Ulrich bundles on surfaces

Motivations of Ulrich bundles naturally came from linear algebra and commutative algebra. Thanks to
Eisenbud and Schreyer, such an algebraic notion dives into the geometric world, and become a key object
to observe the cone of cohomology tables and Cayley-Chow forms. They asked whether every projective
variety supports an Ulrich bundle, which is still wildly open even for smooth surfaces. In the very recent
years, Ulrich bundles have received a lot of attention, and many constructions of rank 2 Ulrich bundles
on surfaces are studied as well. In this talk, we review various motivations of Ulrich bundles and ideas for
the construction of rank 2 Ulrich bundles on surfaces.