Librería: Readify Books, New Castle, DE, Estados Unidos de America
EUR 13,85
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Añadir al carritoPaperback. Condición: NEW. International Edition, Brand New, ISBN and Cover same but contents similar to U.S. Edition, We ship from multiple Locations including India. Legal to use despite any disclaimer, We ship to PO , APO and FPO adresses in U.S.A. Ship from multiple Locations including India Choose Expedited Shipping for FASTER DELIVERY.Customer Satisfaction Guaranteed.
Idioma: Inglés
Publicado por Basel. Springer/Birkhäuser, 2010
ISBN 10: 3034604998 ISBN 13: 9783034604994
Librería: Druckwaren Antiquariat, Salzwedel, Alemania
Miembro de asociación: GIAQ
EUR 20,00
Cantidad disponible: 1 disponibles
Añadir al carritoBroschiert. Condición: Gut. 274 S. ill. engl. Einbd. etw. berieben, ansonsten sehr gut erh. ISBN: 9783034604994 Sprache: Englisch Gewicht in Gramm: 710.
EUR 3,99
Cantidad disponible: 1 disponibles
Añadir al carritohardcover. Condición: Gut. 454 Seiten; BirkhÃuser - 2. Auf. 1990 : Joseph Silk - gb - 10-49-5 OH-MI7I-YOG3 Sprache: Deutsch Gewicht in Gramm: 1.
Idioma: Inglés
Publicado por Cham, Springer International Publishing/Birkhäuser., 2019
ISBN 10: 3319991744 ISBN 13: 9783319991740
Librería: Universitätsbuchhandlung Herta Hold GmbH, Berlin, Alemania
Original o primera edición
EUR 12,00
Cantidad disponible: 1 disponibles
Añadir al carrito1st edition 2019. 23.5 cm x 15.5 cm. X, 100 p. Softcover. Einband bestoßen, daher Mängelexemplar gestempelt, sonst sehr guter Zustand. Imperfect copy due to slightly bumped cover, apart from this in very good condition. Stamped. Applied and Numerical Harmonic Analysis; Lecture Notes in Applied and Numerical Harmonic Analysis. Sprache: Englisch.
Idioma: Inglés
Publicado por Cham, Springer International Publishing : Imprint: Birkhäuser., 2021
ISBN 10: 3030847993 ISBN 13: 9783030847999
Librería: Universitätsbuchhandlung Herta Hold GmbH, Berlin, Alemania
Original o primera edición
EUR 14,00
Cantidad disponible: 2 disponibles
Añadir al carrito1st ed. 2021. X, 151 p. Softcover. Versand aus Deutschland / We dispatch from Germany via Air Mail. Einband bestoßen, daher Mängelexemplar gestempelt, sonst sehr guter Zustand. Imperfect copy due to slightly bumped cover, apart from this in very good condition. Stamped. Research Perspectives CRM Barcelona, 15. Sprache: Englisch.
Idioma: Inglés
Publicado por Cham, Springer/Birkhäuser., 2018
ISBN 10: 3319727087 ISBN 13: 9783319727080
Librería: Universitätsbuchhandlung Herta Hold GmbH, Berlin, Alemania
EUR 17,00
Cantidad disponible: 1 disponibles
Añadir al carritoVI, 338 p. Softcover. Einband bestoßen, daher Mängelexemplar gestempelt, sonst sehr guter Zustand. Imperfect copy due to slightly bumped cover, apart from this in very good condition. Stamped. Pageoph Topical Volumes. Sprache: Englisch.
Idioma: Inglés
Publicado por Birkhauser Press/ Springer Verlag, 1986, 1986
ISBN 10: 0817632689 ISBN 13: 9780817632687
Librería: Sutton Books, Norwich, VT, Estados Unidos de America
Miembro de asociación: IOBA
EUR 44,30
Cantidad disponible: 1 disponibles
Añadir al carritohbk various paginations dj is shelf wornwith closed tear now in protective sleeve text is excellent as new nb vol 1 only.
Idioma: Inglés
Publicado por Birkhauser Press/ Springer Verlag, 1986, 1986
ISBN 10: 0817632689 ISBN 13: 9780817632687
Librería: Sutton Books, Norwich, VT, Estados Unidos de America
Miembro de asociación: IOBA
EUR 44,30
Cantidad disponible: 1 disponibles
Añadir al carritohbk various paginations dj is shelf worn now in protective sleeve text is excellent as new nb vol 1 only.
