Michael Atiyah

Michael Atiyah (1929–2019) was a British mathematician whose work connected geometry, topology, and mathematical physics. He co-authored the Atiyah–Singer index theorem and helped establish K-theory as a tool in topology.

K-theory: A branch of mathematics that studies vector bundles and related structures through algebraic invariants. Atiyah helped develop topological K-theory and used it across geometry and topology.
Atiyah–Singer index theorem: A theorem equating the analytical index of an elliptic operator with a topological index. Atiyah and Isadore Singer proved this theorem, linking analysis and topology.
Isadore Singer: An American mathematician known for work in geometry, analysis, and mathematical physics. Singer co-authored the index theorem with Atiyah and shared the 2004 Abel Prize with him.
Abel Prize: An international mathematics award established by the Norwegian government in 2002. Atiyah and Singer received its 2004 prize for their index theorem and related work.
Index theorem: A theorem relating an operator’s analytical index to topological data. The Atiyah–Singer theorem became a landmark instance of this broad principle.
Atiyah–Bott fixed-point theorem: A formula relating fixed points of a group action to traces on cohomology. Atiyah and Raoul Bott developed this influential fixed-point formula.
Raoul Bott: A Hungarian-American mathematician whose work shaped topology and differential geometry. Bott collaborated with Atiyah on fixed-point theory and other geometric results.
Fields Medal: A mathematics prize awarded every four years to young mathematicians for outstanding work. Atiyah received the 1966 medal for contributions spanning topology and analysis.
Differential geometry: The study of smooth spaces using calculus and geometric structures. Atiyah’s work connected differential geometry with topology and analysis.
Atiyah–Hirzebruch spectral sequence: A spectral sequence connecting generalized cohomology theories with ordinary cohomology. Atiyah and Friedrich Hirzebruch introduced this tool for computing topological K-theory.

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Baum–Connes conjectureRelated: The conjecture generalizes index-theoretic ideas associated with his work.