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Quadratic Forms and Their Applications Quadratic Forms and Their Applications Proceedings of the Conference on Quadratic Forms and Their Applications July 5{9, 1999 University College Dublin Eva Bayer-Fluckiger David Lewis Andrew Ranicki Editors Published as Contemporary Mathematics 272, A.M.S. (2000) vii Contents Preface ix Conference lectures x Conference participants xii Conference photo xiv Galois cohomology of the classical groups Eva Bayer-Fluckiger 1 Symplectic lattices Anne-Marie Berge¶ 9 Universal quadratic forms and the ¯fteen theorem J.H. Conway 23 On the Conway-Schneeberger ¯fteen theorem Manjul Bhargava 27 On trace forms and the Burnside ring Martin Epkenhans 39 Equivariant Brauer groups A. FrohlichÄ and C.T.C. Wall 57 Isotropy of quadratic forms and ¯eld invariants Detlev W. Hoffmann 73 Quadratic forms with absolutely maximal splitting Oleg Izhboldin and Alexander Vishik 103 2-regularity and reversibility of quadratic mappings Alexey F. Izmailov 127 Quadratic forms in knot theory C. Kearton 135 Biography of Ernst Witt (1911{1991) Ina Kersten 155 viii Generic splitting towers and generic splitting preparation of quadratic forms Manfred Knebusch and Ulf Rehmann 173 Local densities of hermitian forms Maurice Mischler 201 Notes towards a constructive proof of Hilbert's theorem on ternary quartics Victoria Powers and Bruce Reznick 209 On the history of the algebraic theory of quadratic forms Winfried Scharlau 229 Local fundamental classes derived from higher K-groups: III Victor P. Snaith 261 Hilbert's theorem on positive ternary quartics Richard G. Swan 287 Quadratic forms and normal surface singularities C.T.C. Wall 293 ix Preface These are the proceedings of the conference on \Quadratic Forms And Their Applications" which was held at University College Dublin from 5th to 9th July, 1999. The meeting was attended by 82 participants from Europe and elsewhere. There were 13 one-hour lectures surveying various appli- cations of quadratic forms in algebra, number theory, algebraic geometry, topology and information theory. In addition, there were 22 half-hour lec- tures on more specialized topics. The papers collected together in these proceedings are of various types. Some are expanded versions of the one-hour survey lectures delivered at the conference. Others are devoted to current research, and are based on the half-hour lectures. Yet others are concerned with the history of quadratic forms. All papers were refereed, and we are grateful to the referees for their work. This volume includes one of the last papers of Oleg Izhboldin who died unexpectedly on 17th April 2000 at the age of 37. His untimely death is a great loss to mathematics and in particular to quadratic form theory. We shall miss his brilliant and original ideas, his clarity of exposition, and his friendly and good-humoured presence. The conference was supported by the European Community under the auspices of the TMR network FMRX CT-97-0107 \Algebraic K-Theory, Linear Algebraic Groups and Related Structures". We are grateful to the Mathematics Department of University College Dublin for hosting the con- ference, and in particular to Thomas Unger for all his work on the TEX and web-related aspects of the conference. Eva Bayer-Fluckiger, Besan»con David Lewis, Dublin Andrew Ranicki, Edinburgh October, 2000 x Conference lectures 60 minutes. A.-M. Berge,¶ Symplectic lattices. J.J. Boutros, Quadratic forms in information theory. J.H. Conway, The Fifteen Theorem. D. Hoffmann, Zeros of quadratic forms. C. Kearton, Quadratic forms in knot theory. M. Kreck, Manifolds and quadratic forms. R. Parimala, Algebras with involution. A. Pfister, The history of the Milnor conjectures. M. Rost, On characteristic numbers and norm varieties. W. Scharlau, The history of the algebraic theory of quadratic forms. J.-P. Serre, Abelian varieties and hermitian modules. M. Taylor, Galois modules and hermitian Euler characteristics. C.T.C. Wall, Quadratic forms in singularity theory. 