The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory (Inter-universal Teichmüller Theory Summit 2016)

Abstract

Inter-universal Teichmüller theory may be described as a construction of certain canonical deformations of the ring structure of a number field equipped with certain auxiliary data, which includes an elliptic curve over the number field and a prime number ≥ 5. In the present paper, we survey this theory by focusing on the rich analogies between this theory and the classical computation of the Gaussian integral. The main common features that underlie these analogies may be summarized as follows: · the introduction of two mutually alien copies of the object of interest; · the computation of the effect -- i.e., on the two mutually alien copies of the object of interest -- of two-dimensional changes of coordinates by considering the effect on infinitesimals; · the passage from planar cartesian to polar coordinates and the resulting splitting, or decoupling, into radial -- i.e., in more abstract valuation-theoretic terminology, "value group" -- and angular -- i.e., in more abstract valuation-theoretic terminology, "unit group" -- portions; · the straightforward evaluation of the radial portion by applying the quadraticity of the exponent of the Gaussian distribution; · the straightforward evaluation of the angular portion by considering the metric geometry of the group of units determined by a suitable version of the natural logarithm function. [Here, the intended sense of the descriptive "alien" is that of its original Latin root, i.e., a sense of abstract, tautological "otherness".] After reviewing the classical computation of the Gaussian integral, we give a detailed survey of inter-universal Teichmüller theory by concentrating on the common features listed above. The paper concludes with a discussion of various historical aspects of the mathematics that appears in inter-universal Teichmüller theory.