Varieties to Schemes: Generalizing Geometric Objects. Part 1

Classical Algebraic Geometry has been concerned with the solutions of systems of polynomials throughout a large part of the history of mathematics. In recent decades, a shift has occurred, where the scope of the subject of AG was enlarged to study new objects, known as schemes. This series of talks will focus on the development of the definition of an abstract scheme from the classical notion of an affine variety, as well as the reasons this leap in generality was made. Assuming that time is on our side, similar examples of generalization in the field of differential geometry will be discussed, and some of the features shared by the generalization processes will be described. A little background in point set topology and abstract algebra would benefit the audience but is certainly not necessary to attend.