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Edward Witten works on theoretical physics and its connections with geometry and topology. His Institute for Advanced Study profile identifies the researcher.^[ref]

His Monopoles and Four-Manifolds by Edward Witten used ideas from supersymmetric gauge theory to introduce a new description of smooth four-manifold invariants. The Seiberg–Witten equations replace a nonabelian instanton-counting problem with equations involving an Abelian gauge field and a spinor. The contribution demonstrates how a physical dual description can reveal a more tractable mathematical structure.^[ref]

For a closed oriented smooth four-manifold $X$ with $b_2^{+}(X)>1$, the Seiberg–Witten invariant of a four-manifold associates an integer-valued invariant, with a choice of homology orientation, to a Spin-c structure. It is obtained from the moduli space of solutions of the Seiberg–Witten equations modulo gauge transformations. When the expected dimension is zero, it is a signed count of solutions after a generic perturbation. Positive-dimensional cases require suitable insertions; when $b_2^{+}=1$, chamber data matters. These analytic counts can distinguish smooth structures and, through Taubes nonvanishing theorem, obstruct decompositions of symplectic manifolds of dimension four.^[ref]

For a closed oriented smooth 3-manifold $Y$, the three-dimensional Seiberg–Witten equations involve a unitary connection and a spinor field associated with a Spin-c structure. A Dirac operator equation couples the spinor field to the unitary connection, while a curvature equation couples the unitary connection back to a quadratic expression in the spinor field. After perturbing and quotienting by gauge transformations, suitable signed counts, with corrections where required, define Seiberg-Witten invariant of a three-manifold.

The hypotheses and conventions matter. For $b_1(Y)>1$, the resulting function on Spin-c structures has a description by refined Reidemeister torsion; for $b_1(Y)=1$, chamber choices enter, and rational homology spheres require additional correction terms. For example, on $S^1\times S^2$ the first Betti number is one, so an unqualified count omitting the chamber data is incomplete. The invariants connect nonlinear differential equations with computable topological invariants.^[ref]

The dependence on perturbations, including wall crossing when $b_1=1$ and a correction term when $b_1=0$, is made explicit in the analytic construction.^[ref]