Geometric and Algebraic Topology

Geometric and algebraic topology investigates the global shape and structure of spaces that cannot be distinguished by local measurements alone, asking when two manifolds are truly the same and what algebraic data captures their differences. Tools like Floer homology and knot invariants translate geometric problems — such as whether a curve in three-dimensional space can be continuously unknotted, or how symplectic structures constrain the dynamics on a manifold — into computable algebraic objects. Active research pushes into quantum topology, where ideas from physics inform new invariants, and into hyperbolic geometry, where the rigid metric structure of three-manifolds interacts with more flexible topological classifications. Open questions include a full understanding of the concordance group of knots, the extent to which Floer-theoretic invariants detect exotic smooth structures, and the topological origins of knotting observed in biological molecules such as proteins.

Works

86,034

Total citations

721,606

Keywords

Symplectic TopologyKnot InvariantsHolomorphic DisksFloer HomologyContact GeometryGroup Theory