Seminarium "The Trans-Carpathian Seminar on Geometry & Physics"

The set of isospectral hermitian matrices, i.e. a (partial) flag variety, is one of fundamental mathematical objects, playing a role in various parts of mathematics from algebraic geometry to combinatorics. One of most succesful ways to understand them is to study the action of the maximal torus of diagonal matrices in SU(n). Special hermitian matrices with fixed spectrum, i.e. the ones for which some of the off-diagonal entries are zero, have been studied in theory of integrable systems. They have interesting topology and beautiful symmetries. The classically studied case was that of tridiagnonal matrices, i.e a_{ij}= 0 if |i-j|>1. It turned out that other "tridiagonal matrices", for example those for which a_{ij}=0 for i>1, have equally interesting topology and symmetries. Again the good approach is to use the diagonal torus action. However new tools are needed to understand even so simple topological invariants like cohomology. There is also an interesting symplectic aspect to these manifolds which I will describe. This is a joint work with Światosław Gal (Wrocław University).