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The Book of Involutions
The Book of Involutions Max-Albert Knus Alexander Sergejvich Merkurjev Herbert Markus Rost Jean-Pierre Tignol @ @ @ @ @ @ @ @ The Book of Involutions Max-Albert Knus Alexander Merkurjev Markus Rost Jean-Pierre Tignol Author address: Dept. Mathematik, ETH-Zentrum, CH-8092 Zurich,¨ Switzerland E-mail address: [email protected] URL: http://www.math.ethz.ch/~knus/ Dept. of Mathematics, University of California at Los Angeles, Los Angeles, California, 90095-1555, USA E-mail address: [email protected] URL: http://www.math.ucla.edu/~merkurev/ NWF I - Mathematik, Universitat¨ Regensburg, D-93040 Regens- burg, Germany E-mail address: [email protected] URL: http://www.physik.uni-regensburg.de/~rom03516/ Departement´ de mathematique,´ Universite´ catholique de Louvain, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium E-mail address: [email protected] URL: http://www.math.ucl.ac.be/tignol/ Contents Pr´eface . ix Introduction . xi Conventions and Notations . xv Chapter I. Involutions and Hermitian Forms . 1 1. Central Simple Algebras . 3 x 1.A. Fundamental theorems . 3 1.B. One-sided ideals in central simple algebras . 5 1.C. Severi-Brauer varieties . 9 2. Involutions . 13 x 2.A. Involutions of the first kind . 13 2.B. Involutions of the second kind . 20 2.C. Examples . 23 2.D. Lie and Jordan structures . 27 3. Existence of Involutions . 31 x 3.A. Existence of involutions of the first kind . 32 3.B. Existence of involutions of the second kind . 36 4. Hermitian Forms . 41 x 4.A. Adjoint involutions . 42 4.B. Extension of involutions and transfer . -
Publication List, Dated 2010
References [1] V. A. Voevodski˘ı. Galois group Gal(Q¯ =Q) and Teihmuller modular groups. In Proc. Conf. Constr. Methods and Alg. Number Theory, Minsk, 1989. [2] V. A. Voevodski˘ıand G. B. Shabat. Equilateral triangulations of Riemann surfaces, and curves over algebraic number fields. Dokl. Akad. Nauk SSSR, 304(2):265{268, 1989. [3] G. B. Shabat and V. A. Voevodsky. Drawing curves over number fields. In The Grothendieck Festschrift, Vol. III, volume 88 of Progr. Math., pages 199{227. Birkh¨auser Boston, Boston, MA, 1990. [4] V. A. Voevodski˘ı. Etale´ topologies of schemes over fields of finite type over Q. Izv. Akad. Nauk SSSR Ser. Mat., 54(6):1155{1167, 1990. [5] V. A. Voevodski˘ı. Flags and Grothendieck cartographical group in higher dimensions. CSTARCI Math. Preprints, (05-90), 1990. [6] V. A. Voevodski˘ı.Triangulations of oriented manifolds and ramified coverings of sphere (in Russian). In Proc. Conf. of Young Scientists. Moscow Univ. Press, 1990. [7] V. A. Voevodski˘ı and M. M. Kapranov. 1-groupoids as a model for a homotopy category. Uspekhi Mat. Nauk, 45(5(275)):183{184, 1990. [8] V. A. Voevodski˘ıand M. M. Kapranov. Multidimensional categories (in Russian). In Proc. Conf. of Young Scientists. Moscow Univ. Press, 1990. [9] V. A. Voevodski˘ıand G. B. Shabat. Piece-wise euclidean approximation of Jacobians of algebraic curves. CSTARCI Math. Preprints, (01-90), 1990. [10] M. M. Kapranov and V. A. Voevodsky. Combinatorial-geometric aspects of polycategory theory: pasting schemes and higher Bruhat orders (list of results). Cahiers Topologie G´eom.Diff´erentielle Cat´eg., 32(1):11{27, 1991. -
Algebraic K-Theory (University of Washington, Seattle, 1997) 66 Robert S