Idioma: Inglés
Publicado por Cham [u.a.], Birkhäuser Springer., 2015
ISBN 10: 3319146173 ISBN 13: 9783319146171
Librería: Universitätsbuchhandlung Herta Hold GmbH, Berlin, Alemania
EUR 18,00
Cantidad disponible: 2 disponibles
Añadir al carritoVIII, 290 p. Hardcover. Versand aus Deutschland / We dispatch from Germany via Air Mail. Einband bestoßen, daher Mängelexemplar gestempelt, sonst sehr guter Zustand. Imperfect copy due to slightly bumped cover, apart from this in very good condition. Stamped. Stamped. Shelfwear. Operator Theory, 245 : Advances and Applications. Sprache: Englisch.
Publicado por Basel - Boston - Berlin / Berlin - Heidelberg - New York - Tokyo, Birkhäuser Verlag - Springer-Verlag 1990., 1990
Librería: Versandantiquariat Peter Kardos, Zürich, Suiza
EUR 10,00
Cantidad disponible: 1 disponibles
Añadir al carritoDeutsche Erstausgabe. Gr.8°. 1 Blatt, 454 Seiten, 1 Blatt, mit zahlreichen z.T. blattgrossen Abbildungen im Text auf Kunstdruckpapier (davon ein Frontispiz: Ein mit dem "Very Large Array" in Soccoro (New Mexico) aufgenommenes Radiobild der elliptischen Riesengalaxie Messier 87). Illustrierter Original-Pappband (Umschlagmotiv: Eine von der Seite gesehene Spiralgalaxie, der Sombreronebel (M 104/ NGC 4594). Gutes, sauberes Exemplar. Beiliegend: 4 Zeitungsausschnitte zu diesem Buch, Ephemera.
Idioma: Inglés
Publicado por New York, NY, Springer US : Imprint: Birkhäuser., 2021
ISBN 10: 1071615106 ISBN 13: 9781071615102
Librería: Universitätsbuchhandlung Herta Hold GmbH, Berlin, Alemania
EUR 20,00
Cantidad disponible: 18 disponibles
Añadir al carritoXXXI, 530 p. Hardcover. Versand aus Deutschland / We dispatch from Germany via Air Mail. Einband bestoßen, daher Mängelexemplar gestempelt, sonst sehr guter Zustand. Imperfect copy due to slightly bumped cover, apart from this in very good condition. Stamped. Progress in Nonlinear Differential Equations and Their Applications, 96. Sprache: Englisch.
Idioma: Alemán
Publicado por Bln., Springer, Basel, Birkhäuser, (1987)., 1987
Librería: Bojara & Bojara-Kellinghaus OHG, Osnabrück, Alemania
EUR 20,00
Cantidad disponible: 1 disponibles
Añadir al carritoQuer-4°. Mit 240 (tls. farb.) Abb., 31 Diagrammen u. 1 gefalt. Taf. 6, 274 S. OPbd. m. OU. - Gutes Ex. (Buchblock unbedeutend gewellt). 1.
Idioma: Inglés
Publicado por Basel, Springer Basel/Birkhäuser., 2019
ISBN 10: 3764385278 ISBN 13: 9783764385279
Librería: Universitätsbuchhandlung Herta Hold GmbH, Berlin, Alemania
EUR 22,00
Cantidad disponible: 2 disponibles
Añadir al carrito2019. Band 2 25.8 cm x 18.5 cm. XVII, 573 S. Gebunden. Schutzumschlag an hinterer oberen Ecke eingerissen. Versand aus Deutschland / We dispatch from Germany via Air Mail. Einband bestoßen, daher Mängelexemplar gestempelt, sonst sehr guter Zustand. Imperfect copy due to slightly bumped cover, apart from this in very good condition. Stamped. Sprache: Deutsch.
