Searle made his mark at Henry Cartwright School when he was young. His main work focuses on algebraic topology, multivariate complex analysis, and then commutative algebra and algebraic geometry, mainly using the techniques of layer theory and homology algebra. Searle's doctoral thesis studied the Leray-Searle spectral sequence of fibrosis map. Searle and Katan calculated a spherical cohomology group by the method of keel space, which was the main topic of topology at that time.
At the Fields Prize awarding ceremony in 1954, Weil praised Searle's contribution and pointed out that this was the first time that the prize was awarded to a mathematician. Since then, the development of mathematics has confirmed Wyre's emphasis on abstract algebra. Searle then changed his research direction. He obviously thought that homosexual ethics had become too technical.
1950-60s, Searle cooperated with Grothendieck, who was two years younger than him, thus leading to the basic work of algebraic geometry, which was motivated by Ye Wei's conjecture. Searle's two basic papers on algebraic geometry are Algebraic Cohesion (FAC) and Algebraic Geometry and Analytic Geometry (Gé omé Trie Algé brique et Gé omé Trie Analyticique, GAGA).
Searle realized long ago that to solve Wei Yi's conjecture, we must popularize the theory of homology. The key point is that cohomology of condensed layers can't grasp the topological properties of algebraic clusters like singular cohomology of integral coefficients. Searle (1954/55) tried to take values as cohomology of Witt vectors, and this idea was later absorbed by crystal cohomology.
Around 1958, Searle suggested to study the equi-trivial covering of algebraic clusters, which is a kind of covering that is transformed into trivial covering after changing the basis of finite covering. This kind of thought can be regarded as the origin of flat homophonic. Grothendieck and his collaborators finally established a complete theory in SGA4.
Searle often provides counterexamples for some overly optimistic inferences, and he also works closely with Belgian mathematician Pierre Deligne. Linde finally completed the proof of Wei Yi's conjecture.
After 1959, Searle's interest turned to number theory, especially the theory of class domain and the theory of complex multiplication of elliptic curves.
His original contributions are: the idea of algebraic K theory, Galois representation theory of homology on l- series, and Searle conjecture about the representation of module P.