Motivic integration, the Langlands program, and rational points of bounded height
My research domains are algebraic geometry, number theory the Langlands program and their interaction. Number theory has a strong impact on our life, for example the internet, computers, information and code theory will develop, if number theory advances. A big problem in number theory is solving Diophantine equations, i.e. equations of polynomials with integer coefficients. With the development of algebraic geometry, the roots of Diophantine equations can be described by algebraic varieties and we try to show that the arithmetic properties of Diophantine equations depend on the geometric properties of the corresponding algebraic varieties. So the big question for mathematicians is to understand the relation between geometric properties and arithmetic properties. The Langlands program is a big project in mathematics. It conjectures many deep results that connect arithmetic and geometry, and it also helps us to study the distribution of primes (another big problem of number theory). Both of geometry and the Langlands program have strong connections to Physics. In my research plan, I want to work in three subjects: The Langlands program and motivic integration, Non-archimedean geometry, Rational and integer points of varieties and definable sets. The first subject connects properties of integrals in geometric side and arithmetic side . In the second subject, I study geometric side and in the last subject I study arithmetic side.