Non-Euclidean Geometries: This family of geometries was obtained by altering the parallel postulate of classical Euclidean geometry, mainly through the work of Gauss, Lobachevsky and Bolyai in the early nineteenth century. Their discovery had profound implications for general mathematics, philosophy and physics. Current research focuses on the work of early predecessors and on the failed attempts to prove the parallel postulate.
Other Geometries and Topology: The rise of a hierarchy of geometries of ascending levels of generality (metric, projective, Riemannian geometry, etc.) in the nineteenth century and their organization under Kline's Erlanger Program are the subjects of numerous papers and monographs.
Infinitesimal Calculus and Analysis: From Newton and Leibniz and their predecessors to recent developments in the theory of functions and functionals, the history of the calculus is entangled with practically all other developments in mathematics since the seventeenth century. Non-Archimedean and non-standard analysis are mostly twentieth century developments intimately connected to advances in mathematical logic and foundations research.
Algebra and Equations Theory: The rise of algebra in the sixteenth century with the work of such figures as Cardano, Tartaglia and Viete, was initially driven by the need to find solutions to elementary equations of degree 3 and higher. The history of abstract algebra follows the discovery and development of abstract structures, starting with group and invariance theory and advancing to rings, ideals and other novelties of nineteenth century mathematics.