DOCSLIB.ORG

Lecture Notes in Mathematics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Lecture Notes in Mathematics For information about Vols. 1-1145 please contact your bookseller Vol. 1173: H. DeHs, M. Knebusch, Locally Semialgebraic Spaces. XVI, or Springer-Verlag. 329 pages. 1g95, Vol. 1146: 5eminaire d'Aigebre Paul Dubreil et Marie-Paula Malliavin. Vol. 1174: Categories in Continuum Physics, Buffalo 1982. Seminar. Proceedings, 1g63-1984. Edite par M.-P. Malliavin. IV, 420 pages. Edited by F.W. Lawvere and S.H. Schanuel. V, t26 pages. t986. 1985. Vol. 1175: K. Mathiak, Valuations of Skew Fields and Projective Vol. 1147: M. Wschebor, Surfaces Aleatoires. VII, 11t pages. 1985. Hjelmslev Spaces. VII, 116 pages. 1986. Vol. 1t48: Mark A. Kon, Probability Distributions in Quantum Statistical Vol. 1176: R.R. Bruner, J.P. May, J.E. McClure, M. Steinberger, Mechanics. V, 12t pages. 1985. Hoo Ring Spectra and their Applications. VII, 388 pages. 1988. Vol. 1149: Universal Algebra and Lattice Theory. Proceedings, 1984. Vol. 1t77: Representation Theory I. Finite Dimensional Algebras. Edited by S. D. Comer. VI, 282 pages. 1985. Proceedings, t984. Edited by V. Dlab, P. Gabriel and G. Michler. XV, 340 pages. 1g86. Vol. 1150: B. Kawohl, Rearrangements and Convexity of Level Sets in Vol. 1178: Representation Theory II. Groups and Orders. Proceed­ PDE. V, 136 pages. 1985. ings, 1984. Edited by V. Dlab, P. Gabriel and G. Michler. XV, 370 Vol 1151: Ordinary and Partial Differential Equations. Proceedings, pages. 1986. 1984. Edited by B.D. Sleeman and R.J. Jarvis. XIV, 357 pages. 1985. Vol. 1179: Shi J .-Y. The Kazhdan-Lusztig Cells in Certain Affine Weyl Vol. 1152: H. Widom, Asymptotic Expansions for Pseudodifferential Groups. X, 307 pages. 1986. Operators on Bounded Domains. V, 150 pages. 1985. Vol. t180: R. Carmona, H. Kesten, J. B. Walsh, Ecole d'Ete de Vol. 1153: Probability in Banach Spaces V. Proceedings, 1984. Edited Probabilites de Saint-Flour XIV - 1984. Edite par P. L. Hennequin. X, by A. Beck, R. Dudley, M. Hahn, J. Kuelbs and M. Marcus. VI, 457 438 pages. 1986. pages. 1985. Vol. t181: Buildings and the Geometry of Diagrams, Como 1984. Vol. 1154: D. S. Naidu, A. K. Rao, Singular Pertubation Analysis of Seminar. Edited by L. Rosati. VII, 277 pages. 1986. Discrete Control Systems. IX, 195 pages. 1985. Vol. 1182: S. Shelah, Around Classification Theory of Models. VII, 279 Vol. 1155: Stability Problems for Stochastic Models. Proceedings, pages. 1986. 1984. Edited by V. V. Kalashnikov and V. M. Zolotarev. VI, 447 Vol. 1183: Algebra, Algebraic Topology and their Interactions. Procee­ pages. t985. dings, 1983. Edited by J.-E. Roos. XI, 396 pages. 1986. Vol. 1t56: Global Differential Geometry and Global Analysis 1984. Vol. 1184: W. Arendt, A. Grabosch, G. Greiner, U. Groh, H.P. Lotz, Proceedings, 1984. Edited by D. Ferus, R.B. Gardner, S. Helgason U. Moustakas, R. Nagel, F. Neubrander, U. Schlotterbeck, One­ and U. Simon. V, 339 pages. 1985. parameter Semigroups of Positive Operators. Edited by R. Nagel. X, 460 pages. 1986. Vol. 1t57: H. Levine, Classifying Immersions into!R4 over Stable Maps 2 of 3-Manifolds into!R • V, 163 pages. 