Publicado por Birkhäuser im Springer Verlag, 2020., 2020
Librería: Antiquariat Carl Wegner, Berlin, B, Alemania
Miembro de asociación: GIAQ
Original o primera edición
EUR 26,00
Cantidad disponible: 2 disponibles
Añadir al carritoSoftcover. 23,5 x 15,5 cm. Originalbroschur, vollständig mit den Seiten (2) 481 - 640 der fortlaufenden Jahrgangspaginierung, mit acht schwarzweiß-Photos und zwei Diagrammen. Gutes Exemplar Artikel: Michael Markert - Modellierte Individualentwicklung. Humanembryologische Praktiken an der Universität Göttingen in der zweiten Hälfte des 20. Jahrhunderts / Aleksei Lokhmatov - Auf dem Weg zur Einheit: Jozef Chalasinski und die Suche nach einer erlaubten Genealogie der Soziologie im Nachkriegspolen 1945-1951 / Christopher Neumaier - Technological Solutions and Contested Interpretations of Scientific Results: Risk Assessment of Diesel Emissions in the U.S. and West Germany 1977 - 1995 / Essay Review / Rezensionen. Texte in deutsch und englisch. -- Bitte Portokosten außerhalb EU erfragen! / Please ask for postage costs outside EU! / S ' il vous plait demander des frais de port en dehors de l ' UE! // Bitte beachten Sie auch unsere Fotos! / Please also note our photos! / Veuillez noter nos photos -- Lesen Sie etwas Schönes auf einer Bank in der Frühlingssonne! Wir haben die passende Lektüre. -- Wir kaufen Ihre werthaltigen Bücher! K00185-507542.
Idioma: Inglés
Publicado por Birkhäuser Basel, Springer Basel Sep 1988, 1988
ISBN 10: 3764322306 ISBN 13: 9783764322304
Librería: BuchWeltWeit Ludwig Meier e.K., Bergisch Gladbach, Alemania
EUR 32,05
Cantidad disponible: 1 disponibles
Añadir al carritoTaschenbuch. Condición: Neu. Neuware -These notes are based on lectures given in the semmar on 'Coding Theory and Algebraic Geometry' held at Schloss Mickeln, Diisseldorf, November 16-21, 1987. In 1982 Tsfasman, Vladut and Zink, using algebraic geometry and ideas of Goppa, constructed a seqeunce of codes that exceed the Gilbert-Varshamov bound. The result was considered sensational. Furthermore, it was surprising to see these unrelated areas of mathematics collaborating. The aim of this course is to give an introduction to coding theory and to sketch the ideas of algebraic geometry that led to the new result. Finally, a number of applications of these methods of algebraic geometry to coding theory are given. Since this is a new area, there are presently no references where one can find a more extensive treatment of all the material. However, both for algebraic geometry and for coding theory excellent textbooks are available. The combination ofthe two subjects can only be found in a number ofsurvey papers. A book by C. Moreno with a complete treatment of this area is in preparation. We hope that these notes will stimulate further research and collaboration of algebraic geometers and coding theorists. G. van der Geer, J.H. van Lint Introduction to CodingTheory and Algebraic Geometry PartI -- CodingTheory Jacobus H. vanLint 11 1. Finite fields In this chapter we collect (without proof) the facts from the theory of finite fields that we shall need in this course. 88 pp. Englisch.
Idioma: Inglés
Publicado por Springer Nature, Birkhäuser, 2015
ISBN 10: 1493936816 ISBN 13: 9781493936816
Librería: Late Knight Books, Philadelphia, PA, Estados Unidos de America
EUR 57,58
Cantidad disponible: 1 disponibles
Añadir al carritoCompletely revised and greatly expanded, the new edition of this text takes readers who have been exposed to only basic courses in analysis through the modern general theory of random processes and stochastic integrals as used by systems theorists, electronic engineers and, more recently, those working in quantitative and mathematical finance. Building upon the original release of this title, this text will be of great interest to research mathematicians and graduate students working in those fields, as well as quants in the finance industry. New features of this edition include: End of chapter exercises; New chapters on basic measure theory and Backward SDEs; Reworked proofs, examples and explanatory material; Increased focus on motivating the mathematics; Extensive topical index.
Idioma: Inglés
Publicado por Birkhäuser, Springer International Publishing, 2017
ISBN 10: 331942811X ISBN 13: 9783319428116
Librería: McBook, Freiburg, Alemania
EUR 26,00
Cantidad disponible: 1 disponibles
Añadir al carritoCondición: gebraucht. 364 S., original Pappeinband/ gebunden, guter Zustand. Versand in stabiler Buchverpackung mit der Deutschen Post oder DHL. Rechnung liegt bei.
