Axiomatic Enriched and Motivic Homotopy Theory

The present volume is based on a number of the lectures given during the workshop. The ASI was the opening workshop of the four month programme "New Contexts for Stable Homotopy Theory" which explored several themes in greater depth.

Axiomatic  Enriched and Motivic Homotopy Theory

The NATO Advanced Study Institute "Axiomatic, enriched and rna tivic homotopy theory" took place at the Isaac Newton Institute of Mathematical Sciences, Cambridge, England during 9-20 September 2002. The Directors were J.P.C.Greenlees and I.Zhukov; the other or ganizers were P.G.Goerss, F.Morel, J.F.Jardine and V.P.Snaith. The title describes the content well, and both the event and the contents of the present volume reflect recent remarkable successes in model categor ies, structured ring spectra and homotopy theory of algebraic geometry. The ASI took the form of a series of 15 minicourses and a few extra lectures, and was designed to provide background, and to bring the par ticipants up to date with developments. The present volume is based on a number of the lectures given during the workshop. The ASI was the opening workshop of the four month programme "New Contexts for Stable Homotopy Theory" which explored several themes in greater depth. I am grateful to the Isaac Newton Institute for providing such an ideal venue, the NATO Science Committee for their funding, and to all the speakers at the conference, whether or not they were able to contribute to the present volume. All contributions were refereed, and I thank the authors and referees for their efforts to fit in with the tight schedule. Finally, I would like to thank my coorganizers and all the staff at the Institute for making the ASI run so smoothly. J.P.C.GREENLEES.

Motivic Homotopy Theory and Refined Enumerative Geometry

MR2175638 [16] Fabien Morel, On the motivic π0 of the sphere spectrum, Axiomatic, enriched and motivic homotopy theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 131, Kluwer Acad. Publ., Dordrecht, 2004, pp.

Motivic Homotopy Theory and Refined Enumerative Geometry

This volume contains the proceedings of the Workshop on Motivic Homotopy Theory and Refined Enumerative Geometry, held from May 14–18, 2018, at the Universität Duisburg-Essen, Essen, Germany. It constitutes an accessible yet swift introduction to a new and active area within algebraic geometry, which connects well with classical intersection theory. Combining both lecture notes aimed at the graduate student level and research articles pointing towards the manifold promising applications of this refined approach, it broadly covers refined enumerative algebraic geometry.

Stable Homotopy Around the Arf Kervaire Invariant

[91] P.G. Goerss: (Pre-)Sheaves of spectra over the moduli stack of formal group laws; Axiomatic, Enriched and Motivic Homotopy Theory (ed. J.P.C. Green- lees) NATO Science Series II #131 (2004) 101–131. [92] P.G. Goerss and J.F. ...

Stable Homotopy Around the Arf Kervaire Invariant

Were I to take an iron gun, And ?re it o? towards the sun; I grant ‘twould reach its mark at last, But not till many years had passed. But should that bullet change its force, And to the planets take its course, ‘Twould never reach the nearest star, Because it is so very far. from FACTS by Lewis Carroll [55] Let me begin by describing the two purposes which prompted me to write this monograph. This is a book about algebraic topology and more especially about homotopy theory. Since the inception of algebraic topology [217] the study of homotopy classes of continuous maps between spheres has enjoyed a very exc- n n tional, central role. As is well known, for homotopy classes of maps f : S ?? S with n? 1 the sole homotopy invariant is the degree, which characterises the homotopy class completely. The search for a continuous map between spheres of di?erent dimensions and not homotopic to the constant map had to wait for its resolution until the remarkable paper of Heinz Hopf [111]. In retrospect, ?nding 3 an example was rather easy because there is a canonical quotient map from S to 3 1 1 2 theorbitspaceofthe freecircleactionS /S =CP = S .

Handbook of Homotopy Theory

Strong convergence in the motivic Adams spectral sequence. arXiv:1901.03399, 2018. ... A comparison of motivic and classical stable homotopy theories. ... Axiomatic, enriched and motivic homotopy theory, NATO Sci. Ser. II Math. Phys.