1985. Vol. 1185: Group Theory, Beijing 1984. Proceedings. Edited by Tuan H. F. V, 403 pages. 1986. Vol. 1158: Stochastic Processes - Mathematics and Physics. Procee­ dings, t984. Edited by S. Albeverio, Ph. Blanchard and L. Streit. VI, Vol. 1t86: Lyapunov Exponents. Proceedings, 1984. Edited by L. 230 pages. t986. Arnold and V. Wihstutz. VI, 374 pages. 1986. Vol. t187: Y. Diers, Categories of Boolean Sheaves of Simple Vol. 1159: Schrodinger Operators, Como t984. Seminar. Edited by S. Algebras. VI, t68 pages. 1986. Graffi. VIII, 272 pages. 1986. Vol. 1188: Fonctions de Plusieurs Variables Complexes V. Seminaire, Vol. 1160: J.-C. van der Meer, The Hamiltonian Hopi Bifurcation. VI, 1979--85. Edite par Fran<;;ois Norguet. VI, 306 pages. 1986. t15 pages. 1985. Vol. 1t89: J. Lukes, J. Maly, L. Zajicek, Fine Topology Methods in Real Vol. 1161: Harmonic Mappings and Minimal Immersions, Montecatini Analysis and Potential Theory. X, 472 pages. t986. 1984. Seminar. Edited by E. Giusti. VII, 285 pages. Jg85. Vol. 1190: Optimization and Related Fields. Proceedings, 1984. Vol. 1162: S.J.L. van Eijndhoven, J. de Graaf, Trajectory Spaces, Edited by R. Conti, E. De Giorgi and F. Giannessi. VIII, 419 pages. IV, 272 pages. Generalized Functions and Unbounded Operators. t986. 1985. Vol. 1191: A.R. Its, V.Yu. Novokshenov, The lsomonodromic Defor­ Vol. 1163: Iteration Theory and its Functional Equations. Proceedings, Equations. IV, 313 pages. 1984. Edited by R. Liedl, L. Reich and Gy. Targonski. VIII, 23t pages. mation Method in the Theory of Painleve 1986. 1985. Vol. 1t92: Equadiff 6. Proceedings, t985. Edited by J. Vosmansky and Vol. 1164: M. Meschiari, J.H. Rawnsley, S. Salamon, Geometry M. Zlamal. XXIII, 404 pages. 1986. Seminar "Luigi Bianchi" II - 1984. Edrted by E. Vesentini. VI, 224 pages. 1985. Vol. t193: Geometrical and Statistical Aspects of Probability in Banach Spaces. Proceedings, t985. Edited by X. Femique, B. Vol. 1165: Seminar on Deformations. Proceedings, 1g82f84. Edited Heinkel, M.B. Marcus and P.A. Meyer. IV, 128 pages. t986. by J. -lawrynowicz. IX, 331 pages. 1985. Vol. 1194: Complex Analysis and Algebraic Geometry. Proceedings, Vol. 1166: Banach Spaces. Proceedings, 1984. Edited by N. Kalton 1985. Edited by H. Grauert. VI, 235 pages. t986. VI, 1g9 pages. 1g85. and E. Saab. Vol.t195: J.M. Barbosa, A. G. Colares, Mtnimal Surfaces in IRa X, t24 Vol. 1167: Geometry and Topology. Proceedings, 1983-84. Edited by pages. t986. J. Alexander and J. Harer. VI, 292 pages. 1985. Vol. tt96: E. Casas-Aivero, S. Xamb6-Descamps, The Enumerative Vol. 1168: S.S. Agaian, Hadamard Matric9s and their Applications. Ill, Theory of Conics after Halphen. IX, t30 pages. t986. 227 pages. 1985. Vol. t197: Ring Theory. Proceedings, t985. Edited by F.M.J. van Vol. 1169: W.A. Light, E.W. Cheney, Approximation Theory in Tensor Oystaeyen. V, 231 pages. t986. Product Spaces. VII, 157 pages. 1g95, Vol. 1198: Seminaire d'Analyse, P. Lelong- P. Dolbeau~- H. Skoda. Vol. 1170: B.S. Thomson, Real Functions. VII, 229 pages. 1985. Seminar 1983/84. X, 260 pages. 