Idioma: Inglés
Publicado por Birkhäuser Architecture / Springer, 2003
ISBN 10: 3764364947 ISBN 13: 9783764364946
Librería: Elizabeth's Bookshops, Fremantle, WA, Australia
EUR 19,00
Cantidad disponible: 1 disponibles
Añadir al carritoHardcover. Condición: Fine. ARCHITECTURENew conceptual strategies for structures, hybrid materials and constructions, computer-aided design and fabrication, and the adaptation of techniques from aeroplane design and ship building: these new technical developments in design and production are radically changing the use of metals in building. yesterday's standards and processes are rapidly becoming out-dated, and it is high time that the most recent advances are documented in a systematic overview.\nCase studies examine significant recent buildings, both completed and in design, which illustrate ground-breaking concepts. These include Frank O. Gehry's Guggenheim Museum at Bilbao and the Experience Music project in Seattle, the Millennium Bridge by Norman Foster, Sendai Mediatheque by Toyo Ito, the Seattle Library by Rem Koolhaas. Each project is comprehensively documented in image and text, paying particular attention to the application and processing of metal and steel.pp. 144 illust #060125 Elizabeth's Bookshops have been one of Australia's premier independent book dealers since 1973. Elizabeth's family-owned business operates four branches in Perth CBD, Fremantle (WA), and Newtown (NSW). All orders are dispatched within 24 hours from our Fremantle Warehouse. All items can be viewed at Elizabeth's Bookshop Warehouse, 23 Queen Victoria Street\, Fremantle WA. (Please note: Over standard weight or size. Orders via ABE Books or Biblio or from Overseas may incur additional postage charges. We will contact you prior to processing order to request your approval or contact us to confirm postage cost.).
Idioma: Inglés
Publicado por Birkhäuser, Springer Aug 2023, 2023
ISBN 10: 3031314905 ISBN 13: 9783031314902
Librería: Wegmann1855, Zwiesel, Alemania
EUR 37,44
Cantidad disponible: 2 disponibles
Añadir al carritoTaschenbuch. Condición: Neu. Neuware.
Idioma: Inglés
Publicado por Springer Birkhauser, New York Heidelberg Dordrecht London, 2014
ISBN 10: 1493914731 ISBN 13: 9781493914739
Librería: Emile Kerssemakers ILAB, Heerlen, Holanda
EUR 40,00
Cantidad disponible: 1 disponibles
Añadir al carrito26 cm. original hardcover. xxiv,500 pp. ills, diagrams. bibliography. index. -(some signs of use, otherwise (very) good). 1080g.
Idioma: Inglés
Publicado por Birkhäuser, Springer Aug 2025, 2025
ISBN 10: 3031941357 ISBN 13: 9783031941351
Librería: Wegmann1855, Zwiesel, Alemania
EUR 48,14
Cantidad disponible: 1 disponibles
Añadir al carritoTaschenbuch. Condición: Neu. Neuware -- 1. Introduction.- 2. Review of Linear Algebra.- 3. Multilinear Forms.- 4. Inner Products.- 5. Tensors.- 6. Some Physical Tensors.- 7. Solutions to Exercises.
Idioma: Alemán
Publicado por Birkhäuser, Springer Okt 2025, 2025
ISBN 10: 3031881141 ISBN 13: 9783031881145
Librería: BuchWeltWeit Ludwig Meier e.K., Bergisch Gladbach, Alemania
EUR 19,99
Cantidad disponible: 1 disponibles
Añadir al carritoTaschenbuch. Condición: Neu. Neuware -Das Ziel dieses kompakten und einführenden Lehrbuches ist es, im Rahmen des Stoffumfangs einer vierstündigen Mathematikvorlesung wesentliche stochastische Modelle der Versicherungsmathematik systematisch einzuführen und in ihren Grundzügen zu analysieren.Das Buch gibt eine Einführung u.a. in die Themen: Lebensversicherungsmathematik, insbesondere zufällige Zahlungsströme und Satz von Hattendorff Schadenversicherungsmathematik, insbesondere Risiko- und Ruintheorie, auch für Großschäden Prämienberechnungsprinzipien, Risikomaße und ErfahrungstarifierungDas Buch legt einen Schwerpunkt auf die Erläuterung klassischer und teils fortgeschrittener stochastischer und maßtheoretischer Methoden und Ergebnisse unter Rückgriff auf möglichst vollständige Beweise.Geeignet ist das Buch für fortgeschrittene Bachelor- sowie Master-Studierende mathematischer Studiengänge mit Vorkenntnissen in Maß-, Integrations- und Wahrscheinlichkeitstheorie. 252 pp. Deutsch.
