  We compute tautological integrals over Quot schemes on curves and surfaces.
After obtaining several explicit formulas over Quot schemes of dimension 0
quotients on curves (and finding a new symmetry), we apply the results to
tautological integrals against the virtual fundamental classes of Quot schemes
of dimension 0 and 1 quotients on surfaces (using also universality, torus
localization, and cosection localization). The virtual Euler characteristics of
Quot schemes of surfaces, a new theory parallel to the Vafa-Witten Euler
characteristics of the moduli of bundles, is defined and studied. Complete
formulas for the virtual Euler characteristics are found in the case of
dimension 0 quotients on surfaces. Dimension 1 quotients are studied on K3
surfaces and surfaces of general type with connections to the Kawai-Yoshioka
formula and the Seiberg-Witten invariants respectively. The dimension 1 theory
is completely solved for minimal surfaces of general type admitting a
nonsingular canonical curve. Along the way, we find a new connection between
weighted tree counting and multivariate Fuss-Catalan numbers which is of
independent interest.