Handbook of Homotopy Theory

The Handbook of Homotopy Theory provides a panoramic view of an active area in mathematics that is currently seeing dramatic solutions to long-standing open problems, and is proving itself of increasing importance across many other mathematical disciplines. The origins of the subject date back to work of Henri Poincaré and Heinz Hopf in the early 20th century, but it has seen enormous progress in the 21st century. A highlight of this volume is an introduction to and diverse applications of the newly established foundational theory of ¥ -categories. The coverage is vast, ranging from axiomatic to applied, from foundational to computational, and includes surveys of applications both geometric and algebraic. The contributors are among the most active and creative researchers in the field. The 22 chapters by 31 contributors are designed to address novices, as well as established mathematicians, interested in learning the state of the art in this field, whose methods are of increasing importance in many other areas.

Proceedings Of The International Congress Of Mathematicians 2010 Icm 2010 In 4 Volumes Vol I Plenary Lectures And Ceremonies Vols Ii iv Invited Lectures

Global methods in homotopy theory. ... The connection between May's axioms for a triangulated tensor product and Happel's description of the derived ... In Axiomatic, enriched and motivic homotopy theory, volume 131 of NATO Sci. Ser.

Proceedings Of The International Congress Of Mathematicians 2010  Icm 2010   In 4 Volumes    Vol  I  Plenary Lectures And Ceremonies  Vols  Ii iv  Invited Lectures

ICM 2010 proceedings comprises a four-volume set containing articles based on plenary lectures and invited section lectures, the Abel and Noether lectures, as well as contributions based on lectures delivered by the recipients of the Fields Medal, the Nevanlinna, and Chern Prizes. The first volume will also contain the speeches at the opening and closing ceremonies and other highlights of the Congress.

Bousfield Classes and Ohkawa s Theorem

Johnson, D.C., Yosimura, Z.: Torsion in Brown-Peterson homology and Hurewicz homomorphisms. ... Jouanolou, J.P.: Une Suite exacte de Mayer-Vietoris en K-Theorie algebrique. ... Axiomatic, Enriched and Motivic Homotopy Theory.

Bousfield Classes and Ohkawa s Theorem

This volume originated in the workshop held at Nagoya University, August 28–30, 2015, focusing on the surprising and mysterious Ohkawa's theorem: the Bousfield classes in the stable homotopy category SH form a set. An inspiring, extensive mathematical story can be narrated starting with Ohkawa's theorem, evolving naturally with a chain of motivational questions: Ohkawa's theorem states that the Bousfield classes of the stable homotopy category SH surprisingly forms a set, which is still very mysterious. Are there any toy models where analogous Bousfield classes form a set with a clear meaning? The fundamental theorem of Hopkins, Neeman, Thomason, and others states that the analogue of the Bousfield classes in the derived category of quasi-coherent sheaves Dqc(X) form a set with a clear algebro-geometric description. However, Hopkins was actually motivated not by Ohkawa's theorem but by his own theorem with Smith in the triangulated subcategory SHc, consisting of compact objects in SH. Now the following questions naturally occur: (1) Having theorems of Ohkawa and Hopkins-Smith in SH, are there analogues for the Morel-Voevodsky A1-stable homotopy category SH(k), which subsumes SH when k is a subfield of C?, (2) Was it not natural for Hopkins to have considered Dqc(X)c instead of Dqc(X)? However, whereas there is a conceptually simple algebro-geometrical interpretation Dqc(X)c = Dperf(X), it is its close relative Dbcoh(X) that traditionally, ever since Oka and Cartan, has been intensively studied because of its rich geometric and physical information. This book contains developments for the rest of the story and much more, including the chromatics homotopy theory, which the Hopkins–Smith theorem is based upon, and applications of Lurie's higher algebra, all by distinguished contributors.

A Handbook of Model Categories

In: Axiomatic, enriched and motivic homotopy theory. vol. 131. NATO Sci. Ser. II Math. Phys. Chem. Kluwer Acad. Publ., Dordrecht, pp. 3–28 (2004) 138. Elmendorf, A.D., Kriz, I., Mandell, M.A., May, J.P.: Rings, modules, and algebras in ...