Idioma: Inglés
Publicado por Springer Basel, Birkhäuser, 2014
ISBN 10: 3034808836 ISBN 13: 9783034808835
Librería: AHA-BUCH GmbH, Einbeck, Alemania
EUR 21,39
Cantidad disponible: 1 disponibles
Añadir al carritoBuch. Condición: Neu. Gkiokas, Thanasis Ilustrador. Druck auf Anfrage Neuware - Printed after ordering - This graphic novel is both a historical novel as well as an entertaining way of using mathematics to solve a crime. The plot, the possible motive of every suspect, and the elements of his or her character are based on actual historical figures.The 2nd International Congress of Mathematicians is being held in Paris in 1900. The main speaker, the renowned Professor X, is found dead in the hotel dining room. Foul play is suspected. The greatest mathematicians of all time (who are attending the Congress) are called in for questioning. Their statements to the police, however, take the form of mathematical problems. The Chief Inspector enlists the aid of a young mathematician to help solve the crime. Do numbers always tell the truth Or don't they.
Librería: ReadAmericaBooks, Holland, MI, Estados Unidos de America
Original o primera edición
EUR 83,29
Cantidad disponible: 1 disponibles
Añadir al carritoHardcover. Condición: New. 1st Edition. 233+ pp. Book/jacket condition: New/na (as issued). 1st/First Edition, 1st/First Printing. All orders processed/shipped from MI of WI, USA.
Idioma: Inglés
Publicado por Springer, Birkhäuser Apr 2022, 2022
ISBN 10: 3030713652 ISBN 13: 9783030713652
Librería: BuchWeltWeit Ludwig Meier e.K., Bergisch Gladbach, Alemania
EUR 64,19
Cantidad disponible: 1 disponibles
Añadir al carritoTaschenbuch. Condición: Neu. Neuware -Many researchers dread writing. They find it laborious - even painful - to put their scholarly work into words. They get bogged down in the study, and lose track of the story. And they produce uninspiring papers that fail to resonate with readers or reviewers. This book offers an antidote to this problem: brief, accessible lessons that guide researchers to write clear and compelling scientific manuscripts. The book is divided into three sections - Story, Craft, and Community. The Story section offers advice on getting the balance of study and story just right, introducing strategies for tackling each section of a scientific manuscript. The Craft section considers the grammatical and rhetorical tools of the trade, showing how they can be wielded for maximum impact. And the Community section offers suggestions for writing collaboratively, supporting other writers, and navigating peer review.Each section features multiple short and pragmatic lessons, peppered with illustrative examples. Readers can use the chapters collectively to build holistic writing skills, or dip in and out to refine specific elements of the craft. Rooted in a coaching philosophy, we aim to unlock our readers' potential as writers through instruction, reflection, and example.And we hope to inspire researchers to face writing with joy.This work is clearly written and easily understandable. Its many practical examples, tools, and exercises make an effective toolbox of support for scholarly writers. This will be invaluable to new scholars and help established scholars as well. The inclusion of examples specific to the health arena and the clear, elegantly simple explanations add strength and relevance to this work.Toni Ungaretti, Johns Hopkins School of Education, Baltimore, MD, USAThis book is the most original perspective I have ever read about the craft of writing. As its title suggests, it is inspiring.Brownie Anderson, NBME, Philadelphia, PA, USA 248 pp. Englisch.
Idioma: Alemán
Publicado por Birkhäuser - Berlin ; Heidelberg : Springer, Basel ; Boston, 1990
ISBN 10: 3764324015 ISBN 13: 9783764324018
Librería: Gebrauchtbücherlogistik H.J. Lauterbach, Gummersbach, NRW, Alemania
EUR 14,44
Cantidad disponible: 1 disponibles
Añadir al carritoBroschiert. Condición: Gut. 28x21cm; XVI, 470 S. Broschiert. 617 Ill. u. graph. Darst. Zustand: Gut bis Sehr Gut min. gebräunt (Innen); Einband (Außen) hat geringe Gebrauchsspuren; * Die Photos sind original von uns erstellt worden, u.a. erkennbar an einem kleinen weißen Stück Papier im oberen Schnitt. Ab und an verwenden Suchmaschinen Verlagsphotos, bei den Portalen selbst, werden aber nur unsere Originalphotos gezeigt.