1986. Vol. 1171: Polynomes Orthogonaux et Applications. Proceedings, Vol. 1199: Analytic Theory of Continued Fractions II. Proceedings, 1984. Edite par C. Brezinski, A. Draux, A. P. Magnus, P. Maroni etA. 1985. Edited by W.J. Thron. VI, 299 pages. 1986. Ronveaux. XXXVII, 584 pages. 1985. Vol. 1200: V. D. Milman, G. Schechtman, Asymptotic Theory of Finite Vol. 1172: Algebraic Topology, Gottingen 1984. Proceedings. Edited Dimensional Normed Spaces. With an Appendix by M. Gromov. VIII, by L. Smith. VI, 209 pages. 1985. 156 pages. 1986. continued on page 45 7 Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann 1370 G. Carlsson R.L. Cohen H. R. Miller D.C. Ravenel (Eds.) Algebraic Topology Proceedings of an International Conference held in Arcata, California, July 27 - August 2, 1986 Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Editors Gunnar Carlsson Princeton University, Department of Mathematics Princeton, NJ 08544, USA Ralph Cohen Stanford University, Department of Mathematics Stanford, CA 94305, USA Haynes Miller Massachusetts Institute of Technology, Room 2-237 Cambridge, MA 02139, USA Douglas Ravenel University of Rochester, Department of Mathematics Rochester, NY 14620, USA Mathematics Subject Classification (1980): 55-06 ISBN 3-540-51118-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-51118-0 Springer-Verlag New York Berlin Heidelberg Thiswork is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1989 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 PREFACE An international conference on algebraic topology was held July 27 to August 2, 1986, at Humboldt State University in Arcata, California, prior to ICM86 in Berkeley. The conference served in part to celebrate the silver jubilee of the journal Topology, and the sixtieth birthday of Edgar H. Brown. It was supported by a grant from the National Science Foundation and a bequest from Pergammon Press, publishers of Topology. The Proceedings contain papers submitted by conference participants. All were refereed, and we take this occasion to thank the referees. We would also like to thank the staff of the Jolly Giant Conference Center for its assistance in organizing this conference. Gunnar Carlsson Ralph Cohen Haynes Miller Doug Ravenel CONTENTS James, I.M. Topology: Past. Present and Future 1 Whitehead, G. The Work of Edgar H. Brown, Jr. in Topology 10 Adem, A. Homology Representations of Finite Transformation Groups 15 Anick. D.J. Homotopy Exponents for Spaces of Category Two 24 Bendersky, M. On the Complex Bordism of Classifying Davis. D.M. Spaces 53 Berrick. A. J. On Equivariant Maps and Immersions of Real Randall. D. Projective spaces 57 Berstein. I. Cogroups which are not Suspensions Harper J.R. 63 Boyer.Ch.P. Instantons and Homotopy Mann. B.M 87 Brown, E.H. Jr. On Real Homotopy Theory Szczarba. R.H. 103 Cohen. F.R. Some Remarks on the Space 1m J Peterson, F.P. 117 Davis. D.M. A New Spectrum Related to 7-connected Mahowald. M. Cobordism 126 Davis. M.W. Aspherical Manifolds without Smooth or PL Hausmann, J.-C. Structure 135 Dwyer. W.G. Homology with Simplicial Coefficients Kan. D.M.
Recommended publications
  • Topological Pattern Recognition for Point Cloud Data∗

