Goresky–MacPherson

Goresky–MacPherson is the joint attribution for the collaborative work of two mathematicians whose joint contributions reshaped modern Topology, Algebraic geometry, and Singularity theory. Their publications introduced techniques linking Morse theory with stratified spaces, producing foundational tools used across Poincaré-related research, Hodge theory, and the study of perverse sheaves in the context of Picard-style problems. Their methods interact with structures studied by researchers connected to Atiyah, Deligne, Grothendieck, Verdier, and have influenced work in Donaldson theory, Seiberg–Witten theory, and modern Representation theory.

History

The collaboration arose during the late 1970s and early 1980s when problems in Lefschetz theory, Calabi–Yau degenerations, and analyses of Whitney-stratified spaces prompted interactions among researchers in Princeton University, Institute for Advanced Study, and departments influenced by Weyl-era topology. Their work built on antecedents including Morse, Thom, Milnor, Whitney, and drew on categorical perspectives from Grothendieck and duality formalized by Verdier. The pair developed concepts in the milieu of conferences such as meetings of the American Mathematical Society, colloquia at MIT, and seminars at Brown University. Their publications influenced subsequent results by Deligne, MacPherson, Goresky, and later contributions linked to Kontsevich, Witten, Donaldson, and Atiyah.

Intersection Homology

Intersection homology, as formulated by the collaborators, modifies classical homology theory to accommodate singular varieties and stratified pseudomanifolds, resolving failures of Poincaré duality in singular settings. The theory uses stratifications in the spirit of Whitney and employs perversity functions related to ideas in Deligne's mixed Hodge modules and Verdier duality. Intersection homology sits alongside sheaf theory frameworks promoted by Grothendieck, and connects to perverse sheaves developed further by Beilinson, Bernstein, and Deligne. This framework has been instrumental in understanding phenomena appearing in the study of Hodge decomposition, mixed Hodge theory, and invariants arising in work by Milnor and Arnold on singularities.

Goresky–MacPherson Formula

The eponymous formula expresses relationships between intersection homology groups and local data on strata, generalizing classical formulas from Lefschetz and Alexander duality settings. It refines earlier results related to Poincaré duality by incorporating contributions from stratified pieces analogous to the role of Euler characteristic in smooth manifolds studied by Euler and later formalized in contexts influenced by Hirzebruch and Atiyah–Singer. The formula enabled computations in cases treated by Thom and Milnor for links of singularities and informed comparisons with invariants used in Donaldson invariants and Seiberg–Witten invariants employed in four-manifold topology analyzed by Kronheimer and Mrowka.

Applications

Applications of their work appear across Algebraic geometry, Singularity theory, and mathematical physics. Intersection homology and the formula are used in the study of moduli space compactifications such as those in Deligne–Mumford theory, analyses of Shimura varieties and Baily–Borel-type boundaries, and computations in Representation theory related to Kazhdan–Lusztig and Springer correspondence. In mathematical physics their ideas support the rigorous understanding of spaces appearing in String theory, Mirror symmetry, and the work of Witten on topological quantum field theories related to Chern–Simons and supersymmetric gauge theories. They also inform computational approaches used in Singular-related studies of algebraic varieties and algorithms in Sheaf cohomology problems encountered by researchers at institutions such as Stanford University and University of Cambridge.

Key Results and Theorems

Key results include the formal definition of intersection homology groups satisfying a version of Poincaré duality for stratified pseudomanifolds, stratified versions of Morse inequalities influenced by Morse and Thom, and formulae connecting local intersection homology of links to global invariants analogous to Lefschetz phenomena. These theorems interact with the theory of perverse sheaves by Beilinson et al., the decomposition theorem used by Beilinson, Bernstein, and Deligne, and results in Hodge theory extended by Deligne and Schmid. Their work underpins advances by Lusztig in representation theory, informs calculations in intersection cohomology used in equivariant cohomology contexts developed by Atiyah and Bott, and continues to influence contemporary research by scholars connected to Institute for Advanced Study and major conferences of the IMU.

Category:Topology