ein neues Gedankenreich wird, und solche Vorkommnisse sind dort nicht bloß Umst¨urze,sondern f¨uhren wie eine Himmelsleiter in die H¨ohe. (Robert Musil, Der Mann ohne Eigenschaften) Mathematics is not politics, but the universe of mathematics has been influencing in a subtle and silent way, but revolutionary in result, the discourses of human beings in this century. Mathematics is the most dangerous science. Here I do not think of people exploiting mathematical knowledge for their own purposes, I have
Recommended publications
  • Hodge Decomposition for Higher Order Hochschild Homology

    Hodge Decomposition for Higher Order Hochschild Homology

    ANNALES SCIENTIFIQUES DE L’É.N.S. TEIMURAZ PIRASHVILI Hodge decomposition for higher order Hochschild homology Annales scientifiques de l’É.N.S. 4e série, tome 33, no 2 (2000), p. 151-179 <http://www.numdam.org/item?id=ASENS_2000_4_33_2_151_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 2000, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Sclent. EC. Norm. Sup., 4° serie, t 33, 2000, p. 151 a 179. HODGE DECOMPOSITION FOR HIGHER ORDER HOCHSCHILD HOMOLOGY TEIMURAZ PIRASHVILI ABSTRACT. - Let r be the category of finite pointed sets and F be a functor from r to the category of vector spaces over a characteristic zero field. Loday proved that one has the natural decomposition 7TnF(S1) ^ ©n^ Q^^CF), n ^ 0. We show that for any d ^ 1, there exists a similar decomposition for 7^nF(Sd). Here 3d is a simplicial model of the d-dimensional sphere. The striking point is, that the knowledge of the decomposition for 7VnF(S1) (respectively 7TnF(S2)) completely determines the decomposition of 7^nF(Sd) for any odd (respectively even) d. These results can be applied to the cohomology of the mapping space Xs , where X is a c^-connected space.
  • Arxiv:Math/9201256V1 [Math.RT] 1 Jan 1992 Naso Lblaayi N Geometry and Vol

    Arxiv:Math/9201256V1 [Math.RT] 1 Jan 1992 Naso Lblaayi N Geometry and Vol

    Annals of Global Analysis and Geometry Vol. 8, No. 3 (1990), 299–313 THE MOMENT MAPPING FOR UNITARY REPRESENTATIONS Peter W. Michor Institut f¨ur Mathematik der Universit¨at Wien, Austria Abstract. For any unitary representation of an arbitrary Lie group I construct a moment mapping from the space of smooth vectors of the representation into the dual of the Lie algebra. This moment mapping is equivariant and smooth. For the space of analytic vectors the same construction is possible and leads to a real analytic moment mapping. Table of contents 1.Introduction . .. .. .. .. .. .. .. .. .. .. 1 2.Calculusofsmoothmappings . 2 3.Calculusofholomorphicmappings . 5 4.Calculusofrealanalyticmappings . 6 5.TheSpaceofSmoothVectors . 7 6.Themodelforthemomentmapping . 8 7. Hamiltonian Mechanics on H∞ .................... 9 8. The moment mapping for a unitary representation . .... 11 9.Therealanalyticmomentmapping . 13 1. Introduction arXiv:math/9201256v1 [math.RT] 1 Jan 1992 With the help of the cartesian closed calculus for smooth mappings as explained in [F-K] we can show, that for any Lie group and for any unitary representation its restriction to the space of smooth vectors is smooth. The imaginary part of the hermitian inner product restricts to a ”weak” symplectic structure on the vector space of smooth vectors. This gives rise to the Poisson bracket on a suitably chosen space of smooth functions on the space of smooth vectors. The derivative of the representation on the space of smooth vectors is a symplectic action of the Lie algebra, which can be lifted to a Hamiltonian action, i.e. a Lie algebra homomorphism from the Lie algebra into the function space with the Poisson bracket.
  • Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic

    Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic

    Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic Richard Blute University of Ottawa ongoing discussions with Geoff Crutwell, Thomas Ehrhard, Alex Hoffnung, Christine Tasson June 20, 2011 Richard Blute Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic Goals Develop a theory of (smooth) manifolds based on differential linear logic. Or perhaps develop a differential linear logic based on manifolds. Convenient vector spaces were recently shown to be a model. There is a well-developed theory of convenient manifolds, including infinite-dimensional manifolds. Convenient manifolds reveal additional structure not seen in finite dimensions. In particular, the notion of tangent space is much more complex. Synthetic differential geometry should also provide information. Convenient vector spaces embed into an extremely good model. Richard Blute Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic Convenient vector spaces (Fr¨olicher,Kriegl) Definition A vector space is locally convex if it is equipped with a topology such that each point has a neighborhood basis of convex sets, and addition and scalar multiplication are continuous. Locally convex spaces are the most well-behaved topological vector spaces, and most studied in functional analysis. Note that in any topological vector space, one can take limits and hence talk about derivatives of curves. A curve is smooth if it has derivatives of all orders. The analogue of Cauchy sequences in locally convex spaces are called Mackey-Cauchy sequences. The convergence of Mackey-Cauchy sequences implies the convergence of all Mackey-Cauchy nets. The following is taken from a long list of equivalences. Richard Blute Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic Convenient vector spaces II: Definition Theorem Let E be a locally convex vector space.
  • Derived Smooth Manifolds

    Derived Smooth Manifolds

    DERIVED SMOOTH MANIFOLDS DAVID I. SPIVAK Abstract. We define a simplicial category called the category of derived man- ifolds. It contains the category of smooth manifolds as a full discrete subcat- egory, and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local C1-rings that is obtained by patching together homotopy zero-sets of smooth functions on Eu- clidean spaces. We show that derived manifolds come equipped with a stable normal bun- dle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a Pontrjagin-Thom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory. This allows us to define fundamental classes in cobordism for all derived man- ifolds. In particular, the intersection A \ B of submanifolds A; B ⊂ X exists on the categorical level in our theory, and a cup product formula [A] ^ [B] = [A \ B] holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a categorification of intersection theory. Contents 1. Introduction 1 2. The axioms 8 3. Main results 14 4. Layout for the construction of dMan 21 5. C1-rings 23 6. Local C1-ringed spaces and derived manifolds 26 7. Cotangent Complexes 32 8. Proofs of technical results 39 9. Derived manifolds are good for doing intersection theory 50 10. Relationship to similar work 52 References 55 1. Introduction Let Ω denote the unoriented cobordism ring (though what we will say applies to other cobordism theories as well, e.g.
  • Math 395: Category Theory Northwestern University, Lecture Notes

    Math 395: Category Theory Northwestern University, Lecture Notes

    Math 395: Category Theory Northwestern University, Lecture Notes Written by Santiago Can˜ez These are lecture notes for an undergraduate seminar covering Category Theory, taught by the author at Northwestern University. The book we roughly follow is “Category Theory in Context” by Emily Riehl. These notes outline the specific approach we’re taking in terms the order in which topics are presented and what from the book we actually emphasize. We also include things we look at in class which aren’t in the book, but otherwise various standard definitions and examples are left to the book. Watch out for typos! Comments and suggestions are welcome. Contents Introduction to Categories 1 Special Morphisms, Products 3 Coproducts, Opposite Categories 7 Functors, Fullness and Faithfulness 9 Coproduct Examples, Concreteness 12 Natural Isomorphisms, Representability 14 More Representable Examples 17 Equivalences between Categories 19 Yoneda Lemma, Functors as Objects 21 Equalizers and Coequalizers 25 Some Functor Properties, An Equivalence Example 28 Segal’s Category, Coequalizer Examples 29 Limits and Colimits 29 More on Limits/Colimits 29 More Limit/Colimit Examples 30 Continuous Functors, Adjoints 30 Limits as Equalizers, Sheaves 30 Fun with Squares, Pullback Examples 30 More Adjoint Examples 30 Stone-Cech 30 Group and Monoid Objects 30 Monads 30 Algebras 30 Ultrafilters 30 Introduction to Categories Category theory provides a framework through which we can relate a construction/fact in one area of mathematics to a construction/fact in another. The goal is an ultimate form of abstraction, where we can truly single out what about a given problem is specific to that problem, and what is a reflection of a more general phenomenom which appears elsewhere.
  • Lecture Notes 1