    Topological Pattern Recognition for Point Cloud Data∗

    Acta Numerica (2014), pp. 289–368 c Cambridge University Press, 2014 doi:10.1017/S0962492914000051 Printed in the United Kingdom Topological pattern recognition for point cloud data∗ Gunnar Carlsson† Department of Mathematics, Stanford University, CA 94305, USA E-mail: [email protected] In this paper we discuss the adaptation of the methods of homology from algebraic topology to the problem of pattern recognition in point cloud data sets. The method is referred to as persistent homology, and has numerous applications to scientific problems. We discuss the definition and computation of homology in the standard setting of simplicial complexes and topological spaces, then show how one can obtain useful signatures, called barcodes, from finite metric spaces, thought of as sampled from a continuous object. We present several different cases where persistent homology is used, to illustrate the different ways in which the method can be applied. CONTENTS 1 Introduction 289 2 Topology 293 3 Shape of data 311 4 Structures on spaces of barcodes 331 5 Organizing data sets 343 References 365 1. Introduction Deriving knowledge from large and complex data sets is a fundamental prob- lem in modern science. All aspects of this problem need to be addressed by the mathematical and computational sciences. There are various dif- ferent aspects to the problem, including devising methods for (a) storing massive amounts of data, (b) efficiently managing it, and (c) developing un- derstanding of the data set. The past decade has seen a great deal of devel- opment of powerful computing infrastructure, as well as methodologies for ∗ Colour online for monochrome figures available at journals.cambridge.org/anu.
  • Topology and Data

    Topology and Data

    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 46, Number 2, April 2009, Pages 255–308 S 0273-0979(09)01249-X Article electronically published on January 29, 2009 TOPOLOGY AND DATA GUNNAR CARLSSON 1. Introduction An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that the nature of the data we are obtaining is significantly different. For example, it is now often the case that we are given data in the form of very long vectors, where all but a few of the coordinates turn out to be irrelevant to the questions of interest, and further that we don’t necessarily know which coordinates are the interesting ones. A related fact is that the data is often very high-dimensional, which severely restricts our ability to visualize it. The data obtained is also often much noisier than in the past and has more missing information (missing data). This is particularly so in the case of biological data, particularly high throughput data from microarray or other sources. Our ability to analyze this data, both in terms of quantity and the nature of the data, is clearly not keeping pace with the data being produced. In this paper, we will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data.
  • Contemporary Mathematics 220

    Contemporary Mathematics 220

    CONTEMPORARY MATHEMATICS 220 Homotopy Theory via Algebraic Geometry and Group Representations Proceedings of a Conference on Homotopy Theory March 23-27, 1997 Northwestern University Mark Mahowald Stewart Priddy Editors http://dx.doi.org/10.1090/conm/220 Selected Titles in This Series 220 Mark Mahowald and Stewart Priddy, Editors, Homotopy theory via algebraic geometry and group representations, 1998 219 Marc Henneaux, Joseph Krasil'shchik, and Alexandre Vinogradov, Editors, Secondary calculus and cohomological physics, 1998 218 Jan Mandel, Charbel Farhat, and Xiao-Chuan Cai, Editors, Domain decomposition methods 10, 1998 217 Eric Carlen, Evans M. Harrell, and Michael Loss, Editors, Advances in differential equations and mathematical physics, 1998 216 Akram Aldroubi and EnBing Lin, Editors, Wavelets, multiwavelets, and their applications, 1998 215 M. G. Nerurkar, D.P. Dokken, and D. B. Ellis, Editors, Topological dynamics and applications, 1998 214 Lewis A. Coburn and Marc A. Rieffel, Editors, Perspectives on quantization, 1998 213 Farhad Jafari, Barbara D. MacCluer, Carl C. Cowen, and A. Duane Porter, Editors, Studies on composition operators, 1998 212 E. Ramirez de Arellano, N. Salinas, M. V. Shapiro, and N. L. Vasilevski, Editors, Operator theory for complex and hypercomplex analysis, 1998 211 J6zef Dodziuk and Linda Keen, Editors, Lipa's legacy: Proceedings from the Bers Colloquium, 1997 210 V. Kumar Murty and Michel Waldschmidt, Editors, Number theory, 1998 209 Steven Cox and Irena Lasiecka, Editors, Optimization methods in partial differential equations, 1997 208 MichelL. Lapidus, Lawrence H. Harper, and Adolfo J. Rumbos, Editors, Harmonic analysis and nonlinear differential equations: A volume in honor of Victor L. Shapiro, 1997 207 Yujiro Kawamata and Vyacheslav V.
  • On Beta Elements in the Adams-Novikov Spectral Sequence

