# Michael Artin

Wolf Prize Laureate in Mathematics 2013

**Michael Artin**

#### Affiliation at the time of the award:

**M.I.T, USA**

#### Award citation:

**“for fundamental contributions to algebraic geometry, both commutative and non-commutative”.**

#### Prize share:

**Michael Artin**

**George D. Mostow **

Michael Artin is one of the main architects of modern algebraic geometry. His fundamental contributions encompass a bewildering number of areas in this field.

To begin with, the theory of étale cohomology was introduced by Michael Artin jointly with Alexander Grothendieck. Their vision resulted in the creation of one of the essential tools of modern algebraic geometry.

In a remarkable and fundamental paper, Artin and Swinnerton-Dyer prove the Shafarevich-Tate conjecture for a K3 surface which is a pencil of elliptic curves over a finite field. In a very original paper Artin and Swinerton-Dyer proved the conjecture for an elliptic K3 surface.

He also collaborated with Barry Mazur to define étale homotopy- another important tool in algebraic geometry- and more generally to apply ideas from algebraic geometry to the study of diffeomorphisms of compact manifold.

We owe to Michael Artin, in large part, also the introduction of algebraic spaces and algebraic stacks. These objects form the correct category in which to perform most algebro-geometrical constructions, and this category is ubiquitous in the theory of moduli and in modern intersection theory. Artin discovered a simple set of conditions for a functor to be represented by an algebraic space. His ”Approximation Theorem” and his ”Existence Theorem” are the starting points of the modern study of moduli problems Artin’s contributions to the theory of surface singularities are of fundamental importance. In this theory he introduced several concepts that immediately became seminal to the field, such as the concepts of rational singularity and of fundamental cycle.

In yet another example of the sheer originality of his thinking, Artin broadened his reach to lay rigorous foundations to deformation theory. This is one of the main tools of classical algebraic geometry, which is the basis of the local theory of moduli of algebraic varieties.

Finally, his contribution to non-commutative algebra has been enormous. The entire subject changed after Artin’s introduction of algebro-geometrical methods in this field. His characterization of Azumaya algebras in terms of polynomial identities, which is the content of the Artin-Procesi theorem, is one of the cornerstones in non-commutative algebra. The Artin-Stafford theorem stating that every integral projective curve is commutative is one of the most important achievements in non-commutative algebraic geometry.

Artin’s mathematical accomplishments are astonishing for their depth and their scope . He is one of the great geometers of the 20th century.