ICTP Prize Winners 2000

The ICTP Prize in the field of mathematics (in honour of Friedrich Hirzebruch) has been awarded jointly to Sheng-Li Tan (East China Normal University) and T.N. Venkataramana (Tata Institute of Fundamental Research).

Sheng-Li Tan has made highly significant contributions to Algebraic geometry and in particular to the theory of Algebraic surfaces. He has proved several outstanding conjectures of renowned experts in the field.

In his early work he constructed a series of examples of surfaces of general type with a canonical map of odd degree, thus answering questions posed by several mathematicians.

He then proved that in a family of curves of genus greater than two with at most ordinary double points, there are at least five singular curves. This confirmed a conjecture of Beauville, which was open for a number of years and interested several geometers. The methods he had developed are enlightening and have enabled him to improve the known bounds for the height of a point on a curve defined over a field of functions - an important question in arithmetic geometry. He found the best possible result here, thus confirming a conjecture of S. Lang.

He has proved a conjecture of Xiao on the topological behaviour of a singular fibre under base change.

In a recent interesting paper he relates the classical Cayley-Bacharach property with the famous Fujita Conjecture and proves the Cayley-Bacharach property for some algebraic varieties. In a joint work with E. Viehweg he has generalised these results to vector bundles and finds some interesting applications.

He has also confirmed the slope conjecture for moduli spaces of curves of genus 7, 8, 9 and 11.

Sheng-Li Tan is an accomplished mathematician whose work reveals geometrical insight and technical capacity of a high order.

T.N. Venkataramana has made very significant contributions to the area of Algebraic groups and their discrete subgroups. His work covers diverse topics in this important area of mathematics.

Venkataramana's early work deals mostly with structural properties of discrete subgroups of Algebraic groups. His most important contribution here is the work in which he extends the famous result of Margulis on the arithmeticity of lattices in higher rank real and p-adic groups to positive characteristics.

He proved (in collaboration with M.S. Raghunathan) the following much sought after theorem. Any co-compact arithmetic subgroup of the orthogonal group of a non-degenerate quadratic form of signature (n,1) with n different from 3 and 7 admits a subgroup finite index, whose first Betti number is non-zero. This result is proved by connecting it with the congruence subgroup problem, an unusual approach.

He has undertaken an extensive study of the restriction of cohomology classes on a locally symmetric manifold to a totally geodesic closed submanifold. In the case of holomorphic classes (in the hermitian symmetric case, handled in collaboration with L. Clozel) this yielded a proof of the abelianness of the Mumford-Tate groups of certain cohomologies of Shimura Varieties. He came up with very subtle ideas for dealing with the general case enabling him to prove that the Mumford-Tate groups are abelian for many more cases. Using these ideas he also settled in the affirmative a conjecture of Harris and Li.

Venkataramana's papers constitute a very valuable contribution to the present knowledge of various aspects of the theory of discrete groups. His work displays commendable scholarship and originality.