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Fibrewise Stable Rational Homotopy
Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 Fibrewise Stable Rational Homotopy Yves F´elix, Aniceto Murillo and Daniel Tanr´e Abstract In this paper, for a given space B we establish a correspondence between differential graded ∗ modules over C (B; Q) and fibrewise rational stable spaces over B. This correspondence opens the door for topological translations of algebraic constructions made with modules over a commutative differential graded algebra. More precisely, given fibrations E → B and ′ E → B, the set of stable rational homotopy classes of maps over B is isomorphic to ∗ ∗ ′ ∗ ExtC∗(B;Q)`C (E ; Q),C (E; Q)´. In particular, a nilpotent, finite type CW-complex X is a q rational Poincar´ecomplex if there exist non trivial stable maps over XQ from (X × S )Q to q+N (X ∨ S )Q for exactly one N. 1. Introduction Sullivan’s approach to rational homotopy theory is based on an adjoint pair of functors between the category of simplicial sets and the category CDGA of commutative differential graded algebras over Q (henceforth called cdga’s ). This correspondence induces an equivalence between the homotopy category of rational (nilpotent and with finite Betti numbers) simplicial sets and a homotopy category of cdga’s. The fact that any algebraic construction or property in CDGA has a topological translation is the most important feature of rational homotopy theory. Until now, however, there was no known procedure for realizing differential graded modules (henceforth called dgm’s) over cdga’s topologically, even though a number of important topological invariants have been interpreted in terms of dgm’s over the years. -
HE WANG Abstract. a Mini-Course on Rational Homotopy Theory
RATIONAL HOMOTOPY THEORY HE WANG Abstract. A mini-course on rational homotopy theory. Contents 1. Introduction 2 2. Elementary homotopy theory 3 3. Spectral sequences 8 4. Postnikov towers and rational homotopy theory 16 5. Commutative differential graded algebras 21 6. Minimal models 25 7. Fundamental groups 34 References 36 2010 Mathematics Subject Classification. Primary 55P62 . 1 2 HE WANG 1. Introduction One of the goals of topology is to classify the topological spaces up to some equiva- lence relations, e.g., homeomorphic equivalence and homotopy equivalence (for algebraic topology). In algebraic topology, most of the time we will restrict to spaces which are homotopy equivalent to CW complexes. We have learned several algebraic invariants such as fundamental groups, homology groups, cohomology groups and cohomology rings. Using these algebraic invariants, we can seperate two non-homotopy equivalent spaces. Another powerful algebraic invariants are the higher homotopy groups. Whitehead the- orem shows that the functor of homotopy theory are power enough to determine when two CW complex are homotopy equivalent. A rational coefficient version of the homotopy theory has its own techniques and advan- tages: 1. fruitful algebraic structures. 2. easy to calculate. RATIONAL HOMOTOPY THEORY 3 2. Elementary homotopy theory 2.1. Higher homotopy groups. Let X be a connected CW-complex with a base point x0. Recall that the fundamental group π1(X; x0) = [(I;@I); (X; x0)] is the set of homotopy classes of maps from pair (I;@I) to (X; x0) with the product defined by composition of paths. Similarly, for each n ≥ 2, the higher homotopy group n n πn(X; x0) = [(I ;@I ); (X; x0)] n n is the set of homotopy classes of maps from pair (I ;@I ) to (X; x0) with the product defined by composition. -
Remarks on Combinatorial and Accessible Model Categories
Theory and Applications of Categories, Vol. 37, No. 9, 2021, pp. 266{275. REMARKS ON COMBINATORIAL AND ACCESSIBLE MODEL CATEGORIES JIRˇ´I ROSICKY´ Abstract. Using full images of accessible functors, we prove some results about com- binatorial and accessible model categories. In particular, we give an example of a weak factorization system on a locally presentable category which is not accessible. 