A Handbook of Model Categories

This book outlines a vast array of techniques and methods regarding model categories, without focussing on the intricacies of the proofs. Quillen model categories are a fundamental tool for the understanding of homotopy theory. While many introductions to model categories fall back on the same handful of canonical examples, the present book highlights a large, self-contained collection of other examples which appear throughout the literature. In particular, it collects a highly scattered literature into a single volume. The book is aimed at anyone who uses, or is interested in using, model categories to study homotopy theory. It is written in such a way that it can be used as a reference guide for those who are already experts in the field. However, it can also be used as an introduction to the theory for novices.

Algebraic Cycles and Motives

[26] F. Morel: On the motivic stable π0 of the sphere spectrum, in Axiomatic, Enriched and Motivic Homotopy Theory, 219–260, J.P.C. Greenlees (ed.), 2004 Kluwer Academic Publishers. [27] F. Morel: Rationalized motivic sphere spectrum ...

Algebraic Cycles and Motives

This 2007 book is a self-contained account of the subject of algebraic cycles and motives.

Mathematical Reviews

The purpose of this note is to prove an analogous formula in the context of oriented pretheories in place of complex cobordisms . The notion of a pretheory was defined by I. Panin [ in Axiomatic , enriched and motivic homotopy theory ...

Mathematical Reviews


St Petersburg Mathematical Journal

2,1 provide a Chern structure on the homology theory E .. * . The trace homomorphisms f ! for ... Riemann - Roch theorems for oriented cohomology , Axiomatic , Enriched , and Motivic Homotopy Theory , NATO Sci . Ser . II Math . Phys .

St  Petersburg Mathematical Journal


Documenta Mathematica

REFERENCES [ PS ] I. Panin ( after I. Panin and A. Smirnov ) , Oriented cohomology theories of algebraic ... in Proceedings of the NATO ASI “ Axiomatic , enriched and motivic homotopy theory " ( Ed . J. P. C. Greenlees ) , NATO Sci .

Documenta Mathematica


American journal of mathematics

Operads and cosimplicial objects : an introduction , Axiomatic , Enriched and Motivic Homotopy Theory : Proceedings of a NATO Advanced Study Institute at the Isaac Newton Institute for Mathematical Sciences , Kluwer Acad .

American journal of mathematics


Pure and Applied Mathematics Quarterly

In : Axiomatic , enriched and motivic homotopy theory , 261-333 , NATO Sci . Ser . II Math . Phys . Chem . , 131 , Kluwer Acad . Publ . , Dordrecht , 2004 . ( 13 ) M. Rost , Construction of splitting varieties , preprint 1998 ...

Pure and Applied Mathematics Quarterly


Annales Scientifiques de L cole Normale Sup rieure

[ 36 ] TOËN B. , Vezzosi G. , From HAG to DAG : derived moduli spaces , in : J.P.C. Greenlees ( Ed . ) , Axiomatic , Enriched and Motivic Homotopy Theory , Proceedings of the NATO Advanced Study Institute , Cambridge , UK ( 9–20 ...

Annales Scientifiques de L   cole Normale Sup  rieure


Mathematics Today

Professor which are homotopy theoretic enrich- Workshop organisers : Rick Jardine and VP Snaith ( Southampton ) ments of ... further applications of the latest ASI Axiomatic , Enriched and Motivic with practising homotopy theorists .

Mathematics Today


American Book Publishing Record

793.73 Axiomatic , enriched , and motivic homotopy theory . 514.24 Ayurveda . 615.538 B B2B e - commerce with WebSphere Commerce Business Edition V5.4 . 658.872028 Babies and bosses . 331.25 Baby boot camp . 242.6431 Baby's first book .

American Book Publishing Record


Peterson s Graduate Programs in the Physical Sciences Mathematics Agricultural Sciences the Environment and Natural Resources 2007

Dwyer , W. G. Localizations , axiomatic , enriched and motivic homotopy theory . In Proceedings of the NATO ASI , pp . 3–28 , ed . J. P. C. Greenless . Kluwer , 2004 . Hind , R. K. Lagrangian spheres in S ? x S. Geom . Funct . Anal .

Peterson s Graduate Programs in the Physical Sciences  Mathematics  Agricultural Sciences  the Environment and Natural Resources 2007

Offers information on entrance and degree requirements, expenses and financial aid, programs of study, and faculty research specialties.