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Topic 1: Some basic linear preserver problems
Topic 1: Some basic linear preserver problems
Speaker: Prof. Chi-Kwong Li
Speaker: Prof. Chi-Kwong Li
Session 1. Basic introduction.
-History: Linear preservers of determinants, singular matrices, etc.
-Different types of preserver problems: Preservers of functions, sets, relations, etc.
-Variations: additive, multiplicative preservers, etc.
-Tutorial: Work out some proofs of determinant preservers, rank one preservers, etc.
Session 2. Norm preservers.
-Basic examples: Frobenius norm, spectral norm, trace norm, preservers.
-Techniques: Mazur-Ulam theorem, extreme points, duality, group theory techniques.
-UI norms, USI norms, G-invariant norms, non-surjective maps, say, from M_2 to M_3.
-Tutorial: Work out different proofs for the spectral norm, trace norm, etc.
Session 3. Linear preservers of sets and functions determined by eigenvalues.
-Linear maps of functions of eigenvalues.
-Linear maps preserving sets such as invertible, singular, nilpotent matrices, etc.
-Tutorial: Work out some simple cases.
Session 4. Idempotent preservers.
-Linear maps T : M_n(F) -> M_r(F) such that T(A)^2 = 0_r whenever A^2 = 0.
-Linear maps T: M_n(F) -> M_r(F) such that f(T(A)) = 0_r whenever f(A) = 0 for a polynomial.
-Tutorial: Work out some simple cases, say, when n=r, and F is the reals or complexes.
Topic 2: General preservers
Speaker: Prof. Peter Šemrl
Topic 2: General preservers
Speaker: Prof. Peter Šemrl
Session 1. Basic introduction.
-Formulation of problems: replacing linearity and preserving property by a single weaker condition, general preservers, commutativity preservers, order preservers.
-Fundamental theorem of projective geometry.
-Wigner's theorem.
-Tutorial: work out some simple cases.
Session 2. Adjacency preservers.
-Fundamental theorem of geometry of matrices, maximal adjacent sets.
-Fundamental theorem of geometry of hermitian matrices, connection with the fundamental theorem of chronogeometry.
-Applications.
-Tutorial: work out some proofs on maximal adjacent sets, work out some simple cases.
Session 3. Commutativity preservers.
-Commutativity preservers on hermitian matrices.
-Commutativity preservers on the algebra of all n x n matrices.
-Tutorial: work out some proofs on commutants and second commutants.
Session 4. Order preserving maps on hermitian matrices.
-Order preservers on hermitian matrices.
-Order preservers on matrix intervals.
-Loewner's theorem.
-Tutorial: work out some simple cases.
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