Jan 182014

The 2014 Wolf Prize in Mathematics is awarded to Peter Sarnak, for his deep contributions in analysis, number theory, geometry, and combinatorics.

Peter Sarnak is on the permanent faculty at the School of Mathematics of the Institute for Advanced Study, Princeton, NJ, USA.

Peter Clive Sarnak (born December 18, 1953) graduated University of the Witwatersrand (B.Sc. 1975) and Stanford University (Ph.D. 1980), under the direction of Paul Cohen.

Prof. Sarnak is a mathematician of an extremely broad spectrum with a far-reaching vision. He has impacted the development of several mathematical fields, often by uncovering deep and unsuspected connections. In analysis, he investigated eigenfunctions of quantum mechanical Hamiltonians which correspond to chaotic classical dynamical systems in a series of fundamental papers. He formulated and supported the “Quantum Unique Ergodicity Conjecture” asserting that all eigenfunctions of the Laplacian on negatively curved manifolds are uniformly distributed in phase space. Sarnak’s introduction of tools from number theory into this domain allowed him to obtain results which had seemed out of reach and paved the way for much further progress, in particular the recent works of E. Lindenstrauss and N. Anantharaman. In his work on L-functions (jointly with Z. Rudnick) the relationship of contemporary research on automorphic forms to random matrix theory and the Riemann hypothesis is brought to a new level by the computation of higher correlation functions of the Riemann zeros. This is a major step forward in the exploration of the link between random matrix theory and the statistical properties of zeros of the Riemann zeta function going back to H. Montgomery and A. Odlyzko. In 1999 it culminates in the fundamental work, jointly with N. Katz, on the statistical properties of low-lying zeros of families of L-functions. Sarnak’s work (with A. Lubotzky and R. Philips) on Ramanujan graphs had a huge impact on combinatorics and computer science. Here again he used deep results in number theory to make surprising and important advances in another discipline.

By his insights and his readiness to share ideas he has inspired the work of students and fellow researchers in many areas of mathematics.

Jan 092013

Prof. George Mostow, Yale, USA:For his fundamental and pioneering contribution to geometry and Lie group theory.

Prof. Michael Artin, M.I.T, USA: For his fundamental contributions to algebraic geometry, both commutative and non-commutative.

George D. Mostow made a fundamental and pioneering contribution to geometry and Lie group theory. His most celebrated accomplishment in this fields is the discovery of the completely new rigidity phenomenon in geometry, the Strong Rigidity Theorems. These theorems are some of the greatest achievements in mathematics in the second half of the 20th century. This established a deep connection between continuous and discrete groups, or equivalently, a remarkable connection between topology and geometry. Mostow’s rigidity methods and techniques opened a floodgate of investigations and results in many related areas of mathematics. Mostow’s emphasis on the “action at infinity” has been developed by many mathematicians in a variety of directions. It had a huge impact in geometric group theory, in the study of Kleinian groups and of low dimensional topology , in work connecting ergodic theory and Lie groups. Mostow’s contribution to mathematics is not limited to strong rigidity theorems. His work on Lie groups and their discrete subgroups which was done during 1948-1965 was very influential. Mostow’s work on examples of nonarithmetic lattices in two and three dimensional complex hyperbolic spaces (partially in collaboration with P. Delinge) is brilliant and lead to many important developments in mathematics. In Mostow’s work one finds a stunning display of a variety of mathematical disciplines. Few mathematicians can compete with the breadth, depth, and originality of his works.

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. Using étale cohomology Artin showed that the finiteness of the Brauer group of a surface fibered by curves is equivalent to the Birch and Swinerton-Dyer conjecture for the Jacobian of a general fiber. In a very original paper Artin and Swinerton-Dyer proved the conjecture for an elliptic \(K_3\) 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.