We will discuss the deformation, obstruction, and intersection theory of the Quot scheme of a vector bundle on a curve and explaing how this is related to sheaf theoretic curve counting on local curves. As time permits, we will discuss relative versions of this Quot scheme where quotients are required to be "well-behaved" near specified points of the curve, so that they can be "glued". Our approach uses logarithmic geometry both to interpret "well-behaved" and to keep track of vanishing cycles in degenerations.