相关论文: A young person's guide to the Hopf fibration

The Hopf fibration is an important object in mathematics and physics. A landmark discovery in topology and a fundamental object in the theory of Lie groups, the Hopf fibration has a wide variety of physical applications including magnetic…

We describe the construction of the slice fibration of a given one.

We give a rather informal introduction to the theory of mixed Hodge modules for young mathematicians.

Heinz Hopf's famous fibrations of the 2n+1-sphere by great circles, the 4n+3-sphere by great 3-spheres, and the 15-sphere by great 7-spheres have a number of interesting properties. Besides providing the first examples of homotopically…

In this survey, we remind some fibrations structure theorems (also called Milnor's fibrations) recently proved in the real and complex case, in the local and global settings. We give several Poincar\'e-Hopf type formulae which relates the…

This is a preprint version of a chapter for Handbook of Algebra.

Here we study geodesics connecting two given points on odd-dimensional spheres respecting the Hopf fibration. This geodesic boundary value problem is completely solved in the case of 3-dimensional sphere and some partial results are…

We give a description up to homeomorphism of $S^3$ and $S^2$ as classifying spaces of small categories, such that the Hopf map $S^3\to{}S^2$ is the realization of a functor.

The Hopf fibration is an example of a texture: a topologically stable, smooth, global configuration of a field. Here we demonstrate the controlled sculpting of the Hopf fibration in nematic liquid crystals through the control of point…

An affine Hopf fibration is a fibration of n-dimensional real affine space by p-dimensional pairwise skew affine subspaces. An example is a fibration of 3-space by pairwise skew lines, the result of the central projection of the classical…

Following a brief review on the history and the current development of fiber optics, the significance of teaching fiber optics for science and non-science major college students is addressed. Several experimental demonstrations designed to…

This note gives a brief and `crash' introduction to the method of Homogenization with the use of wave equation and diffusion equation with periodic in space coefficients as instructive examples. We expose the method with the use of an…

Astounding properties of biological sensors can often be mapped onto a dynamical system in the vicinity a bifurcation. For mammalian hearing, a Hopf bifurcation description has been shown to work across a whole range of scales, from…

These are slightly informal lecture notes intended for graduate students about the standard local theory of holomorphic foliations and vector fields. Though the material presented here is well-known some of the proofs differs slightly from…

We construct an explicit diffeomorphism taking any fibration of a sphere by great circles into the Hopf fibration, using elementary geometry--indeed the diffeomorphism is a local (differential) invariant, algebraic in derivatives.

Given any smooth fibration of the unit 3-sphere by great circles, we show that the distribution of 2-planes orthogonal to the great circle fibres is a tight contact structure, a fact well known in the special case of the Hopf fibrations.…

By combining tools from different areas of mathematics, we obtain 3D visualizations of elliptic curves over different fields that faithfully capture the underlying algebra and geometry.

The Hopf fibration mapping circles on a 3-sphere to points on a 2-sphere is well known to topologists. While the 2-sphere is embedded in 3-space, four-dimensional Euclidean space is needed to visualize the 3-sphere. Visualizing objects in…

With the help of a new type of functionals we study manifolds diffeomorphic to $S^2\times S^2$ and establish, in particular, the Hopf conjecture.

We present the explicit formulae relating Hopf maps with Wigner's little groups. They, particularly, explain simple action of group on a fiber for the first and second Hopf fibrations, and present most simplified form for the third one.…