    On Beta Elements in the Adams-Novikov Spectral Sequence

    Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 On β-elements in the Adams-Novikov spectral sequence Hirofumi Nakai and Douglas C. Ravenel Dedicated to Professor Takao Matumoto on his sixtieth birthday Abstract In this paper we detect invariants in the comodule consisting of β-elements over the Hopf algebroid (A(m + 1);G(m + 1)) defined in[Rav02], and we show that some related Ext groups vanish below a certain dimension. The result obtained here will be extensively used in [NR] to extend the range of our knowledge for π∗(T (m)) obtained in[Rav02]. Contents 1. Introduction ................ 1 2. The construction of Bm+1 ............. 5 3. Basic methods for finding comodule primitives ........ 8 4. 0-primitives in Bm+1 .............. 11 5. 1-primitives in Bm+1 .............. 13 6. j-primitives in Bm+1 for j > 1 ........... 16 7. Higher Ext groups for j = 1 ............ 20 Appendix A. Some results on binomial coefficients ........ 21 Appendix B. Quillen operations on β-elements ........ 24 References ................. 26 1. Introduction In this paper we describe some tools needed in the method of infinite descent, which is an approach to finding the E2-term of the Adams-Novikov spectral sequence converging to the stable homotopy groups of spheres. It is the subject of [Rav86, Chapter 7], [Rav04, Chapter 7] and [Rav02]. We begin by reviewing some notation. Fix a prime p. Recall the Brown-Peterson spectrum BP . Its homotopy groups and those of BP ^ BP are known to be polynomial algebras π∗(BP ) = Z(p)[v1; v2 :::] and BP∗(BP ) = BP∗[t1; t2 :::]: In [Rav86, Chapter 6] the second author constructed intermediate spectra 0 ::: S(p) = T (0) / T (1) / T (2) / T (3) / / BP with T (m) is equivalent to BP below the dimension of vm+1.
  • Mathematics People, Volume 51, Number 6

    Mathematics People, Volume 51, Number 6

    Mathematics People 2004–2005 AMS Centennial Fellowships Awarded The AMS has awarded two Centennial Fellowships for 2004–2005. The recipients are JINHO BAIK of the University of Michigan and NITU KITCHLOO of Johns Hopkins Univer- sity. The amount of each fellowship is $60,000. The Cen- tennial Fellows also receive an expense allowance of $1,700 and a complimentary Society membership for one year. Jinho Baik Jinho Baik received his Ph.D. in 1999 from the Courant Institute of Mathematical Sciences, New York University, under the direction of Percy Deift. He was a Veblen Jinho Baik Nitu Kitchloo Instructor at Princeton University and the Institute for Advanced Study during 1999–2002, and an assistant pro- loop groups, and Kac-Moody groups. His current interests fessor at Princeton University during 2002–2003. Since involve studying the topology of symplectomorphism 2002 he has been an assistant professor at the University groups. He is also working on the topology of certain in- of Michigan. finite loop spaces arising from spectra related to complex Baik has been applying methods from integrable sys- cobordism. tems to probability, combinatorics, and statistical physics. Please note: Information about the competition for the In particular, asymptotic problems on random matrices, 2005–2006 AMS Centennial Fellowships will be published random permutations and orthogonal polynomials, and in the “Mathematics Opportunities” section of an upcom- their interrelations have been his main research interest. ing issue of the Notices. He plans to use the fellowship to visit the Courant In- stitute of Mathematical Sciences. —Allyn Jackson Nitu Kitchloo Nitu Kitchloo received his Ph.D.
  • ON BEYOND HATCHER! in the Fall Semester of 2012, Constantin