http://dx.doi.org/10.1090/pspum/067 Selected Titles in This Series 67 Wayne Raskind and Charles Weibel, Editors, Algebraic K-theory (University of Washington, Seattle, 1997) 66 Robert S. Doran, Ze-Li Dou, and George T. Gilbert, Editors, Automorphic forms, automorphic representations, and arithmetic (Texas Christian University, Fort Worth, 1996) 65 M. Giaquinta, J. Shatah, and S. R. S. Varadhan, Editors, Differential equations: La Pietra 1996 (Villa La Pietra, Florence, Italy, 1996) 64 G. Ferreyra, R. Gardner, H. Hermes, and H. Sussmann, Editors, Differential geometry and control (University of Colorado, Boulder, 1997) 63 Alejandro Adem, Jon Carlson, Stewart Priddy, and Peter Webb, Editors, Group representations: Cohomology, group actions and topology (University of Washington, Seattle, 1996) 62 Janos Kollar, Robert Lazarsfeld, and David R. Morrison, Editors, Algebraic geometry—Santa Cruz 1995 (University of California, Santa Cruz, July 1995) 61 T. N. Bailey and A. W. Knapp, Editors, Representation theory and automorphic forms (International Centre for Mathematical Sciences, Edinburgh, Scotland, March 1996) 60 David Jerison, I. M. Singer, and Daniel W. Stroock, Editors, The legacy of Norbert Wiener: A centennial symposium (Massachusetts Institute of Technology, Cambridge, October 1994) 59 William Arveson, Thomas Branson, and Irving Segal, Editors, Quantization, nonlinear partial differential equations, and operator algebra (Massachusetts Institute of Technology, Cambridge, June 1994) 58 Bill Jacob and Alex Rosenberg, Editors, K-theory and algebraic geometry: Connections with quadratic forms and division algebras (University of California, Santa Barbara, July 1992) 57 Michael C. Cranston and Mark A. Pinsky, Editors, Stochastic analysis (Cornell University, Ithaca, July 1993) 56 William J. Haboush and Brian J. -
Essential Dimension, Spinor Groups and Quadratic Forms 3
ESSENTIAL DIMENSION, SPINOR GROUPS, AND QUADRATIC FORMS PATRICK BROSNANy, ZINOVY REICHSTEINy, AND ANGELO VISTOLIz Abstract. We prove that the essential dimension of the spinor group Spinn grows exponentially with n and use this result to show that qua- dratic forms with trivial discriminant and Hasse-Witt invariant are more complex, in high dimensions, than previously expected. 1. Introduction Let K be a field of characteristic different from 2 containing a square root of −1, W(K) be the Witt ring of K and I(K) be the ideal of classes of even-dimensional forms in W(K); cf. [Lam73]. By abuse of notation, we will write q 2 Ia(K) if the Witt class on the non-degenerate quadratic form q defined over K lies in Ia(K). It is well known that every q 2 Ia(K) can be expressed as a sum of the Witt classes of a-fold Pfister forms defined over K; see, e.g., [Lam73, Proposition II.1.2]. If dim(q) = n, it is natural to ask how many Pfister forms are needed. When a = 1 or 2, it is easy to see that n Pfister forms always suffice; see Proposition 4.1. In this paper we will prove the following result, which shows that the situation is quite different when a = 3. Theorem 1.1. Let k be a field of characteristic different from 2 and n ≥ 2 be an even integer. Then there is a field extension K=k and an n-dimensional quadratic form q 2 I3(K) with the following property: for any finite field extension L=K of odd degree qL is not Witt equivalent to the sum of fewer than 2(n+4)=4 − n − 2 7 3-fold Pfister forms over L. -
Quillen's Work in Algebraic K-Theory
J. K-Theory 11 (2013), 527–547 ©2013 ISOPP doi:10.1017/is012011011jkt203 Quillen’s work in algebraic K-theory by DANIEL R. GRAYSON Abstract We survey the genesis and development of higher algebraic K-theory by Daniel Quillen. Key Words: higher algebraic K-theory, Quillen. Mathematics Subject Classification 2010: 19D99. Introduction This paper1 is dedicated to the memory of Daniel Quillen. In it, we examine his brilliant discovery of higher algebraic K-theory, including its roots in and genesis from topological K-theory and ideas connected with the proof of the Adams conjecture, and his development of the field into a complete theory in just a few short years. We provide a few references to further developments, including motivic cohomology. Quillen’s main work on algebraic K-theory appears in the following papers: [65, 59, 62, 60, 61, 63, 55, 57, 64]. There are also the papers [34, 36], which are presentations of Quillen’s results based on hand-written notes of his and on communications with him, with perhaps one simplification and several inaccuracies added by the author. Further details of the plus-construction, presented briefly in [57], appear in [7, pp. 84–88] and in [77]. Quillen’s work on Adams operations in higher algebraic K-theory is exposed by Hiller in [45, sections 1-5]. Useful surveys of K-theory and related topics include [81, 46, 38, 39, 67] and any chapter in Handbook of K-theory [27]. I thank Dale Husemoller, Friedhelm Waldhausen, Chuck Weibel, and Lars Hesselholt for useful background information and advice. 1. -
