1. The Unity of Mathematics 2. The concept of Space 2.1. Projective Geometry 2.2. The Angel of Geometry and the Devil of Algebra 2.3. Non-Euclidean Geometry 2.4. Symmetries 2.5. Line element and Riemannian geometry 2.6. Noncommutative Geometry 2.7. Grothendieck’s Motives 2.8. Topos theory 3. Fundamental Tools 3.1. Positivity 3.2. Cohomology 3.3. Calculus 3.4. Trace and Index Formulas 3.5. Abeli...

The history of relativity is structured for most commentators by two landmark discoveries due to Albert Einstein: the special theory (1905) and the general theory of relativity (1915). To get from one theory to the other, we know that Einstein relied on a certain number of fundamental concepts, such as the equivalence principle, and a few key mathematical techniques, for instance, the absolute ...

Non-Euclidean geometry can be taught to students using astronomical images. By photographs of the Moon taken with a smartphone through simple telescope, we were able introduce these concepts high-school and lower-level college students. We teach how calculate lengths mountain ranges or areas craters on Moon's surface ideas geodesics spherical triangles. Students see that accurate measurements c...

This attractive little volume consists of two major components, together with a number of shorter sections. The major components are labeled " Introduction " and " Appendix " respectively , both designations grossly understating their content. The " Appendix " consists of Bolyai's revolutionary tract, with the subtitle " THE SCIENCE ABSOLUTE OF SPACE Independent of the Truth or Falsity of Eucli...

In this paper, we study fuzzy substructures in connection withHv-structures. The original idea comes from geometry, especially from thetwo dimensional Euclidean vector space. Using parameters, we obtain a largenumber of hyperstructures of the group-like or ring-like types. We connect,also, the mentioned hyperstructures with the theta-operations to obtain morestrict hyperstructures, as Hv-groups...

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consist of an incomplete set of distances and the output is a set of points in Euclidean space realizing those given distances. We survey the theory of Euclidean distance geometry and its most important applications, with special emphasis o...