    Lecture Notes 1

    Convenient Calculus and Differential Geometry in Infinite Dimensions Peter W. Michor Norwegian Summer School on Analysis and Geometry Bergen, Norway. June 24-28, 2013 Some words on smooth convenient calculus Traditional differential calculus works well for finite dimensional vector spaces and for Banach spaces. Beyond Banach spaces, the main difficulty is that composition of linear mappings stops to be jointly continuous at the level of Banach spaces, for any compatible topology. For more general locally convex spaces we sketch here the convenient approach as explained in [Fr¨olicher-Kriegl1988] and [Kriegl-Michor 1997]. The c1-topology Let E be a locally convex vector space. A curve c : R ! E is called smooth or C 1 if all derivatives exist and are continuous. Let 1 C (R; E) be the space of smooth functions. It can be shown that 1 the set C (R; E) does not depend on the locally convex topology of E, only on its associated bornology (system of bounded sets). The final topologies with respect to the following sets of mappings into E coincide: 1 1. C (R; E). 2. The set of all Lipschitz curves (so that c(t)−c(s) f t−s : t 6= s; jtj; jsj ≤ Cg is bounded in E, for each C). 3. The set of injections EB ! E where B runs through all bounded absolutely convex subsets in E, and where EB is the linear span of B equipped with the Minkowski functional kxkB := inffλ > 0 : x 2 λBg. 4. The set of all Mackey-convergent sequences xn ! x (there exists a sequence 0 < λn % 1 with λn(xn − x) bounded).
  • Notes, We Review Useful Concepts in Convexity and Normed Spaces

    Notes, We Review Useful Concepts in Convexity and Normed Spaces

    Mastermath, Spring 2019 Lecturers: D. Dadush, J. Briet Geometric Functional Analysis Lecture 0 Scribe: D. Dadush Review of Convexity and Normed Spaces In these notes, we review useful concepts in convexity and normed spaces. As a warning, many of the statements made here will not be proved. For detailed proofs, one is recommended to look up reference books in functional and convex analysis. The intent here is for these notes to serve as a reference for the basic language for the course. The reader may thus skim the material upon first reading, and then reread more carefully later during the course if the relevant concepts remain unclear. 1 Basic Convexity In this section, we will focus on convexity theorems in a finite dimensional real vector space, which will identify with Rn or subspaces theoreof. For vectors x, y 2 Rn, we define the standard T n n inner product hx, yi := x y = ∑i=1 xiyi. For two sets A, B ⊆ R and scalars c, d 2 R, we define the Minkowski sum cA + dB = fca + db : a 2 A, b 2 Bg. A convex set K ⊆ Rn is a set for which the line segment between any two points is in the set, or formally for which 8x, y 2 K, l 2 [0, 1], lx + (1 − l)y 2 K. A useful equivalent definition is that a non-empty set K is convex if 8a, b ≥ 0, aK + bK = (a + b)K. If K = −K, K is said to be symmetric. If K is compact with non-empty interior, then K is a convex body.
  • On the Smoothness of Functors

