The theory of mixed motives provides one of the most fertile grounds for investigations in number theory via its connection to the theory of L-functions of algebraic varieties. Although the study of motives is essentially homological in nature, Deligne has defined a motivic fundamental group attached to varieties over number fields. Coordinate functions on such fundamental groups are related to special functions like multiple polylogarithms and, thereby, also to special values of L-functions. On the other hand, the proposer has discovered a direct connection between motivic fundamental groups and Diophantine geometry, somewhat along the lines suggested by Gronthendieck's `anabelian' philosophy. The technical tools involve p-adic integration, p-adic Hodge theory, and the global study of Galois cohomology. He proposes to continue research into this connection, aiming towards homotopy-theoretic proofs of well-known theorems, such as those of Faltings or Wiles, and eventually new higher-dimensional results in the line of the conjectures of Serge Lang on hyperbolic varieties.
The deep relationship between geometry and arithmetic is a venerable topic of study going back at least to ruler and compass constructions of special numbers in ancient Greece. The modern manifestation of this tradition is the subject of arithmetic geometry, an area of mathematics that has yielded some of the most profound mathematical results of the previous century. This proposal describes several related ideas for obtaining new results in the theory of algebraic equations using ideas at the interface of linear and non-linear geometry.
