transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Arithmetic differential geometry is an approach to arithmetic which looks to find analogs of constructions in differential geometry, for example, arithmetic jet spaces. It has been developed primarily by Alexandru Buium.
According to this approach, the classical derivatives of differential geometry are replaced by p-derivations for prime $p$, such as the Fermat quotient
Buium explains that when working with a single prime his approach is consistent with Borger's absolute geometry, which is described as “an algebraization of our analytic theory” (Buium 17, p. 24). However, in the case of multiple primes Borger requires Frobenius lifts to commute, and this diverges from the non-vanishing ‘curvature’ Buium derives from non-commuting lifts. For him, the (“manifold” corresponding to) the integers, $\mathbb{Z}$, is “intrinsically curved”.
Alexandru Buium, Foundations of arithmetic differential geometry, 2017, AMS, Mathematical Surveys and Monographs Vol. 222, (AMS, Preface and Introduction).
Alexandru Buium, Arithmetic differential geometry, May 19, 2017, talk slides
Last revised on July 25, 2023 at 05:22:58. See the history of this page for a list of all contributions to it.