1. Introduction Twenty years ago, M. Hovey asked for examples of model categories which are not cofi- brantly generated. This is the same as asking for examples of weak factorization systems which are not cofibrantly generated. One of the first examples was given in [1]: it is a weak factorization system (L; R) on the locally presentable category of posets where L consists of embeddings. The reason is that posets injective to embeddings are precisely complete lattices which do not form an accessible category. Hence L is not cofibrantly generated. Since then, the importance of accessible model categories and accessible weak factorization systems has emerged, and, the same question appears again, i.e., to give an example of a weak factorization system (L; R) on a locally presentable category which is not accessible. Now, L-injective objects do not necessarily form an accessible category but only a full image of an accessible functor. Such full images are accessible only under quite restrictive assumptions (see [5]). But, for an accessible weak factorization system, L-injective objects form the full image of a forgetful functor from algebraically L-injective objects (see Corollary 3.3). Moreover, such full images are closed under reduced prod- ucts modulo κ-complete filters for some regular cardinal κ (see Theorem 2.5). -
CURRICULUM VITAE Jonathan Andrew Scott
CURRICULUM VITAE Jonathan Andrew Scott [email protected] Address: 6-53 Maclaren Street, Ottawa ON K2P 0K3 Canada Current Affiliation: Department of Mathematics and Statistics, University of Ottawa Telephone (office): (613) 562-5800 ext. 2082 Telephone (home): (613) 236-4812 Education 2000 Ph.D. in Mathematics, University of Toronto. Supervisor: Steve Halperin. 1994 M.Sc. in Mathematics, University of Toronto. 1993 B.Sc. (Honours) in Mathematical Physics, Queen’s University at Kingston. Research Interests Operads, category theory, E∞ algebras, cohomology operations, loop space homology, Bockstein spectral sequence. Publications 1. K. Hess, P.-E. Parent, and J. Scott, A chain coalgebra model for the James map, Homology, Homotopy Appl. 9 (2007) 209–231, arXiv:math.AT/0609444. 2. K. Hess, P.-E. Parent, J. Scott, and A. Tonks, A canonical enriched Adams-Hilton model for simplicial sets, Adv. Math. 207 (2006) 847–875, arXiv:math.AT/0408216. 3. J. Scott, Hopf algebras up to homotopy and the Bockstein spectral sequence, Algebr. Geom. Topol. 5 (2005) 119–128, arXiv:math.AT/0412207. 4. J. Scott, A torsion-free Milnor-Moore theorem, J. London Math. Soc. (2) 67 (2003) 805–816, arXiv:math.AT/0103223. 5. J. Scott, Algebraic structure in the loop space homology Bockstein spectral sequence, Trans. Amer. Math. Soc. 354 (2002) 3075–3084, arXiv: math.AT/9912106. 6. J. Scott, A factorization of the homology of a differential graded Lie algebra, J. Pure Appl. Alg. 167 (2002) 329–340. Submitted Manuscripts 1. K. Hess and J. Scott, Homotopy morphisms and Koszul resolutions of operads 2. K. Hess, P.-E. Parent, and J. -
Mite Beometrles Ana Corn Binatorial Designs
AMERICAN MATHEMATICAL SOCIETY mite beometrles ana Corn binatorial Designs Proceedings of the AMS Special Session in Finite Geometries and Combinatorial Designs held October 29-November 1, 1987 http://dx.doi.org/10.1090/conm/111 Titles in This Series Volume 1 Markov random fields and their 19 Proceedings of the Northwestern applications, Ross Kindermann and homotopy theory conference, Haynes J. Laurie Snell R. Miller and Stewart B. Priddy, Editors 2 Proceedings of the conference on 20 Low dimensional topology, Samuel J. integration, topology, and geometry in Lomonaco, Jr., Editor linear spaces, William H. Graves, Editor 21 Topological methods in nonlinear 3 The closed graph and P-closed functional analysis, S. P. Singh, graph properties in general topology, S. Thomaier, and B. Watson, Editors T. R. Hamlett and L. L. Herrington 22 Factorizations of b" ± 1, b = 2, 4 Problems of elastic stability and 3, 5, 6, 7, 10, 11, 12 up to high vibrations, Vadim Komkov, Editor powers, John Brillhart, D. H. Lehmer, 5 Rational constructions of modules J. L. Selfridge, Bryant Tuckerman, and for simple Lie algebras, George B. S. S. Wagstaff, Jr. Seligman 23 Chapter 9 of Ramanujan's second 6 Umbral calculus and Hopf algebras, notebook-Infinite series identities, Robert Morris, Editor transformations, and evaluations, Bruce C. Berndt and Padmini T. Joshi 7 Complex contour integral representation of cardinal spline 24 Central extensions, Galois groups, and functions, Walter Schempp ideal class groups of number fields, A. Frohlich 8 Ordered fields and real algebraic geometry, D. W. Dubois and T. Recio, 25 Value distribution theory and its Editors applications, Chung-Chun Yang, Editor 9 Papers in algebra, analysis and 26 Conference in modern analysis statistics, R. -
