# An Essay on the Prize in Mathematical Sciences 2006

David Mumford and Wentsun Wu both started their careers in pure mathematics (algebraic geometry and topology respectively) but each then made a substantial move towards applied mathematics in the direction of computer science.

Mumford worked on computer aspects of vision and Wu on computer proofs in the field of Geometry. In both cases their pioneering contributions to research and in the development of the field were outstanding. Many leading scientists in these areas were trained by them or followed in their footsteps.

Mumford's early work, for which he received the Fields Medal in 1974, was in algebraic geometry and especially the study of algebraic curves. This is an old and central subject in mathematics with contributions from many of the great names of the past. Despite this, much remained to be done and Mumford's great achievement was to revitalise and push forward the theory of moduli. Algebraic Curves depend on an important integer, the genus g. For g = 0 the curve is rational, for g = 1 it is elliptic and depends on an additional continuous parameter or modulus. For g > 2 there are 3g-3 moduli, forming a (complicated) space whose features give us information about the totality of all curves. Mumford laid the foundations for a systematic and fruitful study of this moduli space. This has been widely influential even, surprisingly, in the physics of string theory.

After two decades in this field, Mumford made a drastic switch to computer vision, where he used his mathematical abilities and insight to make original and fundamental contributions. He helped to provide a conceptual framework and to provide examples of specific solutions that can in principle be generalized to a range of problems. His 1985 paper with Shah on variational approaches to signal processing was recently awarded a prize by the Institute of Electrical and Electronics Engineers (IEEE).

Mumford's many original contributions to pattern theory and vision research were described in his 1999 book Two and Three Dimensional Patterns of the Face (A K Peters Co.) and the forthcoming Pattern Theory through Examples.

Wentsun Wu was one of the geometers strongly influenced by Chern Shiing-Shen (Shaw Laureate in 2004). His early work, in the post-war period, centred on the topology of manifolds which underpins differential geometry and the area where the famous Chern classes provide important information. Wu discovered a parallel set of invariants, now called the Wu classes, which have proved almost equally important. Wu went on to use his classes for a beautiful result on the problem of embedding manifolds in Euclidean Space.

In the 1970's Wu turned his attention to questions of computation, in particular the search for effective methods of automatic machine proofs in geometry. In 1977 Wu introduced a powerful mechanical method, based on Ritt's concept of characteristic sets. This transforms a problem in elementary geometry into an algebraic statement about polynomials which lends itself to effective computation.

This method of Wu completely revolutionized the field, effectively provoking a paradigm shift. Before Wu the dominant approach had been the use of AI search methods, which proved a computational dead end. By introducing sophisticated mathematical ideas Wu opened a whole new approach which has proved extremely effective on a wide range of problems, not just in elementary geometry.

Wu also returned to his early love, topology, and showed how the rational homotopy theory of Dennis Sullivan could be treated algorithmically, thus uniting the two areas of his mathematical life.

In his 1994 Basic Principles in Mechanical Theorem Proving in Geometry (Springer), and his 2000 Mathematics Mechanization (Science Press), Wu described his revolutionary ideas and subsequent developments. Under his leadership Mathematics Mechanization has expanded in recent years into a rapidly growing discipline, encompassing research in computational algebraic geometry, symbolic computation, computer theorem proving and coding theory.

Although the mathematical careers of Mumford and Wu have been parallel rather than contiguous they have much in common. Beginning with the traditional mathematical field of geometry, contributing to its modern development and then moving into the new areas and opportunities which the advent of the computer has opened up, they demonstrate the breadth of mathematics. Together they represent a new role model for mathematicians of the future and are deserved winners of the Shaw Prize.

Mathematical Sciences Selection Committee

The Shaw Prize

12 September 2006, Hong Kong