Related papers: A young person's guide to the Hopf fibration
A summary of an experimental course on fractals is given that was held for young learners at age 12. The course was a part of Epsilon camp, a program designed for very gifted students who have already demonstrated high interest in studying…
These are lecture notes of a course given in Pisa, SNS, in february 2002. They provide a classification of holomorphic foliations of nongeneral type on compact Kaehler surfaces.
Self-reflection is important metacognitive skill, enabling students to build coherence into their learning and embed content in a broader context. While various pedagogical techniques exist to encourage student reflection, little research…
We classify global surfaces of section for the Reeb flow of the standard contact form on the 3-sphere, defining the Hopf fibration. As an application, we prove the degree-genus formula for complex projective curves, using an elementary…
Noncommutative geometry of quantised contact spheres introduced by Omori, Maeda, Miyazaki and Yoshioka is studied. In particular it is proven that these spheres form a noncommutative Hopf fibration in the sense of Hopf-Galois extensions.…
In this paper, we describe a new surprising example of a fibration of the Clifford torus S3 x S3 in the 7-sphere by great 3-spheres, which is fiberwise homogeneous but whose fibers are not parallel to one another. In particular it is not…
Lecture notes for an introductory course in elementary particles.
We explain the setup for using the pde2path libraries for Hopf bifurcation and continuation of branches of periodic orbits and give implementation details of the associated demo directories. See [Uecker, Comm. in Comp. Phys., 2019] for a…
This note is the written version of conversations with young colleagues on unofficial history, general ideas, unexpected facts and open problems concerning tilting theory.
Let p be a finite regular covering on a 2-sphere with at least three branch points. In this paper, we construct a local signature for the class of fibrations whose general fibers are isomorphic to the covering p.
This is a set of expository lecture notes created originally for a graduate course on holomorphic curves taught at ETH Zurich and the Humboldt University Berlin in 2009/2010. The notes are still incomplete, but due to recent requests from…
We present a simple, and robust method for writing long period fiber gratings with low polarization dependent losses. Our method is based on utilizing mechanical vibrations of the tapered fiber while pooling it. Our method enables real-time…
This article is a survey on recent contributions to an effective version of Bautin's theory about the bifurcation of periodic orbits (limit cycles). The analysis of Hopf bifurcations of higher order is possible by use of the return mapping.…
The purpose of this short note is to show the interplay between math outreach and conducting original research, in particular how each can build off the other.
These are the notes from my courses on the arithmetic of quadratic forms.
Hopf bifurcations have been studied perturbatively under two broad headings, viz., super-critical and sub-critical. The criteria for occurrences of such bifurcations have been investigated using the renormalization group. The procedure has…
We classify global surfaces of section for flows on 3-manifolds defining Seifert fibrations. We discuss branched coverings -- one way or the other -- between surfaces of section for the Hopf flow and those for any other Seifert fibration of…
We show that some of centeral fibers of degenerations of hyperelliptic curves are realized as those trigonal curves. In particular, any hyperelliptic curve can be the central fiber of a degeneration of trigonal curves.
This is a book aimed at graduate students and researchers in symplectic geometry, based on a course I taught in 2019. The primary message is that the base of a Lagrangian torus fibration inherits an integral affine structure, which you can…
The Hopf fibration has inspired any number of geometric structures in physical systems, in particular in chiral liquid crystalline materials. Because the Hopf fibration lives on the three sphere, $\mathbb{S}^3$, some method of projection or…