James H. Olsen
James H. Olsen Contact Department of Mathematics Information North Dakota State University Voice: (614) 292-2572 205 Dreese Labs Fax: (614) 292-7596 2015 Neil Avenue E-mail: [email protected] Columbus, OH 43210 USA WWW: math.ndsu.nodak.edu/faculty/olsen/ Research Ergodic theory Interests Education The University of Minnesota, Minneapolis, Minnesota USA Ph.D., Mathematics, 1968 • Dissertation Title: Dominated Estimates of Lp Contractions, 1 < p < ∞. • Advisor: Professor Rafael V. Chacon • Area of Study: Ergodic Theory M.S., Mathematics, 1967 • Advisor: Professor Rafael V. Chacon • Area of Study: Ergodic Theory N.C. State College, Raleigh, North Carolina USA B.S., Electrical Engineering, 1963 Awards and Eta Kappa Nu Honors • Member Scabbard and Blade National Honor Society • First Sergeant Academic and North Dakota State University, Fargo, ND USA Professional Experience Professor of Mathematics 1984 to present Associate Professor of Mathematics 1974 to 1984 Assistant Professor of Mathematics 1970 to 1974 Mathematical Research Division, National Security Agency, Fort George G. Meade, Maryland, USA Research Mathematician 1969 to 1970 • Active duty with the U.S. Army. • Achieved the rank of captain. The University of Minnesota, Minneapolis, Minnesota USA Instructor, School of Mathematics Fall 1968 Nuclear Effects Lab, White Sands Missle Range, New Mexico USA Electronics Engineer Summer 1964 1 of 5 Publications Dominated estimates of positive contractions, (with R.V. Chacon), Proceedings of the American Mathematical Society, 20, 226-271 (1969). Dominated estimates of convex combinations of commuting isometries, Israel Journal of Mathematics, 11, 1-13 (1972). Estimates of convex combinations of commuting isometries, Z. Wahrscheinlicheits- theorie verw. Geb. 26, 317-324 (1973). -
Rational Homotopy Theory: a Brief Introduction
Contemporary Mathematics Rational Homotopy Theory: A Brief Introduction Kathryn Hess Abstract. These notes contain a brief introduction to rational homotopy theory: its model category foundations, the Sullivan model and interactions with the theory of local commutative rings. Introduction This overview of rational homotopy theory consists of an extended version of lecture notes from a minicourse based primarily on the encyclopedic text [18] of F´elix, Halperin and Thomas. With only three hours to devote to such a broad and rich subject, it was difficult to choose among the numerous possible topics to present. Based on the subjects covered in the first week of this summer school, I decided that the goal of this course should be to establish carefully the founda- tions of rational homotopy theory, then to treat more superficially one of its most important tools, the Sullivan model. Finally, I provided a brief summary of the ex- tremely fruitful interactions between rational homotopy theory and local algebra, in the spirit of the summer school theme “Interactions between Homotopy Theory and Algebra.” I hoped to motivate the students to delve more deeply into the subject themselves, while providing them with a solid enough background to do so with relative ease. As these lecture notes do not constitute a history of rational homotopy theory, I have chosen to refer the reader to [18], instead of to the original papers, for the proofs of almost all of the results cited, at least in Sections 1 and 2. The reader interested in proper attributions will find them in [18] or [24]. The author would like to thank Luchezar Avramov and Srikanth Iyengar, as well as the anonymous referee, for their helpful comments on an earlier version of this article. -
President's Report