30 minutes. A. Arutyunov, Quadratic forms and abnormal extremal problems: some results and unsolved problems. P. Balmer, The Witt groups of triangulated categories, with some applica- tions. G. Berhuy, Hermitian scaled trace forms of ¯eld extensions. P. Calame, Integral forms without symmetry. P. Chuard-Koulmann, Elements of given minimal polynomial in a central simple algebra. M. Epkenhans, On trace forms and the Burnside ring. L. Fainsilber, Quadratic forms and gas dynamics: sums of squares in a discrete velocity model for the Boltzmann equation. C. Frings, Second trace form and T2-standard normal bases. J. Hurrelbrink, Quadratic forms of height 2 and di®erences of two P¯ster forms. M. Iftime, On spacetime distributions. A. Izmailov, 2-regularity and reversibility of quadratic mappings. S. Joukhovitski, K-theory of the Weil transfer functor. xi V. Mauduit, Towards a Drinfeldian analogue of quadratic forms for poly- nomials. M. Mischler, Local densities and Jordan decomposition. V. Powers, Computational approaches to Hilbert's theorem on ternary quartics. S. PumplunÄ , The Witt ring of a Brauer-Severi variety. A. Queguiner¶ , Discriminant and Cli®ord algebras of an algebra with in- volution. U. Rehmann, A surprising fact about the generic splitting tower of a qua- dratic form. C. Riehm, Orthogonal representations of ¯nite groups. D. Sheiham, Signatures of Seifert forms and cobordism of boundary links. V. Snaith, Local fundamental classes constructed from higher dimensional K-groups. K. Zainoulline, On Grothendieck's conjecture about principal homoge- neous spaces. xii Conference participants A.V. Arutyunov, Moscow [email protected] P. Balmer, Lausanne [email protected] E. Bayer-Fluckiger, Besan»con [email protected] K.J. Becher, Besan»con [email protected] A.-M. Berg¶e,Talence [email protected] G. Berhuy, Besan»con [email protected] J.J. Boutros, Paris [email protected] L. Broecker, Muenster [email protected] Ph. Calame, Lausanne [email protected] P. Chuard-Koulmann, Louvain-la-Neuve [email protected] J.H. Conway, Princeton [email protected] J.-Y. Degos, Talence [email protected] S. Dineen, Dublin [email protected] Ph. Du Bois, Angers [email protected] G. Elencwajg, Nice [email protected] M. Elhamdadi, Trieste [email protected] M. Elomary, Louvain-la-Neuve [email protected] M. Epkenhans, Paderborn [email protected] L. Fainsilber, Goteborg [email protected] D.A. Flannery, Cork dfl[email protected] C. Frings, Besan»con [email protected] R.S. Garibaldi, Zurich [email protected] S. Gille, Muenster [email protected] M. Gindraux, Neuchatel [email protected] R. Gow, Dublin [email protected] F. Gradl, Duisburg [email protected] Y. Hellegouarch, Caen [email protected] D. Ho®mann, Besan»con [email protected] J. Hurrelbrink, Baton Rouge [email protected] K. Hutchinson, Dublin [email protected] M. Iftime, Suceava [email protected] A.F. Izmailov, Moscow [email protected] S. Joukhovitski, Bonn [email protected] M. Karoubi, Paris [email protected] C. Kearton, Durham [email protected] M. Kreck, Heidelberg [email protected] T.J. La®ey, Dublin thomas.la®[email protected] C. Lamy, Paris [email protected] A. Leibak, Tallinn [email protected] D. Lewis, Dublin [email protected] J.-L. Loday, Strasbourg [email protected] M. Mackey, Dublin [email protected] xiii M. Marjoram, Dublin [email protected] V. Mauduit, Dublin [email protected] A. Mazzoleni, Lausanne [email protected] S. McGarraghy, Dublin [email protected] G. McGuire, Maynooth [email protected] M. Mischler, Lausanne [email protected] M. Monsurro, Besan»con [email protected] J. Morales, Baton Rouge [email protected] C. Mulcahy, Atlanta [email protected] H. Munkholm, Odense [email protected] R. Parimala, Besan»con [email protected] O. Patashnick, Chicago [email protected] S. Perret, Neuchatel [email protected] A. P¯ster, Mainz p¯[email protected] V. Powers, Atlanta [email protected] S. PumplÄun,Regensburg [email protected] A. Qu¶eguiner,Paris [email protected] A. Ranicki, Edinburgh [email protected] U. Rehmann, Bielefeld [email protected] H. Reich, Muenster [email protected] C. Riehm, Hamilton, Ontario [email protected] M. Rost, Regensburg [email protected] D. Ryan, Dublin [email protected] W. Scharlau, Muenster [email protected] C. Scheiderer, Duisburg [email protected] J.-P. Serre, Paris [email protected] D. Sheiham, Edinburgh [email protected] F. Sigrist, Neuchatel [email protected] V. Snaith, Southampton [email protected] M. Taylor, Manchester [email protected] J.-P. Tignol, Louvain-la-Neuve [email protected] U. Tipp, Ghent [email protected] D.A. Tipple, Dublin [email protected] M. Tuite, Galway [email protected] T. Unger, Dublin [email protected] C.T.C. Wall, Liverpool [email protected] S. Yagunov, London Ontario [email protected] K. Zahidi, Ghent [email protected] K. Zainoulline, St. Petersburg [email protected] xiv Conference photo 17 16 42 27 15 54 41 26 53 40 68 25 67 52 13 51 39 38 66 12 14 50 24 11 37 65 49 36 10 64 23 63 35 48 9 34 47 22 62 61 33 8 21 60 46 32 59 31 20 58 7 45 6 57 30 56 44 5 55 29 19 43 4 28 18 3 2 (1) A. Ranicki,1 (2) J.H. Conway, (3) G. Elencwajg, (4) K.
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