Idioma: Inglés
Publicado por Birkhauser, a Part of Springer Science+Business Media LLC, Boston, 2010
ISBN 10: 0817637923 ISBN 13: 9780817637927
Librería: Tangible Tales, Eugene, OR, Estados Unidos de America
EUR 86,39
Cantidad disponible: 1 disponibles
Añadir al carritoPaperback. Condición: Very Good. Reprint. Thick trade paperback, a reprint of the 1997 edition. A very good copy with minor shelf wear and a small line of soiling spineward being the only noted flaws. 495 pp., a highly respected introduction to geometric representation theory.
EUR 26,74
Cantidad disponible: 1 disponibles
Añadir al carritoTaschenbuch. Condición: Neu. Druck auf Anfrage Neuware - Printed after ordering - This book is a conceptual exploration of numbers, offering a unique perspective that blends mathematical insights with philosophical reflections.It includes chapters on specific numbers to illustrate broader theoretical issues, with a notable emphasis on the concept of infinity. The content is designed to be accessible and enjoyable, avoiding heavy mathematical terminology and notation to maintain readability. The book also incorporates historical and biographical snippets, along with occasional irreverent comments, providing an approachable introduction to the world of numbers.This book is for anyone wanting to know more about the philosophy of numbers, including those who may not have a background in mathematics but are curious about the subject.
Librería: AHA-BUCH GmbH, Einbeck, Alemania
EUR 26,74
Cantidad disponible: 1 disponibles
Añadir al carritoTaschenbuch. Condición: Neu. Druck auf Anfrage Neuware - Printed after ordering - This book collects the notes of the lectures given at the Advanced Course on Crossed Products, Groupoids, and Rokhlin dimension, that took place at the Centre de Recerca Matemàtica (CRM) from March 13 to March 17, 2017. The notes consist of three series of lectures. The first one was given by Dana Williams (Dartmouth College), and served as an introduction to crossed products of C\*-algebras and the study of their structure. The second series of lectures was delivered by Aidan Sims (Wollongong), who gave an overview of the theory of topological groupoids (as a model for groups and group actions) and groupoid C\*-algebras, with particular emphasis on the case of étale groupoids. Finally, the last series was delivered by Gábor Szabó (Copenhagen), and consisted of an introduction to Rokhlin type properties (mostly centered around the work of Hirshberg, Winter and Zacharias) with hints to the more advanced theory related to groupoids.
Librería: AHA-BUCH GmbH, Einbeck, Alemania
EUR 26,70
Cantidad disponible: 1 disponibles
Añadir al carritoTaschenbuch. Condición: Neu. Druck auf Anfrage Neuware - Printed after ordering - This book is an introduction to two higher-categorical topics in algebraic topology and algebraic geometry relying on simplicial methods.Moerdijk's lectures offer a detailed introduction to dendroidal sets, which were introduced by himself and Weiss as a foundation for the homotopy theory of operads. The theory of dendroidal sets is based on trees instead of linear orders and has many features analogous to the theory of simplicial sets, but it also reveals new phenomena. For example, dendroidal sets admit a closed symmetric monoidal structure related to the Boardman-Vogt tensor product of operads. The lecture notes start with the combinatorics of trees and culminate with a suitable model structure on the category of dendroidal sets. Important concepts are illustrated with pictures and examples.The lecture series by Toën presents derived algebraic geometry. While classical algebraic geometry studies functors from the category of commutative rings to the category of sets, derived algebraic geometry is concerned with functors from simplicial commutative rings (to allow derived tensor products) to simplicial sets (to allow derived quotients). The central objects are derived (higher) stacks, which are functors satisfying a certain up-to-homotopy descent condition. These lectures provide a concise and focused introduction to this vast subject, glossing over many of the technicalities that make the subject's research literature so overwhelming.Both sets of lectures assume a working knowledge of model categories in the sense of Quillen. For Toën's lectures, some background in algebraic geometry is also necessary.