Connections on principal bundles, gauge groups, Yang-Mills connections, (anti-)self-dual connections, stability of vector bundles, Donaldson invariants, the Seiberg-Witten equations and invariants, the Bogomolnyi (monopole) equation, Chern-Simons invariant, Donaldson-Thomas theory, relations to Gromov-Witten theory, applications to low-dimensional topology.

Gauge theory is a subfield of mathematical physics, differential geometry, and differential topology. The aim is to study the geometry and topology of a space by examining an appropriate moduli space of connections (and possibly spinors) which satisfy certain PDE. These PDE frequently have their origins in physics.

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