TAN KAH KEE Science Award

Mathematics and physics

Ngaiming Mok

**Mathematics and physics Complex differential geometry and its applications**

**Abstract：**Complex differential geometry is a core field of modern mathematics which plays an important role in function theory, algebraic geometry, number theory and other branches of mathematics. Mok has made a series of fundamental contributions in this field and related fields. The concept of the variety of minimal rational tangents (VMRT) which he introduced has become a widely used research tool. In collaboration with Hwang he established the rigidity of irreducible Hermitian symmetric spaces of the compact type under K？hler deformation, and settled the celebrated Lazarsfeld problem in algebraic geometry. Schanuel's conjecture is the core conjecture in transcendental number theory which extends significantly the classical Lindemann theorem, and the Ax-Schanuel conjecture is its analogue in the context of function fields. By means of complex differential geometry including the Mok-Zhong compactification theorem, and the concept of the Chow scheme and its associated universal family in algebraic geometry, Mok proved the Ax-Lindemann theorem for all lattices in the rank-1 case, and that perspective is also applicable to the situation of Shimura varieties. In 2019 Mok-Pila-Tsimerman established the Ax-Schanuel theorem on all Shimura varieties. In the latter article, the authors successfully overcome the big gap between products of modular curves and Shimura varieties of higher dimension, and the article has become a landmark achievement of the successful collaboration between complex geometry and arithmetic geometry.

**Awardee：**Ngaiming Mok is a mathematician born in May, 1956. He obtained his Ph.D. degree in 1980 from Stanford University, USA, started his career right afterward at Princeton University, and was later Professor at Universit de Paris (Orsay) and Columbia University. He returned to Hong Kong to take up the position of Chair Professor in Mathematics in 1994. Starting in 1999 he has been Director of the Institute of Mathematical Research, and he has been an Endowed Professor since 2011.

Ngaiming Mok has dedicated his research work to the areas of Several Complex Variables, Complex Differential Geometry and Algebraic Geometry. In the area of Complex Differential Geometry he made use of the Ricci flow and the theory of rational curves to resolve the Generalized Frankel Conjecture. He introduced the algebro-geometrization of complete K？hler manifolds and established with a co-author the compactification theorem for complete K？hler manifolds of finite volume under certain curvature assumptions. In Algebraic Geometry he introduced in 1998 with a co-author the differential geometric theory of varieties of minimal rational tangents (VMRT), proving the rigidity of irreducible Hermitian symmetric spaces of the compact type under K？hler deformation, and resolved afterward a series of related classical open problems. With collaborators he resolved in 2019 the Ax-Schanuel Conjecture on Shimura varieties in the area of Arithmetic Geometry. In 2007 he was awarded the State Natural Science Award (Class II). Based on his fundamental contributions to Several Complex Variables and related achievements he was awarded the Bergman Prize in 2009. In 2002-2014 he was on the Editorial Board of Inventiones Mathematicae, and he was appointed Member of the Fields Committee for the International Congress of Mathematicians in 2010. Currently he is on the Editorial Boards of Mathematische Annalen and other research journals in Mathematics. In 2015 he was elected Academician of the Chinese Academy of Sciences, and in 2017 he was elected Fellow of the Hong Kong Academy of Sciences.