The massage cites from: http://www.abelprisen.no/en/prisvinnere/2009/

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2009 to

Institut des Hautes Études Scientifiques, Bures-sur-Yvette, Frankrike

Riemannian geometry developed from the study of curved surfaces and their higher dimensional analogues and has found applications, for instance, in the theory of general relativity. Gromov played a decisive role in the creation of modern global Riemannian geometry. His solutions of important problems in global geometry relied on new general concepts, such as the convergence of Riemannian manifolds and a compactness principle, which now bear his name.

Gromov is one of the founders of the feld of global symplectic geometry. Holomorphic curves were known to be an important tool in the geometry of complex manifolds. However, the environment of integrable complex structures was too rigid. In a famous paper in 1985, he extended the concept of holomorphic curves to J-holomorphic curves on symplectic manifolds. This led to the theory of Gromov-Witten invariants, which is now an extremely active subject linked to modern quantum feld theory. It also led to the creation of symplectic topology and gradually penetrated and transformed many other areas of mathematics.

Gromov’s work on groups of polynomial growth introduced ideas that forever changed the way in which a discrete infnite group is viewed. Gromov discovered the geometry of discrete groups and solved several outstanding problems. His geometrical approach rendered complicated combinatorial arguments much more natural and powerful.

Mikhail Gromov is always in pursuit of new questions and is constantly thinking of new ideas for solutions of old problems. He has produced deep and original work throughout his career and remains remarkably creative. The work of Gromov will continue to be a source of inspiration for many future mathematical discoveries.

]]>

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2008, worth NOK 6,000,000 (close to US$ 1.2. mill, EURO 750,000) to

## John Griggs ThompsonGraduate Research Professor, University of Florida |
## and |
## Jacques TitsProfessor Emeritus, |

**"for their profound achievements in algebra and in particular for shaping modern group theory"**

Modern algebra grew out of two ancient traditions in mathematics, the art of solving equations, and the use of symmetry as for example in the patterns of the tiles of the Alhambra. The two came together in late eighteenth century, when it was frst conceived that the key to understanding even the simplest equations lies in the symmetries of their solutions. This vision was brilliantly realised by two young mathematicians, Niels Henrik Abel and Evariste Galois, in the early nineteenth century. Eventually it led to the notion of a group, the most powerful way to capture the idea of symmetry. In the twentieth century, the group theoretical approach was a crucial ingredient in the development of modern physics, from the understanding of crystalline symmetries to the formulation of models for fundamental particles and forces.

In mathematics, the idea of a group proved enormously fertile. Groups have striking properties that unite many phenomena in different areas. The most important groups are fnite groups, arising for example in the study of permutations, and linear groups, which are made up of symmetries that preserve an underlying geometry. The work of the two laureates has been complementary: John Thompson concentrated on fnite groups, while Jacques Tits worked predominantly with linear groups.

Thompson revolutionised the theory of fnite groups by proving extraordinarily deep theorems that laid the foundation for the complete classifcation of fnite simple groups, one of the greatest achievements of twentieth century mathematics. Simple groups are atoms from which all fnite groups are built. In a major breakthrough, Feit and Thompson proved that every non-elementary simple group has an

even number of elements. Later Thompson extended this result to establish a classifcation of an important kind of fnite simple group called an N-group. At this point, the classifcation project came within reach and was carried to completion by others. Its almost incredible conclusion is that all fnite simple groups belong to certain standard families, except for 26 sporadic groups. Thompson and his

students played a major role in understanding the fascinating properties of these sporadic groups, including the largest, the so-called Monster.

Tits created a new and highly infuential vision of groups as geometric objects. He introduced what is now known as a Tits building, which encodes in geometric terms the algebraic structure of linear groups. The theory of buildings is a central unifying principle with an amazing range of applications, for example to the classifcation of algebraic and Lie groups as well as fnite simple groups, to Kac-

Moody groups (used by theoretical physicists), to combinatorial geometry (used in computer science), and to the study of rigidity phenomena in negatively curved spaces. Tits’s geometric approach was essential in the study and realisation of the sporadic groups, including the Monster. He also established the celebrated “Tits alternative”: every fnitely generated linear group is either virtually

solvable or contains a copy of the free group on two generators. This result has inspired numerous variations and applications.

The achievements of John Thompson and of Jacques Tits are of extraordinary depth and infuence. They complement each other and together form the backbone of modern group theory.

]]>

THE 2008 WOLF FOUNDATION PRIZE IN MATHEMATICS

The Prize Committee for Mathematics has unanimously decided that the 2008 Wolf Prize will be jointly awarded to:

Pierre R. Deligne

Institute for Advanced Study

Princeton, New Jersey, USA

for his work on mixed Hodge theory; the Weil conjectures; the Riemann-Hilbert correspondence; and for his contributions to arithmetic.

Phillip A. Griffiths

Institute for Advanced Study

Princeton, New Jersey, USA

for his work on variations of Hodge structures; the theory of periods of abelian integrals; and for his contributions to complex differential geometry.

David B. Mumford

Brown University

Providence, Rhode Island, USA

for his work on algebraic surfaces; on geometric invariant theory; and for laying the foundations of the modern algebraic theory of moduli of curves and theta functions.

]]>

The massage is cited from: http://www.wolffund.org.il/cat.asp?id=23&cat_title=MATHEMATICS

THE 2006/7 WOLF FOUNDATION PRIZE IN MATHEMATICS

The Prize Committee for Mathematics has unanimously decided that the 2006/7 Wolf Prize will be jointly awarded to:

Stephen Smale

University of California at Berkeley

Berkeley, California, USA

for his groundbreaking contributions that have played a fundamental role in shaping differential topology, dynamical systems, mathematical economics, and other subjects in mathematics

and

Harry Furstenberg

The Hebrew University of Jerusalem

Jerusalem, Israel

for his profound contributions to ergodic theory, probability, topological dynamics, analysis on symmetric spaces and homogenous flows.

]]>