PhD in Pure and Applied Mathematics

The purpose of this course is twofold; on the one hand the basics on the problem of the study of linear systems of matrices will be given, from classical results to the state of the art. On the other hand, a series of connections with other fields and applications will be explained.

Contents:

• Spaces of matrices of constant and bounded rank, compression spaces, primitive spaces.
• Connection with the theory of sheaves and vector bundles, and use of their invariants (such as Chern and Segre classes).
• Connection with secant varieties of Segre and Veronese varieties, and secants of Grassmannians.
• Maximal dimension of spaces of matrices of constant and bounded rank, and classification problems (from the classical works of Westick and Eisenbud and Harris to today's research and open problems).
• Applications to differential geometry (varieties with degenerate dual and second fundamental form from works of Ilic and Landsberg), mathematical physics (congruences of lines and hyperbolic systems of conservation laws from works of Agafonov and Ferapontov), numerical analysys (uniform determinantal representations from works of van Doornmalen, Draisma, Hochstenbach, Plesteniak and myself).

Here:

https://staff.polito.it/ada.boralevi/teaching.html

you can find some slides I prepared for a similar course.