higher geometry / derived geometry
Ingredients
Concepts
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
Constructions
Examples
derived smooth geometry
Theorems
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)
There are two circle constants used in geometry, the more common circle constant $\pi$ representing the straight angle, and the less common circle constant $\tau = 2 \pi$ representing a full turn.
Leonard Euler original definition of the circle constant $\pi$ was the constant valued as 6.283…, which is equivalent in today’s notion as $\tau$. Euler’s definition was often used in competition with the definition 3.1415.. until the late 1700s, when mathematicians collectively decided on standardizing $\pi$ as 3.1415…
However, in 2010, Michael Hartl proposed that the constant 6.283… which equals $2 \pi$ is much better for mathematics than $\pi$, as it would get rid of a significant number of factors of 2 from many equations, and is intuitively better for teaching and understanding subjects such as trigonometry and geometry, and proposed using the symbol $\tau$ for $2 \pi$. Since then, a few mathematicians such as Peter Harremoës have used $\tau$ instead of $\pi$ for research mathematics, and $\tau$ has been incorporated into a number of programming languages such as Python and Rust.
Euler, Leonard (1746). Nova theoria lucis et colorum. Opuscula varii argumenti, p.169-244.
Segner, Johann Andreas von (1761). Cursus Mathematicus: Elementorum Analyseos Infinitorum Elementorum Analyseos Infinitorvm (in Latin). Renger. p. 374. “Si autem π notet peripheriam circuli, cuius diameter eſt 2”
Hartl, Michael (2010). The Tau Manifesto
Tran, Andy (2017). Tau: The True Circle Constant. (slides)
Harremoës, Peter (2017). “Bounds on tail probabilities for negative binomial distributions”. Kybernetika. 52 (6): 943–966. arXiv:1601.05179. doi:10.14736/kyb-2016-6-0943. S2CID 119126029.
Last revised on May 16, 2022 at 21:58:07. See the history of this page for a list of all contributions to it.