synthetic differential geometry, deformation theory
infinitesimally thickened point
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
Formal geometry is a highly overloaded term in mathematics, used in number of conceptually similar ways, usually meaning that we work in setup in which some crucial details of geometry or analysis are not present or satisfied, e.g.
we work with functions on “manifolds” but the functions do not necessarily converge, the geometry is rather based on topological algebras of formal power series; this is the formal geometry of Grothendieck school and the main notion is that of a formal scheme (or more general ind-schemes). There are also noncommutative versions like Kapranov's noncommutative geometry.
Gelfand’s formal geometry: study infinite dimensional manifolds of jet bundles and related objects coming from usual differential geometry, geometry of formal differential operators, study of related objects from homological algebra, including Gelfand’s formal manifold (homological vector field)
we talk about neighborhoods,or localizations, morphisms of spaces, but not about spectra and points (a part of noncommutative geometry is done in such style) – this is sometimes called “pseudogeometry”
in algebraic geometry: formal spectrum of an adic noetherian ring
See MathOverflow: formal-geometry, how-do-i-describe-the-gl-n-torsor-attached-to-a-smooth-morphism-of-relative-dimen,
For characteristic $p\gt 0$ case see
In a similar formal context Gelfand and collaborators introduced $(\mathfrak{A},\mathcal{D})$-systems
The $(\mathfrak{A},\mathcal{D})$-systems were partly motivated by the calculus of variations, formalizing further the setting of works of Gelfand and Dorfman.
Examples of sequences of local structures
geometry | point | first order infinitesimal | $\subset$ | formal = arbitrary order infinitesimal | $\subset$ | local = stalkwise | $\subset$ | finite |
---|---|---|---|---|---|---|---|---|
$\leftarrow$ differentiation | integration $\to$ | |||||||
smooth functions | derivative | Taylor series | germ | smooth function | ||||
curve (path) | tangent vector | jet | germ of curve | curve | ||||
smooth space | infinitesimal neighbourhood | formal neighbourhood | germ of a space | open neighbourhood | ||||
function algebra | square-0 ring extension | nilpotent ring extension/formal completion | ring extension | |||||
arithmetic geometry | $\mathbb{F}_p$ finite field | $\mathbb{Z}_p$ p-adic integers | $\mathbb{Z}_{(p)}$ localization at (p) | $\mathbb{Z}$ integers | ||||
Lie theory | Lie algebra | formal group | local Lie group | Lie group | ||||
symplectic geometry | Poisson manifold | formal deformation quantization | local strict deformation quantization | strict deformation quantization |