The investigator proposes several problems in arithmetic geometry and transcendental number theory that explore the arithmetic nature of special values of analytic functions. In one project, the investigator plans to study transcendence and algebraic independence of numbers in positive characteristic arising from Anderson-Drinfeld motives. These transcendence questions are closely connected to Galois groups of Frobenius difference equations, and in particular the investigator proposes to pursue the interaction between transcendence theory and Galois theory to study function field multiple zeta values, periods and logarithms of Drinfeld modules, and higher dimensional versions. The investigator proposes to study Fourier coefficients of modular forms and their relationships with finite field hypergeometric functions and Calabi-Yau manifolds. He plans also to discover arithmetic properties of the extension groups of elliptic curves and their connections with Galois representations.
One of the fundamental branches of modern mathematics, number theory serves as the basis for many applications, including cryptography and coding theory. The proposed research considers questions involving the interplay between arithmetic and analytic aspects of number theory, which have their beginnings in classical work of Euler and Gauss and which continue to have important implications today to the essential understanding of problems both in number theory and in many other branches of mathematics. Several parts of the project lead naturally to problems for graduate and undergraduate research.
