In the two-volume set 'A Selection of Highlights' we present basics of mathematics in an exciting and pedagogically sound way. This volume examines many fundamental results in Geometry and Discrete Mathematics along with their proofs and their history. In the second edition we include a new chapter on Topological Data Analysis and enhanced the chapter on Graph Theory for solving further classical problems such as the Traveling Salesman Problem.

This second edition gives a thorough introduction to the vast field of Abstract Algebra with a focus on its rich applications. It is among the pioneers of a new approach to conveying abstract algebra starting with rings and fields, rather than with groups. Our teaching experience shows that examples of groups seem rather abstract and require a certain formal framework and mathematical maturity that would distract a course from its main objectives. Our philosophy is that the integers provide the most natural example of an algebraic structure that students know from school. A student who goes through ring theory first, will attain a solid background in Abstract Algebra and be able to move on to more advanced topics. The centerpiece of our book is the development of Galois Theory and its important applications, such as the solvability by radicals and the insolvability of the quintic, the fundamental theorem of algebra, the construction of regular n-gons and the famous impossibilities: squaring the circling, doubling the cube and trisecting an angle. However, our book is not limited to the foundations of abstract algebra but concludes with chapters on applications in Algebraic Geometry and Algebraic Cryptography.