  Quivers, gauge theories and singular geometries are of great interest in both
mathematics and physics. In this note, we collect a few open questions which
have arisen in various recent works at the intersection between gauge theories,
representation theory, and algebraic geometry. The questions originate from the
study of supersymmetric gauge theories in different dimensions with different
supersymmetries. Although these constitute merely the tip of a vast iceberg, we
hope this guide can give a hint of possible directions in future research. This
is an invited contribution to a special volume of Proyecciones, E. Gasparim,
Ed., and it is the hope that the questions are specific enough for research
projects aimed at PhD students.