Volume 38, Number 4 NEWSLETTER July–August 2008 President’s Report Dear Colleagues: I am delighted to announce that our new executive director is Maeve Lewis McCarthy. I am very excited about what AWM will be able to accomplish now that she is in place. (For more about Maeve, see the press release on page 7.) Welcome, Maeve! Thanks are due to the search committee for its thought and energy. These were definitely required because we had some fabulous candidates. Thanks also to Murray State University, Professor McCarthy’s home institution, for its coopera- tion as we worked out the details of her employment with AWM. The AWM Executive Committee has voted to give honorary lifetime mem- IN THIS ISSUE berships to our founding presidents, Mary Gray and Alice T. Schafer. In my role as president, I am continually discovering just how extraordinary AWM is 7 McCarthy Named as an organization. Looking back at its early history, I find it hard to imagine AWM Executive Director how AWM could have come into existence without the vision, work, and persist- ence of these two women. 10 AWM Essay Contest Among newly elected members of the National Academy of Sciences in the physical and mathematical sciences are: 12 AWM Teacher Partnerships 16 MIT woMen In maTH Emily Ann Carter Department of Mechanical and Aerospace Engineering and the Program in 19 Girls’ Angle Applied and Computational Mathematics, Princeton University Lisa Randal Professor of theoretical physics, Department of Physics, Harvard University Elizabeth Thompson Department of Statistics, University of Washington, Seattle A W M The American Academy of Arts and Sciences has also announced its new members. -
Editorial Board George E
Advances in Mathematics http://www.ees.elsevier.com/aim Elsevier Editorial Office 525 B Street, Suite 1900 San Diego, CA 92101-4495, USA E-mail: [email protected] Telephone: (619) 699-6854 Fax: (619) 699-6700 Edited by Michael J. Hopkins Tomasz S. Mrowka Harvard University Massachusetts Institute of Technology Cambridge, MA 02138, USA Cambridge, MA 02139-4307, USA Gang Tian Princeton University Princeton, New Jersey 08544, USA Editorial Board George E. Andrews Charles Fefferman Pennsylvania State University Princeton University University Park, PA 16802, USA Princeton, NJ 08544, USA Alexandra Bellow Daniel S. Freed Northwestern University University of Texas at Austin Evanston, IL 60201, USA Austin, TX 78712, USA David Ben-Zvi Mathematics Department Adriano Garsia University of Texas at Austin University of California, San Diego Austin, TX 78712-0257, USA La Jolla, CA 92037, USA Roman Bezrukavnikov I.M. Gelfand Massachusetts Institute of Technology Rutgers University Massachusetts, USA New Brunswick, NJ 08903, USA Sara C. Billey Department of Mathematics Mark Hovey University of Washington Wesleyan University Seattle, WA 98195-4350, USA Middletown, CT 06459, USA Felix Browder Vaughan Jones Hill Center, Bush Campus University of California, Berkeley Rutgers University Berkeley, CA 94720, USA New Brunswick, NJ 08903, USA Luis Caffarelli Gil Kalai RLM 8.100 Institute of Mathematics University of Texas at Austin Hebrew University, Givat Ram Austin, TX 78712, USA Jerusalem 91904, Israel Alain Connes Ludmil Katzarkov Institut Hautes Études Scientifiques University of California, Irvine 35, Rue de Chartres Irvine, CA 92697, USA 91440 Bûres-sur-Yvette, France Andreas Dress Takahiro Kawai Research Institute for Mathematical Sciences Postfach 8640 Kyoto University Universität Bielefeld D-4800 Bielefeld 1, Germany Kitashirakawa, Sakyo-ku Kyoto 60601, Japan Kenneth Falconer H. -
Comments on [5], [14], [22], [30], and [31]
Comments on [5], [14], [22], [30], and [31] Donald L. Burkholder Department of Mathematics University of Illinois Urbana, IL 61801 [email protected] During the two decades after the publication of the pi- oneering papers Proof of the ergodic theorem and Proof of the quasi-ergodic hypothesis by Birkhoff [Bi] and von Neumann [vN], but before the publication of A general ergodic theorem by Calder´onin 1953, there were new proofs and generalizations of the results of Birkhoff and von Neumann given by many mathematicians. These in- clude, among others, Khintchine, Hopf, Kolmogorov, F. Riesz, Yoshida, Kakutani, Wiener, Dunford, Pitt, Doob, and Zygmund. Much of this substantial body of work is referenced in [Ho2], [Ka], and [Kr]. Explorations in many new directions have continued since then and have greatly expanded the connections and applications of er- godic theory to other parts of mathematics and beyond. Calder´on's1953 paper [5] played an important role in this development. In the paper, Calder´onstudies the behavior of averages of the form 1 Z F (gx) dg: (1) Nt j j Nt Here (Nt)t>0 is a family of compact symmetric neigh- 1 borhoods of the identity of a locally compact group G of one-to-one measure-preserving transformations g on a measure space E of finite measure, Nt is the left in- j j variant Haar measure of Nt with dg denoting an element of this measure, and F is an integrable function on E such that (g; x) F (gx) has appropriate measurabil- ity. It is not assumed7! that G is abelian but it is as- sumed that G is ergodic in the sense that there exists a family (Nt)t>0 as above satisfying NtNs Nt+s and ⊂ N2t < α Nt with the choice of the constant α not de- pendingj j onj tj. -
