Compact Courses
Introduction in Matrix Geometrical Analysis
by
Prof. S. Gindikin
Rutgers University, New Brunswick, USA
January 27 - February 20, 2009
at
11:30 AM, LH – 1, Department of Mathematics
Indian Institute of Science, Bangalore
Abstract
(Geometry and Analysis in Siegel’s Universe)
What is a multi-dimensional analysis? There is a straight forward way to apply one-dimensional analysis to each of several variables, but it is much more interesting to explore “irreducible” possibilities. One of most successful realizations goes back to Siegel, who considered the “matrix line” of symmetric matrices. We build step-by-step geometry of matrices – analogues of half-lines, segments, projective geometry, quadrics, complexifications, discs, half-planes, Cayley transform, conformal geometry etc. Simultaneously, we develop matrix analogues of analytic objects – Euler integrals, derivatives and antiderivatives, Riemann-Liouville integrals (fractional derivatives), Cauchy integral formulas, residues, etc. We will discuss also some generalizations in a context of analysis on symmetric spaces.
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