Enumerative geometry of LSSM

Linear spaces of symmetric matrices (LSSM) appear naturally in many branches of mathematics. They represent spaces of quadrics, special statistical models, partially symmetric tensors and more. In the first part of the talk we will focus on basic invariants of LSSM, presenting examples and showing possible difficulties. We will motivate introduction of objects known from pure algebraic geometry, such as line bundles, divisors and cohomology rings. Guided by the fact that 'smoothness' and 'properness' are our best friends, we will arrive at the space of complete quadrics. In the second part of the talk we will show how the theory of symmetric functions and cohomology rings can provide answers to very basic questions about LSSM.