The faculty at UMass Amherst and the surrounding colleges have broad interests in geometry and topology that include knot theory (topological and geometric knot invariants), symplectic geometry (holomorphic curves applied to rigidity and dynamics), low-dimensional topology (smooth structures on 4-manifolds, geometric structures on 3-manifolds), orbifold theory (manifolds with local group actions, orbifold Gromov-Witten invariants), string topology (algebraic structures on loop spaces), minimal surfaces (surfaces in R3 with mean curvature zero, modeling soap films), surfaces with constant mean curvature (surfaces modeling soap bubbles and fluid droplets), variational and evolution problems (for harmonic maps,Yamabe metrics, etc.), integrable systems (a tool for studying special surfaces, harmonic maps, etc.), harmonic analysis (to obtain PDE estimates, especially applied to dispersive and hyperbolic analogues of harmonic maps) and mathematical visualization.
Some of the faculty research is focused around the GANG (Geometry, Analysis, Numerics, and Graphics) Center, where visually compelling results are recorded. The faculty (and others) also participate in the weekly Geometry and Topology Seminar and the Valley Geometry Seminar.