    ON BEYOND HATCHER! in the Fall Semester of 2012, Constantin

    ON BEYOND HATCHER! ERIC PETERSON In the Fall semester of 2012, Constantin Teleman encouraged me to run a short seminar which would teach attendees about advanced algebraic topology. UC-Berkeley runs a pair of graduate courses in algebraic topology which more or less go through Allen Hatcher’s Algebraic Topology, but there is a long road from there to the forefront of the field. Getting students research-ready inside of four seminar talks is an impossible task, of course, and so instead my goal is to sketch a picture of some of the major components of the field, so that students know enough of the flavor of the subject to at least identify whether they’re intrigued by its questions and its methods, and then further to know where to look to learn more. To reinforce the impression that there’s a lot going on in algebraic topology, each day will be very distinctly flavored from the others, and no day will require absolutely understanding any day that came before it. In the same vein, few things will be completely proven, but I hope to at least define the topological things under discussion. Some knowledge from related fields (algebraic geometry, primarily) will be assumed without hesitation. Each of these talks is meant to last roughly 50 minutes. The reader should additionally be warned that, due to my limited worldliness, these notes are bent sharply toward what I consider interesting. This means that they are highly algebraic, and they make no mention of exciting geometric things in stable homotopy theory — like string topology, for one example of many.
  • Realising Unstable Modules As the Cohomology of Spaces and Mapping Spaces

    Realising Unstable Modules As the Cohomology of Spaces and Mapping Spaces

    Realising unstable modules as the cohomology of spaces and mapping spaces G´eraldGaudens and Lionel Schwartz UMR 7539 CNRS, Universit´eParis 13 LIA CNRS Formath Vietnam Abstract This report discuss the question whether or not a given unstable module is the mod-p cohomology of a space. One first discuss shortly the Hopf invariant 1 problem and the Kervaire invariant 1 problem and give their relations to homotopy theory and geometric topology. Then one describes some more qualitative results, emphasizing the use of the space map(BZ=p; X) and of the structure of the category of unstable modules. 1 Introduction Let p be a prime number, in all the sequel H∗X will denote the the mod p singular cohomology of the topological space X. All spaces X will be supposed p-complete and connected. The singular mod-p cohomology is endowed with various structures: • it is a graded Fp-algebra, commutative in the graded sense, • it is naturally a module over the algebra of stable cohomology operations which is known as the mod p Steenrod algebra and denoted by Ap. The Steenrod algebra is generated by element Sqi of degree i > 0 if p = 2, β and P i of degree 1 and 2i(p − 1) > 0 if p > 2. These elements satisfy the Adem relations. In the mod-2 case the relations write [a=2] X b − t − 1 SqaSqb = Sqa+b−tSqt a − 2t 0 There are two types of relations for p > 2 [a=p] X (p − 1)(b − t) − 1 P aP b = (−1)a+t P a+b−tP t a − pt 0 1 for a; b > 0, and [a=p] [(a−1)=p] X (p − 1)(b − t) X (p − 1)(b − t) − 1 P aβP b = (−1)a+t βP a+b−tP t+ (−1)a+t−1 P a+b−tβP t a − pt a − pt − 1 0 0 for a; b > 0.
  • Arxiv:1909.13379V2 [Math.AT] 10 Jul 2021 6

