Some Riemannian geometric proofs of the fundamental theorem of algebra
If a (non-constant) polynomial has no zero, then a certain Riemannian metric is constructed on the two dimensional sphere. Several geometric arguments are then shown to contradict this fact. M.S.C. 2010: 12E05, 53C20.
Yet another application of the Gauss-Bonnet Theorem for the sphere
where K and dAg denote the Gaussian curvature and the area element of g, respectively. In particular, this implies the existence of at least an elliptic point of (S, g). In this short note we prove the Fundamental Theorem of Algebra as a consequence of this fact. As far as the authors know, this proof is new. Moreover, it does have the advantage of being directly based on the work of Gauss, who was the first mathematician giving an universally accepted proof of the Fundamental Theorem of Algebra (see [2], [3]). Hence our argument can also have some historical and pedagogical interest. As it is well known, the stereographic projection allows to identify the sphere S with the one-point compactification Ĉ = C ∪ {∞} of the complex plane C. Let us take p(z) = a0 + a1z+ · · ·+ anz n with a0 an 6= 0 and let us assume that p(z) 6= 0 for all z ∈ C. We set p∗(z) = an + an−1z + · · ·a0z , that is, the i-th coefficient of p∗(z) is the (n− i)-th coefficient of p(z). Clearly, p∗(z) = z n p(1/z) for all z ∈ C \ {0}. Put
Heritage Sites of Astronomy and Archaeoastronomy in the context of the UNESCO World Heritage Convention: A Thematic Study
"Following the publication of the first ICOMOS–IAU Thematic Study (“TS1”) in 2010, the IAU requested its Working Group on Astronomy and World Heritage to develop particular case studies in greater detail, so as to explore further and clarify some of the key issues highlighted in TS1 that can arise in the particular case of astronomical heritage sites. In doing so, it would further encourage and aid State Parties in the development of nominations. In collaboration with ICOMOS, nine “extended case studies” were duly prepared for discussion at a workshop held at Mount Cook, New Zealand, in June 2012 and presented at the IAU General Assembly in August of that year. 
 
A particularly complex issue is the recognition and protection of dark skies. Dark sky areas cannot in themselves be considered as potential World Heritage Sites, but a thematic chapter by Michel Cotte considers a range of ways in which dark sky values can be interrelated with broader cultural or natural values of a place and thereby contribute to its overall cultural or natural value and potential OUV. Other issues explored in TS2 include the need to balance archaeoastronomical considerations in the context of broader archaeological and cultural values; the potential for serial nominations, for example among groups of monuments whose astronomical significance is only evident from the group as a whole; and management issues such as preserving the integrity of astronomical sightlines through the landscape. 
 
The case studies included in TS2 include seven-stone antas (prehistoric dolmens) in Portugal and Spain, the thirteen towers of Chankillo in Peru, the astronomical timing of irrigation in Oman, Pic du Midi de Bigorre Observatory in France, Baikonur Cosmodrome in Kazakhstan, and Aoraki–Mackenzie International Dark Sky Reserve in New Zealand. A case study on Stonehenge, already a World Heritage Site, focuses on preserving the integrity of the solstitial sightlines." 
 
Source: UNESCO portal to the heritage of astronomy