Georgia Institute of Technology College of Sciences

Moduli spaces are central objects in modern topology and geometry, serving as powerful tools for extracting invariants from underlying manifolds. Gauge theory provides a prolific source of such spaces, utilizing techniques from geometry, analysis, and algebra to probe their structure. In this talk, we survey key gauge-theoretic moduli spaces with an emphasis on how $S^1$-actions can be used to study their topological properties. In particular, we apply these methods to hyperkähler ALE spaces, discussing their applications to the McKay correspondence and symplectic and contact homologies.