From Simple Algebras to the Block-Kato Conjecture Alexander Merkurjev
Public lecture From simple algebras to the Block-Kato conjecture Alexander Merkurjev Professor at the University of California, Los Angeles Guest of the Atlantic Algebra Centre of MUN Distinguished Lecturer Atlantic Association for Research in the Mathematical Sciences Abstract. Hamilton’s quaternion algebra over the real numbers was the first nontrivial example of a simple algebra discovered in 1843. I will review the history of the study of simple algebras over arbitrary fields that culminated in the proof of the Bloch-Kato Conjecture by the Fields Medalist Vladimir Voevodsky. Time and Place The lecture will take place on June 15, 2016, on St. John’s campus of Memorial University of Newfound- land, in Room SN-2109 of Science Building. Beginning at 7 pm. Awards and Distinctions of the Speaker In 1986, Alexander Merkurjev was a plenary speaker at the International Congress of Mathematicians in Berkeley, California. His talk was entitled "Milnor K-theory and Galois cohomology". In 1994, he gave an invited plenary talk at the 2nd European Congress of Mathematics in Budapest, Hungary. In 1995, he won the Humboldt Prize, a prestigious international prize awarded to the renowned scholars. In 2012, he won the Cole Prize in Algebra, for his fundamental contributions to the theory of essential dimension. Short overview of scientific achievements The work of Merkurjev focuses on algebraic groups, quadratic forms, Galois cohomology, algebraic K- theory, and central simple algebras. In the early 1980s, he proved a fundamental result about the struc- ture of central simple algebras of period 2, which relates the 2-torsion of the Brauer group with Milnor K- theory. -
Essential Dimension of the Spin Groups in Characteristic 2
Essential dimension of the spin groups in characteristic 2 Burt Totaro The essential dimension of an algebraic group G is a measure of the number of parameters needed to describe all G-torsors over all fields. A major achievement of the subject was the calculation of the essential dimension of the spin groups over a field of characteristic not 2, started by Brosnan, Reichstein, and Vistoli, and completed by Chernousov, Merkurjev, Garibaldi, and Guralnick [3, 4, 7], [18, Theorem 9.1]. In this paper, we determine the essential dimension of the spin group Spin(n) for n ≥ 15 over an arbitrary field (Theorem 2.1). We find that the answer is the same in all characteristics. In contrast, for the groups O(n) and SO(n), the essential dimension is smaller in characteristic 2, by Babic and Chernousov [1]. In characteristic not 2, the computation of essential dimension can be phrased to use a natural finite subgroup of Spin(2r + 1), namely an extraspecial 2-group, a central extension of (Z=2)2r by Z=2. A distinctive feature of the argument in characteristic 2 is that the analogous subgroup is a finite group scheme, a central r r extension of (Z=2) × (µ2) by µ2, where µ2 is the group scheme of square roots of unity. In characteristic not 2, Rost and Garibaldi computed the essential dimension of Spin(n) for n ≤ 14 [6, Table 23B], where case-by-case arguments seem to be needed. We show in Theorem 4.1 that for n ≤ 10, the essential dimension of Spin(n) is the same in characteristic 2 as in characteristic not 2. -
Emissary | Spring 2021
