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DTSTART;VALUE=DATE:20231219T171500
DTEND;VALUE=DATE:20231219T171500
UID:14848@agenda.unifr.ch
DESCRIPTION:Every complex symplectic matrix in Sp2n\npCq can be factorized as a\nproduct of the following types of unipotent matrices (in interchanging\norder).\n‚ (i): ˆ\nI B\n0 I\n˙\n, upper triangular with symmetric B “ BT\n.\n‚ (ii): ˆ\nI 0\nC I˙\n, lower triangular with symmetric C “ C\nT\n.\nThe optimal number TpCq of such factors that any matrix in Sp2n\npCq\ncan be factored into a product of T factors has recently been established\nto be 5 by Jin, P. Lin, Z. and Xiao, B.\nIf the matrices depend continuously or holomorphically on a parameter, equivalently their entries are continuous functions on a topological\nspace or holomorphic functions on a Stein space X, it is by no means\nclear that such a factorization by continuous/holomorphic unipotent\nmatrices exists. A necessary condition for the existence is the map\nX Ñ Sp2n\npCq to be null-homotopic. This problem of existence of a\nfactorization is known as the symplectic Vaserstein problem or GromovVaserstein problem. In this talk we report on the results of the speaker\nand his collaborators B. Ivarsson, E. Low and of his Ph.D. student J.\nSchott on the complete solution of this problem, establishing uniform\nbounds Tpd, nq for the number of factors depending on the dimension\nof the space d and the size n of the matrices. It seems difficult to establish the optimal bounds. However we obtain results for the numbers\nTp1, nq, Tp2, nq for all sizes of matrices in joint work with our Ph.D.\nstudents G. Huang and J. Schott. Finally we give an application to\nthe problem of writing holomorphic symplectic matrices as product of\nexponentials.\n
SUMMARY:Factorization of Holomorphic Matrices
CATEGORIES:Colloque / Congrès / Forum
LOCATION:PER 08\, auditoire 2.52\, Chemin du Musée 3\, 1700 Fribourg
URL;VALUE=URI:https://agenda.unifr.ch/e/fr/14848
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