European Congress of Mathematics pp 47-53 | Cite as

# An Introduction to Non-Commutative Mori Theory

## Abstract

Abstract. The classification of algebraic surfaces of Enriques and Kodaira, and its conjectured higher dimensional generalisation by S. Mori et al, leaves as many questions unanswered as it solves; for example: which surfaces are hyperbolic? The highly conceptual nature of Mori’s programme permits its extension to the birational classification of ordinary differential equations, and this in its full generality is the proper definition of non-commutative Mori theory. A remarkable observation of [2] is that not only can much of Mori’s rational curve technology be brought to bear on this problem, but that many of the basic theorems of classification depend not on the dimension of the space on which the ODE is defined, but rather the dimension of its solutions. This suggests that, despite the difficulty of Mori’s programme for higher dimensional varieties, its analogue for foliations by curves in any dimension will succeed. Whence a conjectural classification for ODEs on surfaces, and an exhaustive, yet wholly algebraic understanding of their hyperbolicity by way of the method of dynamic Diophantine approximation. A refined version of the programme leads similarly to the analogue of the Mordell conjecture for algebraic surfaces over function fields, which in [13] it had already succeeded in establishing under the hypothesis of positive index. Innovation in Diophantine approximation would lead to its applicability in arithmetic per se. Other applications are anticipated.

## Keywords

Abelian Variety Mori Theory Algebraic Surface Diophantine Approximation Kodaira Dimension## Preview

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