    Arxiv:1909.13379V2 [Math.AT] 10 Jul 2021 6

    THE TELESCOPE CONJECTURE AT HEIGHT 2 AND THE TMF RESOLUTION A. BEAUDRY, M. BEHRENS, P. BHATTACHARYA, D. CULVER, AND Z. XU Abstract. Mahowald proved the height 1 telescope conjecture at the prime 2 as an application of his seminal work on bo-resolutions. In this paper we study the height 2 telescope conjecture at the prime 2 through the lens of tmf-resolutions. To this end we compute the structure of the tmf-resolution for a specific type 2 complex Z. We find that, analogous to the height 1 case, the E1-page of the tmf-resolution possesses a decomposition into a v2- periodic summand, and an Eilenberg-MacLane summand which consists of bounded v2-torsion. However, unlike the height 1 case, the E2-page of the tmf-resolution exhibits unbounded v2-torsion. We compare this to the work of Mahowald-Ravenel-Shick, and discuss how the validity of the telescope con- jecture is connected to the fate of this unbounded v2-torsion: either the un- bounded v2-torsion kills itself off in the spectral sequence, and the telescope conjecture is true, or it persists to form v2-parabolas and the telescope conjec- ture is false. We also study how to use the tmf-resolution to effectively give low dimensional computations of the homotopy groups of Z. These computations allow us to prove a conjecture of the second author and Egger: the E(2)-local Adams-Novikov spectral sequence for Z collapses. Contents 1. Introduction2 2. Background 11 3. The good/evil decomposition of the E1-term 15 4. Morava stabilizer groups and algebras 20 5.
  • RESEARCH STATEMENT My Research Lies in the Area of Stable Homotopy Theory. This Is the Part of Homotopy Theory Which Studies

    RESEARCH STATEMENT My Research Lies in the Area of Stable Homotopy Theory. This Is the Part of Homotopy Theory Which Studies

    RESEARCH STATEMENT D. CULVER 1. INTRODUCTION My research lies in the area of stable homotopy theory. This is the part of homotopy theory which studies invariants of spaces that are stable under the suspension functor. My research primarily focuses on studying pat- terns that arise in the stable homotopy groups of finite complexes, with a particular emphasis on spheres. I am especially interested in computing infinite periodic families in the stable homotopy groups of spheres. Most of my research can be described as studying these families by using Adams type spectral sequences based on interesting ring spectra. The primary case of interest is the ring spectrum tmf of topological modular forms. I have also become interested in studying the analogous problems in motivic and equi- variant homotopy theory. In addition, I am computing invariants of ring spectra such as topological Hochschild homology, topological cyclic homology, and algebraic K-theory. 2. BACKGROUND A natural tool for approaching the stable homotopy groups of spheres is the classical Adams spectral sequences (ASS). This is a spectral sequence of the form Exts,t , 0 , A (Fp Fp ) = πt s S Zp ∗ ) − ⊗ where p is a prime number. This spectral sequence is particularly good for computing the stable homotopy groups in a finite range, but obscures infinite patterns in the stable homotopy groups of spheres. However, given any ring spectrum E 1, one can always construct an ASS based on E. Under suitable conditions, this spectral sequence takes the form 0 ExtE E (E , E ) = π LE S . ∗ ∗ ∗ ∗ 0 ) Here E E = π (E E) and LE S is a certain localization of the sphere.
  • A Combinatorial Model for the Known Bousfield Classes

    A Combinatorial Model for the Known Bousfield Classes

    A combinatorial model for the known Bousfield classes N. P. STRICKLAND We give a combinatorial construction of an ordered semiring A, and show that it can be identified with a certain subquotient of the semiring of p•local Bousfield classes, containing almost all of the classes that have previously been named and studied. This is a convenient way to encapsulate most of the known results about Bousfield classes. 1 Introduction Fix a prime p, and let L denote the set of Bousfield classes in the p•local stable category (which can be regarded as an ordered semiring in a natural way). This note is an attempt to organise many of the known results about the structure of L in a more coherent way. One of the main open questions about L is Ravenel’s Telescope Conjecture (TC). The statement will be recalled in Remark 4.2. Many people suspect that TC is false, but this has still not been proven. We will define an ordered semiring L which is, in a certain sense, the largest quotient of L in which TC becomes true. We will then define (in an explicit, combinatorial way) another ordered semiring A and a function φ: A→L arXiv:1608.08533v2 [math.AT] 8 Jul 2018 such that the composite φ π A −→L −→ L is an injective homomorphism of ordered semirings. (However, φ itself is probably not a semiring homomorphism, unless TC holds.) For almost all elements x ∈L that have been named and studied, we have π(x) ∈ πφ(A). Thus, A is a good model for the known part of L.
  • Three Keynote Lectures Delivered By: ▫ Prof. Gunnar Carlsson, Stanford

