Here is a simple (to state) linear algebra problem: given $n$ Hermitian matrices with known spectra, what sets of real numbers can be the spectrum of their sum?

In [this beautiful paper][1] the authors address this question using Schubert calculus.  (Admittedly, I'm not sure on which side of the boundary between algebraic topology and algebraic geometry this falls.)


  [1]: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.492.1074&rep=rep1&type=pdf