    On the Smoothness of Functors

    Iranian Journal of Mathematical Sciences and Informatics Vol. 5, No. 1 (2010), pp. 27-39 On the Smoothness of Functors A. Bajravania and A. Rastegarb,∗ aDepartment of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P. O. Box 14115-134, Tehran, Iran b Faculty of Mathematics, Sharif University, P. O. Box 11155, Tehran, Iran E-mail: [email protected] E-mail: [email protected] Abstract. In this paper we will try to introduce a good smoothness no- tion for a functor. We consider properties and conditions from geometry and algebraic geometry which we expect a smooth functor should have. Keywords: Abelian Category, First Order Deformations, Multicategory, Tangent Category, Topologizing Subcategory. 2000 Mathematics subject classification: 14A20, 14A15, 14A22. 1. Introduction Nowadays noncommutative algebraic geometry is in the focus of many basic topics in mathematics and mathematical physics. In these fields, any under consideration space is an abelian category and a morphism between noncom- mutative spaces is a functor between abelian categories. So one may ask to gen- eralize some aspects of morphisms between commutative spaces to morphisms between noncommutative ones. One of the important aspects in commuta- tive case is the notion of smoothness of a morphism which is stated in some languages, for example: by lifting property as a universal language, by projec- tivity of relative cotangent sheaves as an algebraic language and by inducing a surjective morphism on tangent spaces as a geometric language. ∗Corresponding Author Received 25 December 2009; Accepted 10 April 2010 c 2010 Academic Center for Education, Culture and Research TMU 27 28 A.
  • Diffeology, Groupoids & Morita Equivalence

    Diffeology, Groupoids & Morita Equivalence

    Diffeology, Groupoids & Morita Equivalence M.Sc. Thesis in Mathematics Nesta van der Schaaf 22nd June 2020 Supervisor: Prof. dr. N.P. Landsman Second reader: Dr. I.T. Mărcuț Abstract This thesis consists of two parts. The first (Chapters I and II) is a thorough introduction to the theory of diffeology. We provide a ground-up account of the theory from the viewpoint of plots (in contrast to the sheaf-theoretic treatments). This includes a proof that the category Diffeol of diffeological spaces and smooth maps is complete and cocomplete, (locally) Cartesian closed, and a quasitopos. In addition, we treat many examples, including a detailed recollection of the classification of irrational tori. The second part (Chapters III to VI) is a proposal for a framework of diffeological Morita equivalence. We give definitions of diffeological groupoid actions, -bundles, and -bibundles, general- ising the known theory of Lie groupoids and their corresponding notions. We obtain a bicategory DiffeolBiBund of diffeological groupoids and diffeological bibundles. This has no analogue inthe Lie theory, since we put no ‘principality’ restrictions on these bibundles. We then define a new notion of principality for diffeological (bi)bundles, and subsequently obtain a notion of Morita equivalence by declaring that two diffeological groupoids are equivalent if and only if there existsa biprincipal bibundle between them. Our main new result is the following: two diffeological groupoids are Morita equivalent if and only if they are weakly equivalent in the bicategory DiffeolBiBund. Equivalently, this means that a diffeological bibundle is weakly invertible if and only if it is biprincipal. This significantly generalises the original theorem in the Lie groupoid setting, where an analogous state- ment can only be made if we assume one-sided principality beforehand.
  • The Convenient Setting for Denjoy--Carleman Differentiable