    Three Keynote Lectures Delivered By: ▫ Prof. Gunnar Carlsson, Stanford

    Three keynote lectures delivered by: . Prof. Gunnar Carlsson, Stanford University, USA . Prof. Samad Hedayat, University of Illinois at Chicago, USA . Prof. Edriss Titi, Texas A&M University, USA Gunnar Carlsson: Gunnar Carlsson is a Professor of Mathematics at Stanford University. A highly influential mathematician, he is most well-known for his proof of the Segal Burnside Ring conjecture and for his work on applied algebraic topology, especially topological data analysis. He was a Professor at Princeton University before joining the Stanford Mathematics Department in 1991 where he also served as a Chairman. He was invited at the International Congress of Mathematicians to speak about his work in homotopy theory. More recently his work on persistent homology and topological data analysis has opened up new fields of research with a large number of followers. Whether through his numerous publications or keynote addresses, Gunnar Carlsson has been having an enormous influence on present day mathematical research and its applications to as far afield as cancer treatment. He has co-authored two books on mathematical education. He served as a consultant to the education boards of the State of Texas and State of California concerning the Content Standards in Mathematics and currently is a member of Texas Instruments California Advisory Board on K-12 education. He is the recipient of many awards and grants including the Alfred P. Sloan Fellowship, NSF grants, DARPA and Air Force Office of Scientific Research grants. He is the editor of several internationally respected mathematics journals in topology and algebra. More information about Professor Gunnar Carlsson can be found at http://math.stanford.edu/~gunnar/ Samad Hedayat Professor Samad Hedayat is a UIC Distinguished Professor at Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago (UIC).
  • Gunnar Carlsson Professor, Department of Mathematics Stanford University

    Gunnar Carlsson Professor, Department of Mathematics Stanford University

    Colorado State University’s Information Science and Technology Center (ISTeC) presents two lectures by Gunnar Carlsson Professor, Department of Mathematics Stanford University ISTeC Distinguished Lecture In conjunction with the Mathematics Department, the Electrical and Computer Engineering Department, and the Computer Science Department ”The Shape of Data” Monday, October 20, 2014 Reception with refreshments: 10:30 am Lecture: 11:00 am – 12:00 noon Location: LSC Grey Rock • Mathematics Department, Electrical and Computer Engineering Department , and Computer Science Department Special Seminar Sponsored by ISTeC “The Algebraic Geometry of Persistence Barcodes” Tuesday, October 21, 2014 Lecture: 1:00 – 2:00 pm Reception with refreshments: 2:00 pm Location: 221 TILT ISTeC (Information Science and Technology Center) is a university-wide organization for promoting, facilitating, and enhancing CSU’s research, education, and outreach activities pertaining to the design and innovative application of computer, communication, and information systems. For more information please see ISTeC.ColoState.EDU. Abstracts The Shape of Data “Big data” is a term which describes many varied problems in the management of data. These relate to storage, query ca- pability, analysis, and numerous other aspects of the general problem of how to make most effective use of the enormous amounts of data currently being gathered. We will talk about a collection of recently developed methods for the analysis of large, high dimensional, and most importantly, complex data sets. These methods us the mathematical notion of shape, as encoded in topological methods, as a new tool in data analysis. We will discuss these methods, with numerous examples. The Algebraic Geometry of Persistence Barcodes Persistent homology associates to a finite metric space an invariant called a persistence barcode, which often allows one to infer the homology of and underlying space from which the finite sample is obtained.