Spring 2021 EMISSARY M a t h e m a t i c a lSc i e n c e sRe s e a r c hIn s t i t u t e www.msri.org Mathematical Problems in Fluid Dynamics Mihaela Ifrim, Daniel Tataru, and Igor Kukavica The exploration of the mathematical foundations of fluid dynamics began early on in human history. The study of the behavior of fluids dates back to Archimedes, who discovered that any body immersed in a liquid receives a vertical upward thrust, which is equal to the weight of the displaced liquid. Later, Leonardo Da Vinci was fascinated by turbulence, another key feature of fluid flows. But the first advances in the analysis of fluids date from the beginning of the eighteenth century with the birth of differential calculus, which revolutionized the mathematical understanding of the movement of bodies, solids, and fluids. The discovery of the governing equations for the motion of fluids goes back to Euler in 1757; further progress in the nineteenth century was due to Navier and later Stokes, who explored the role of viscosity. In the middle of the twenti- eth century, Kolmogorov’s theory of tur- bulence was another turning point, as it set future directions in the exploration of fluids. More complex geophysical models incorporating temperature, salinity, and ro- tation appeared subsequently, and they play a role in weather prediction and climate modeling. Nowadays, the field of mathematical fluid dynamics is one of the key areas of partial differential equations and has been the fo- cus of extensive research over the years. -
The Mathematics of Andrei Suslin
B U L L E T I N ( N e w Series) O F T H E A M E R I C A N MATHEMATICAL SOCIETY Vo lume 57, N u mb e r 1, J a n u a r y 2020, P a g e s 1–22 https://doi.org/10.1090/bull/1680 Art icle electronically publis hed o n S e p t e mbe r 24, 2019 THE MATHEMATICS OF ANDREI SUSLIN ERIC M. FRIEDLANDER AND ALEXANDER S. MERKURJEV C o n t e n t s 1. Introduction 1 2. Projective modules 2 3. K 2 of fields and the Brauer group 3 4. Milnor K-theory versus algebraic K-theory 7 5. K-theory and cohomology theories 9 6. Motivic cohomology and K-theories 12 7. Modular representation theory 16 Acknowledgments 18 About the authors 18 References 18 0. Introduction Andrei Suslin (1950–2018) was both friend and mentor to us. This article dis- cusses some of his many mathematical achievements, focusing on the role he played in shaping aspects of algebra and algebraic geometry. We mention some of the many important results Andrei proved in his career, proceeding more or less chronolog- ically beginning with Serre’s Conjecture proved by Andrei in 1976 (and simul- taneously by Daniel Quillen). As the reader will quickly ascertain, this article does not do justice to the many mathematicians who contributed to algebraic K- theory and related subjects in recent decades. In particular, work of Hyman Bass, Alexander Be˘ılinson, Spencer Bloch, Alexander Grothendieck, Daniel Quillen, Jean- Pierre Serre, and Christophe Soul`e strongly influenced Andrei’s mathematics and the mathematical developments we discuss. -
On Motivic Spherical Bundles
ON MOTIVIC SPHERICAL BUNDLES FLORIAN STRUNK Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) des Fachbereichs Mathematik/Informatik an der Universit¨atOsnabr¨uck vorgelegt von Florian Strunk aus Lemgo Institut f¨urMathematik Universit¨atOsnabr¨uck Dezember 2012 Contents Introduction 1 1. Unstable motivic homotopy theory 3 1.1. The unstable motivic homotopy category . 5 1.2. Homotopy sheaves and fiber sequences . 15 1.3. Mather's cube theorem and some implications . 24 2. Stable motivic homotopy theory 31 2.1. The stable motivic homotopy category . 31 2.2. A consequence of the motivic Hurewicz theorem . 34 2.3. Motivic connectivity of morphisms . 39 3. The motivic J-homomorphism 44 3.1. Principal bundles and fiber bundles . 44 3.2. Fiber sequences from bundles . 52 3.3. Definition of a motivic J-homomorphism . 53 4. Vanishing results for the J-homomorphism 61 4.1. A relation with Thom classes . 61 4.2. Motivic Atiyah duality and the transfer . 