Taubes, Clifford H.Zhang, Boyu2019-05-202018-052018-05-132018http://nrs.harvard.edu/urn-3:HUL.InstRepos:40050132This thesis studies several compactness problems in gauge theory and explores their applications is low dimensional topology. The first chapter studies a connection between taut foliations and Seiberg-Witten theory. Let Y be a closed oriented 3-manifold and F a smooth oriented foliation on Y. Assume that F does not admit any transverse invariant measure. This chapter constructs an invariant c(F) for F which takes value in the monopole Floer homology group of Y. The invariant is well defined up to a sign. The second chapter proves the deformation invariance for the parity of the number of Klein-bottle leaves in a smooth taut foliation. Given two smooth cooriented taut foliations, assume that every Klein-bottle leaf has non-trivial linear holonomy, and assume that the two foliations can be smoothly deformed to each other through taut foliations, then the parities of the number of Klein-bottle leaves are the same. The third chapter proves that the zero locus of a Z/2 harmonic spinor on a 4 dimensional manifold is 2-rectifiable and has locally finite Minkowski content.application/pdfenMathematicsThesis or Dissertation2019-05-20