Editing Michael Atiyah (section)

===Algebraic geometry (1952–1958)===
{{Main|Algebraic geometry}}
[[File:Twisted cubic curve.png|thumb|right|250px|A [[twisted cubic curve]], the subject of Atiyah's first paper]]
Atiyah's early papers on algebraic geometry (and some general papers) are reprinted in the first volume of his collected works.<ref>{{harvnb|Atiyah|1988a}}</ref>

As an undergraduate Atiyah was interested in classical projective geometry, and wrote his first paper: a short note on [[twisted cubics]].<ref>{{harvnb|Atiyah|1988a|loc=paper 1}}</ref> He started research under [[W. V. D. Hodge]] and won the [[Smith's prize]] for 1954 for a [[Sheaf (mathematics)|sheaf-theoretic]] approach to [[ruled surface]]s,<ref>{{harvnb|Atiyah|1988a|loc=paper 2}}</ref> which encouraged Atiyah to continue in mathematics, rather than switch to his other interests—architecture and archaeology.<ref>{{harvnb|Atiyah|1988a|p= 1}}</ref>
His PhD thesis with Hodge was on a sheaf-theoretic approach to [[Solomon Lefschetz]]'s theory of integrals of the second kind on algebraic varieties, and resulted in an invitation to visit the Institute for Advanced Study in Princeton for a year.<ref>{{harvnb|Atiyah|1988a|loc=papers 3, 4}}</ref> While in Princeton he classified [[vector bundle]]s on an [[elliptic curve]] (extending [[Alexander Grothendieck]]'s classification of vector bundles on a genus 0 curve), by showing that any vector bundle is a sum of (essentially unique) indecomposable vector bundles,<ref>{{harvnb|Atiyah|1988a|loc=paper 5}}</ref> and then showing that the space of indecomposable vector bundles of given degree and positive dimension can be identified with the elliptic curve.<ref>{{harvnb|Atiyah|1988a|loc=paper 7}}</ref> He also studied double points on surfaces,<ref>{{harvnb|Atiyah|1988a|loc=paper 8}}</ref> giving the first example of a [[flop (algebraic geometry)|flop]], a special birational transformation of [[3-fold]]s that was later heavily used in [[Shigefumi Mori]]'s work on [[minimal model (birational geometry)|minimal model]]s for 3-folds.<ref>{{harvnb|Matsuki|2002}}.</ref> Atiyah's flop can also be used to show that the universal marked family of [[K3 surface]]s is not [[Hausdorff space|Hausdorff]].<ref>{{harvnb|Barth|Hulek|Peters|Van de Ven|2004}}</ref>