Algebraic geometry deals with curves or surfaces (or more abstract generalizations of these) which can be viewed both as geometric objects and as solutions to algebraic (specifically, polynomial) equations.
The soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case. Every compact manifold is its own soul. In 1972, Cheeger and Gromoll proved the theorem by the generalization of a 1969 result by Gromoll and Wolfgang Meyer.
An ordinary differential equation is a differential equation containing one or more functions of one independent variable and the derivatives of those functions.
Logic is the discipline that deals with the methods of reasoning. On an elementary level, logic provides rules and techniques for determining whether a given argument is valid. Logical reasoning is used in mathematics to prove theorems, and in computer science to verify the correctness of programs.
Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. Fourier series makes use of the orthogonality relationships of the sine and cosine functions.
“As long as algebra and geometry have been separated, their progress has been slow and their uses limited; but when these two sciences have been united, they have lent each mutual forces, and have marched together towards perfection.”
The heat equation is among the most widely studied topics in pure mathematics, and its analysis is regarded as fundamental to the broader field of partial differential equations. Solutions to the heat equation are sometimes known as caloric functions.
In mathematics, a surface is a geometrical shape that resembles a deformed plane. The most familiar examples arise as boundaries of solid objects in ordinary three-dimensional Euclidean space R3, such as spheres.