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A conjecture of S. Ram\'{\i}rez states that every indecomposable non-degenerate involutive set-theoretic solution to the Yang-Baxter equation with dihedral permutation group of order $2n$ has cardinality $2n$. The conjecture is verified for odd $n$ and disproved for even $n$. The proof for odd $n$ is obtained from the more general result that the permutation group of a finite solution never acts as a Frobenius group.

One of the first remarkable results in the representation theory of artin algebras, due to Auslander and Ringel-Tachikawa, is the characterization of when an artin algebra is representation-finite. In this paper, we investigate aspects of representation-finiteness in the general context of exact categories in the sense of Quillen. In this framework, we introduce "big objects" and prove an Auslander-type "splitting-big-objects" theorem. Our approach generalises and unifies the known results from the literature. As a further application of our methods, we extend the theorems of Auslander and Ringel-Tachikawa to arbitrary dimension, i.e. we characterise when a Cohen-Macaulay order over a complete regular local ring is of finite representation type., Comment: v2: 30 pages, some misprints corrected, references updated, comments are welcome

We develop a method to construct all the indecomposable involutive set-theoretic solutions of the Yang-Baxter equation with a prime-power number of elements and cyclic permutation group. Moreover, we give a complete classification of the indecomposable ones having abelian permutation group and cardinality $pq$ (where $p$ and $q$ are prime numbers not necessarily distinct).

The Local-to-Global-Principle used in the proof of convexity theorems for momentum maps has been extracted as a statement of pure topology enriched with a structure of convexity. We extend this principle to not necessarily closed maps $f\colon X\ra Y$ where the convexity structure of the target space $Y$ need not be based on a metric. Using a new factorization of $f$, convexity of the image is proved without local fiber connectedness, and for arbitrary connected spaces $X$., Comment: 21 pages LaTeX2e; minor revisions, to appear in Topology and its Applications; Dedicated to Alan D. Weinstein, Dennis P. Sullivan, and in memory of Jerrold E. Marsden. arXiv admin note: substantial text overlap with arXiv:1009.2527

We extend the Local-to-Global-Principle used in the proof of convexity theorems for momentum maps to not necessarily closed maps whose target space carries a convexity structure which need not be based on a metric. Using a new factorization of the momentum map, convexity of its image is proved without local fiber connectedness, and for almost arbitrary spaces of definition. Geodesics are obtained by straightening rather than shortening of arcs, which allows a unified treatment and extension of previous convexity results., Comment: 19 pages LaTeX2e, Preprint 2009, see also: Convexity of Momentum Maps: A Topological Analysis, several parts of the content were presented at the Young Topologists Meeting 2010 in Copenhagen, Denmark, June 16-20, 2010, and at Geometry, Mechanics, and Dynamics: A workshop celebrating the 60th birthday of Tudor Ratiu at CIRM, Luminy, France, July 12-16, 2010