Geometric and Algebraic Topology

Geometric and algebraic topology studies the global shape and structure of spaces, asking when two spaces are genuinely different and what invariants can reliably tell them apart. Symplectic and contact geometry bring this program into settings where the spaces carry physical meaning — arising naturally in classical mechanics and string theory — while tools like Floer homology and knot invariants translate difficult geometric questions into tractable algebraic ones by counting solutions to certain differential equations, such as holomorphic disks. Active work is pushing these invariants toward a unified quantum-topological picture, connecting polynomial knot invariants to representation theory and exploring how hyperbolic geometry constrains three-dimensional manifolds. Central open problems include a complete classification of knots through their invariants, understanding the full structure of concordance groups, and determining exactly how symplectic flexibility and rigidity trade off in higher dimensions.

Works

87,913

Total citations

725,134

Keywords

Symplectic TopologyKnot InvariantsHolomorphic DisksFloer HomologyContact GeometryGroup Theory