65 4.3. A motivic version of Brown's trick . 67 4.4. Dualizability of GLn=NT ........................71 A. Appendix 74 A.1. Preliminaries on model categories . 74 A.2. Monoidal model categories . 80 A.3. Simplicial model categories . 82 A.4. Pointed model categories . 86 A.5. Stable model categories . 89 A.6. Fiber and cofiber sequences . 93 References . 96 ON MOTIVIC SPHERICAL BUNDLES 1 Introduction This thesis deals with a motivic version of the J-homomorphism from algebraic topology. The classical J-homomorphism was introduced under a different name in 1942 by Whitehead [Whi42] in order to study homotopy groups of the spheres. -
Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory
http://dx.doi.org/10.1090/pspum/075 Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory Proceedings of Symposia in PURE MATHEMATICS Volume 75 Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory Proceedings of the Bretton Woods Workshop on Multiple Dirichlet Series Bretton Woods, New Hampshire July 11-14, 2005 Solomon Friedberg (Managing Editor) Daniel Bump Dorian Goldfeld Jeffrey Hoffstein Editors American Mathematical Society Providence, Rhode Island >^VDED^% 2000 Mathematics Subject Classification. Primary llFxx, llMxx, 11-02; Secondary 22E50, 22E55. The Bretton Woods Workshop on Multiple Dirichlet Series was supported by a Focussed Research Group grant from the National Science Foundation. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Library of Congress Cataloging-in-Publication Data Bretton Woods Workshop on Multiple Dirichlet Series (2005 : Bretton Woods, N.H.) Multiple Dirichlet series, automorphic forms, and analytic number theory : proceedings of the Bretton Woods Workshop on Multiple Dirichlet Series, July 11-14, 2005, Bretton Woods, New Hampshire / Solomon Friedberg, managing editor... [et al.]. p. cm. - (Proceedings of symposia in pure mathematics ; v. 75) Includes bibliographical references. ISBN-13: 978-0-8218-3963-8 (alk. paper) ISBN-10: 0-8218-3963-2 (alk. paper) 1. Dirichlet series—Congresses. 2. L-functions—Congresses. I. Friedberg, Solomon, 1958— II. Title. QA295.B788 2005 515'.243-dc22 2006049095 Copying and reprinting. Material in this book may be reproduced by any means for edu• cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg• ment of the source is given. -
Study Group: the Bloch–Kato Conjecture on Special Values of L-Functions
Study group: The Bloch{Kato conjecture on special values of L-functions Organisers: Andrew Graham and Dami´anGvirtz Spring term 2021 Introduction In 1979, Deligne noticed how results for special values of L-functions in the num- ber field and elliptic curve cases (the latter being the famous and still open BSD conjecture) could be generalised. He predicted the special value L(M; n) of the L-function attached to a motif M, up to a rational factor, at certain \critical" n 2 Z where the L-function has neither pole nor zero. Beilinson removed the criticality assumption with a conjecture on the leading term L(M; n)∗, again only up to a rational number. The topic of this study group is the conjecture by Bloch and Kato which determines this rational number and thus offers a precise value for L(M; n)∗. In particular, we want to see what the conjecture says in special cases, e.g. the refined Birch and Swinnerton-Dyer conjecture. Organisation The study group meets every Wednesday from 12:00-13:30. A Zoom link will be distributed via the mailing list. An up to date version of this document can be found on the study group website: Schedule 1 Motives (20/01) { Wojtek Wawr´ow Introduce the category of Chow motives, its various cohomological realisa- tion functors (de Rham, Betti, p-adic) and compatible systems of realisations. [Sch94a, Jan95] 2 Motivic cohomology (27/01) { Alex Torzewski Treating the unknown category of mixed motives or the known derived cate- gory of Voevodsky as a black box, motivate motivic cohomology.