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The first book on the systematic algebraic solutions of linear and quadratic equations by the Persian scholar Muhammad ibn Mūsā al-Khwārizmī.

The book is considered to be the foundation of modern algebra and Islamic mathematics.

Serre introduced Čech cohomology of sheaves in this paper, and, despite some technical deficiencies, revolutionized formulations of algebraic geometry.

For example, the long exact sequence in sheaf cohomology allows one to show that some surjective maps of sheaves induce surjective maps on sections; specifically, these are the maps whose kernel (as a sheaf) has a vanishing first cohomology group.

In mathematics, algebraic geometry and analytic geometry are closely related subjects, where analytic geometry is the theory of complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables.

A (mathematical) theory of the relationship between the two was put in place during the early part of the 1950s, as part of the business of laying the foundations of algebraic geometry to include, for example, techniques from Hodge theory.

The first book on group theory, giving a then-comprehensive study of permutation groups and Galois theory. The first comprehensive work on transformation groups, serving as the foundation for the modern theory of Lie groups.

It laid the foundations of Indian mathematics and was influential in South Asia and its surrounding regions, and perhaps even Greece.Borel and Serre's exposition of Grothendieck's version of the Riemann–Roch theorem, published after Grothendieck made it clear that he was not interested in writing up his own result.Grothendieck reinterpreted both sides of the formula that Hirzebruch proved in 1953 in the framework of morphisms between varieties, resulting in a sweeping generalization.Jigu Suanjing (626AD) This book by Tang dynasty mathematician Wang Xiaotong Contains the world's earliest third order equation.Contained rules for manipulating both negative and positive numbers, rules for dealing the number zero, a method for computing square roots, and general methods of solving linear and some quadratic equations, solution to Pell's equation.

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Many of the original techniques used in this paper were used in the eventual classification of finite simple groups.

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