School of Mathematics

To apply the technique of virtual fundamental cycle (chain) in the study of pseudo-holomorphic curve, we need to construct certain structure, which we call Kuranishi strucuture, on its moduli space. In this talk I want to review certain points of its construction.

The Weil height measures the “complexity” of an algebraic number. It vanishes precisely at 0 and at the roots of unity. Moreover, a finite field extension of the rationals contains no elements of arbitrarily small, positive heights. Amoroso, Bombieri, David, Dvornicich, Schinzel, Zannier and others exhibited many infinite field extensions of the rationals with a height gap. For example, the maximal extension of any number field with abelian Galois group has this property. I will present the history of this problem and talk about a new non-abelian example and its application.