    The Convenient Setting for Denjoy--Carleman Differentiable

    THE CONVENIENT SETTING FOR DENJOY–CARLEMAN DIFFERENTIABLE MAPPINGS OF BEURLING AND ROUMIEU TYPE ANDREAS KRIEGL, PETER W. MICHOR, AND ARMIN RAINER Abstract. We prove in a uniform way that all Denjoy–Carleman differen- tiable function classes of Beurling type C(M) and of Roumieu type C{M}, admit a convenient setting if the weight sequence M = (Mk) is log-convex and of moderate growth: For C denoting either C(M) or C{M}, the category of C-mappings is cartesian closed in the sense that C(E, C(F,G)) =∼ C(E × F,G) for convenient vector spaces. Applications to manifolds of mappings are given: The group of C-diffeomorphisms is a regular C-Lie group if C ⊇ Cω, but not better. 1. Introduction Denjoy–Carleman differentiable functions form classes of smooth functions that are described by growth conditions on the Taylor expansion. The growth is pre- scribed in terms of a sequence M = (Mk) of positive real numbers which serves as a weight for the iterated derivatives: for compact K the sets f (k)(x) k : x K, k N ρ k! Mk ∈ ∈ n o are required to be bounded. The positive real number ρ is subject to either a universal or an existential quantifier, thereby dividing the Denjoy–Carleman classes into those of Beurling type, denoted by C(M), and those of Roumieu type, denoted {M} by C , respectively. For the constant sequence M = (Mk) = (1), as Beurling type we recover the real and imaginary parts of all entire functions on the one hand, and as Roumieu type the real analytic functions on the other hand, where 1/ρ plays the role of a radius of convergence.
  • Approximating Euler's Equation on the Full Diffeomorphism Group, And

    Approximating Euler's Equation on the Full Diffeomorphism Group, And

    Approximating Euler's equation on the full diffeomorphism group, and soliton-like solutions Peter W. Michor Geometrical Aspects of Hydrodynamics workshop Simons Center for Geometry and Physics Stony Brook University USA Based on [David Mumford, Peter W. Michor: On Euler's equation and `EPDiff'. Journal of Geometric Mechanics, 5, 3 (2013), 319-344.] May 19 - 23, 2014 Some words on smooth convenient calculus Traditional differential calculus works well for finite dimensional vector spaces and for Banach spaces. Beyond Banach spaces, the main difficulty is that composition of linear mappings stops to be jointly continuous at the level of Banach spaces, for any compatible topology. For more general locally convex spaces we sketch here the convenient approach as explained in [Fr¨olicher-Kriegl1988] and [Kriegl-Michor 1997]. The c1-topology Let E be a locally convex vector space. A curve c : R ! E is called smooth or C 1 if all derivatives exist and are continuous. Let 1 C (R; E) be the space of smooth functions. It can be shown that 1 the set C (R; E) does not depend entirely on the locally convex topology of E, only on its associated bornology (system of bounded sets). The final topologies with respect to the following sets of mappings into E coincide: 1 1. C (R; E). 2. The set of all Lipschitz curves (so that c(t)−c(s) f t−s : t 6= s; jtj; jsj ≤ Cg is bounded in E, for each C). 3. The set of injections EB ! E where B runs through all bounded absolutely convex subsets in E, and where EB is the linear span of B equipped with the Minkowski functional kxkB := inffλ > 0 : x 2 λBg.
  • A Zoo of Diffeomorphism Groups on $$ \Mathbb{R }^{N}$$ Rn

    A Zoo of Diffeomorphism Groups on $$ \Mathbb{R }^{N}$$ Rn

    A zoo of diffeomorphism groups on $$ \mathbb{R }^{n}$$ Rn Peter W. Michor & David Mumford Annals of Global Analysis and Geometry ISSN 0232-704X Ann Glob Anal Geom DOI 10.1007/s10455-013-9380-2 1 23 Your article is protected by copyright and all rights are held exclusively by Springer Science +Business Media Dordrecht. This e-offprint is for personal use only and shall not be self- archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Ann Glob Anal Geom DOI 10.1007/s10455-013-9380-2 A zoo of diffeomorphism groups on Rn Peter W. Michor · David Mumford Received: 25 November 2012 / Accepted: 6 May 2013 © Springer Science+Business Media Dordrecht 2013 n n n Abstract We consider the groups DiffB(R ), Diff H ∞ (R ), and DiffS (R ) of smooth diffeo- n morphisms on R which differ from the identity by a function which is in either B (bounded ∞ = k S in all derivatives), H k≥0 H ,or (rapidly decreasing). We show that all these groups are smooth regular Lie groups.