Groups of order 8 and 16

This document is rather a course about groups than a research paper. However, it can be of interest for many master students in mathematics which are devoted to the p-group classification theory. This paper is inspirated of the course of David Clausen from the university of Puget Sound (USA) on the classification of groups of order 16. His publication was made on GNU Licence. However, his publication contained some typo errors which make difficult the understanding of his method. Moreover, his proof, although interesting, is not the simplest. In my document, while keeping the development that he proposed, I greatly simplified his proof of classification of groups of order 16.Comment: 18 page

Intrinsic symmetry groups of links with 8 and fewer crossings

We present an elementary derivation of the "intrinsic" symmetry groups for knots and links of 8 or fewer crossings. The standard symmetry group for a link is the mapping class group \MCG(S^3,L) or \Sym(L) of the pair $(S^3,L)$. Elements in this symmetry group can (and often do) fix the link and act nontrivially only on its complement. We ignore such elements and focus on the "intrinsic" symmetry group of a link, defined to be the image $\Sigma(L)$ of the natural homomorphism \MCG(S^3,L) \rightarrow \MCG(S^3) \cross \MCG(L). This different symmetry group, first defined by Whitten in 1969, records directly whether $L$ is isotopic to a link $L'$ obtained from $L$ by permuting components or reversing orientations. For hyperbolic links both \Sym(L) and $\Sigma(L)$ can be obtained using the output of \texttt{SnapPea}, but this proof does not give any hints about how to actually construct isotopies realizing $\Sigma(L)$. We show that standard invariants are enough to rule out all the isotopies outside $\Sigma(L)$ for all links except $7^2_6$, $8^2_{13}$ and $8^3_5$ where an additional construction is needed to use the Jones polynomial to rule out "component exchange" symmetries. On the other hand, we present explicit isotopies starting with the positions in Cerf's table of oriented links which generate $\Sigma(L)$ for each link in our table. Our approach gives a constructive proof of the $\Sigma(L)$ groups.Comment: 72 pages, 66 figures. This version expands the original introduction into three sections; other minor changes made for improved readabilit

Compact manifolds with exceptional holonomy

In the classification of Riemannian holonomy groups, the exceptional holonomy groups are G2 in 7 dimensions, and Spin(7) in 8 dimensions. We outline the construction of the first known examples of compact 7- and 8-manifolds with holonomy G2 and Spin(7)

8 Lectures on quantum groups and q-special functions

Lecture notes for an eight hour course on quantum groups and $q$-special functions at the fourth Summer School in Differential Equations and Related Areas, Universidad Nacional de Colombia and Universidad de los Andes, Bogot\'a, Colombia, July 22 -- August 2, 1996. The lecture notes contain an introduction to quantum groups, $q$-special functions and their interplay. After generalities on Hopf algebras, orthogonal polynomials and basic hypergeometric series we work out the relation between the quantum SU(2) group and the Askey-Wilson polynomials out in detail as the main example. As an application we derive an addition formula for a two-parameter subfamily of Askey-Wilson polynomials. A relation between the Al-Salam and Chihara polynomials and the quantised universal enveloping algebra for $su(1,1)$ is given. Finally, more examples and other approaches as well as some open problems are given.Comment: AMS-TeX, 82 page

A supramolecular radical cation: folding-enhanced electrostatic effect for promoting radical-mediated oxidation.

We report a supramolecular strategy to promote radical-mediated Fenton oxidation by the rational design of a folded host-guest complex based on cucurbit[8]uril (CB[8]). In the supramolecular complex between CB[8] and a derivative of 1,4-diketopyrrolo[3,4-c]pyrrole (DPP), the carbonyl groups of CB[8] and the DPP moiety are brought together through the formation of a folded conformation. In this way, the electrostatic effect of the carbonyl groups of CB[8] is fully applied to highly improve the reactivity of the DPP radical cation, which is the key intermediate of Fenton oxidation. As a result, the Fenton oxidation is extraordinarily accelerated by over 100 times. It is anticipated that this strategy could be applied to other radical reactions and enrich the field of supramolecular radical chemistry in radical polymerization, photocatalysis, and organic radical battery and holds potential in supramolecular catalysis and biocatalysis