Improving IMS Meetings
Volume 37 • Issue 9 IMS Bulletin November 2008 Improving IMS Meetings CONTENTS Xuming He writes: Each year, the IMS sponsors, and co-sponsors, a healthy number 1 Improving IMS Meetings of conferences, meetings, workshops or symposiums in probability and statistics. The IMS is a major sponsor of the Joint Statistical Meetings (JSM), each August in North 2 Members’ News: Jeff Wu; Sally Morton America. Every other year (odd years), the IMS Annual Meeting takes place at JSM: the 2009 IMS Annual Meeting will be embedded in the JSM to be held in Washington Journal News 3 DC, August 1–6, 2009. Other significant meetings to take place next year include 4 Meetings: Workshop for the ENAR/IMS Spring Meetings in San Antonio, Texas (March 15–19), the WNAR/ Women in Probability; IMS Meeting in Portland, Oregon (June 14–19), and the first IMS Asia Pacific Rim International Workshop in Meeting in Seoul, Korea (June 28 – July 1). Meeting announcements can be found in Applied Probability each issue of the Bulletin, and also online at http://www.imstat.org/meetings/. 6 Contribute Even with rising costs, enthusiasm about meetings never diminishes in our profes- 7 IMS Awards sion, because we all benefit from face-to-face meetings. We exchange ideas, disseminate new research outcomes, and promote collaborations. So how do we maximize our Laha Awardees 8 rewards each time we take several days off and travel to speak at, or attend, a meeting? 9 Making the Most of Almost all of our meetings are organized by volunteers in our own societies. -
Ergodic Theory in the Perspective of Functional Analysis
Ergodic Theory in the Perspective of Functional Analysis 13 Lectures by Roland Derndinger, Rainer Nagel, GÄunther Palm (uncompleted version) In July 1984 this manuscript has been accepted for publication in the series \Lecture Notes in Mathematics" by Springer-Verlag. Due to a combination of unfortunate circumstances none of the authors was able to perform the necessary ¯nal revision of the manuscript. TÄubingen,July 1987 1 2 I. What is Ergodic Theory? The notion \ergodic" is an arti¯cial creation, and the newcomer to \ergodic theory" will have no intuitive understanding of its content: \elementary ergodic theory" neither is part of high school- or college- mathematics (as does \algebra") nor does its name explain its subject (as does \number theory"). Therefore it might be useful ¯rst to explain the name and the subject of \ergodic theory". Let us begin with the quotation of the ¯rst sentence of P. Walters' introductory lectures (1975, p. 1): \Generally speaking, ergodic theory is the study of transformations and flows from the point of view of recurrence properties, mixing properties, and other global, dynamical, properties connected with asymptotic behavior." Certainly, this de¯nition is very systematic and complete (compare the beginning of our Lectures III. and IV.). Still we will try to add a few more answers to the question: \What is Ergodic Theory ?" Naive answer: A container is divided into two parts with one part empty and the other ¯lled with gas. Ergodic theory predicts what happens in the long run after we remove the dividing wall. First etymological answer: ergodhc=di±cult. Historical answer: 1880 - Boltzmann, Maxwell - ergodic hypothesis 1900 - Poincar¶e - recurrence theorem 1931 - von Neumann - mean ergodic theorem 1931 - Birkho® - individual ergodic theorem 1958 - Kolmogorov - entropy as an invariant 1963 - Sinai - billiard flow is ergodic 1970 - Ornstein - entropy classi¯es Bernoulli shifts 1975 - Akcoglu - individual Lp-ergodic theorem Naive answer of a physicist: Ergodic theory proves that time mean equals space mean.