JuliaCon 2020 | Manifolds in Julia – Manifolds.jl & ManifoldsBase.jl | Ronny Bergmann

From optimization to statistical modeling, many problems involve working with manifolds, smooth sets of points with nonlinear constraints.
We will show how to use these tools to construct new manifolds and implement algorithms on these manifolds.

In many scientific and engineering scenarios, measured data or model variables lie in smooth, nonlinear spaces like the sphere or the group of 3D rotations.
Such spaces often have a manifold structure, that is, they locally can be approximately linearized (i.e. they are locally diffeomorphic to a Hilbert space).
Equipping such manifolds with a (pseudo-)Riemannian metric tensor, which defines a ruler on the manifold, enables local computation of distances and angles.
These features enable construction of efficient algorithms that respect the nonlinear structure, for example for optimization, interpolation, and statistical modeling.

With these tools, you can for example explicitly locally linearize points using inverse retractions and orthonormal bases.
You can also compute intrinsic statistics, such as the Riemannian mean and variance, of data on manifolds.

Examples are the product manifold of two manifolds, the power manifold, the tangent bundle, and Lie groups.
Using a trait-based system, any manifold may be augmented with additional geometric structure, including various metrics, without sacrificing efficiency.

The interface can also be used to develop new, stand-alone projects.
It is minimally restrictive and makes it possible to write algorithms that are independent from any concrete implementations of the manifolds.

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[OlliWilkman]

Cool stuff. Working with light scattering, I often work with data defined on a sphere, a sphere times a circle, or a sphere times a sphere (two spherical directions, incident and outgoing light). I'll have to look into this Manifolds code to see if I can improve some of my things.