The three branches of modern mathematics are algebra, geometry and analysis. The definition of mathematics is to study a set and some structure on it, which is a formal science. Set theory and logic are its foundation and proof is its soul. Because of its close relationship with natural science, especially physics, mathematics is sometimes listed as one of the six basic disciplines of natural science. Undefined concepts in mathematics are set, and everything else is defined. The standard form of mathematics is axiomatic method, that is, a set and a structure on a set are endowed with a set of axioms, and all other theories are deduced and proved by this set of axioms. Structures on a set are defined as some relationships between geometric elements or subsets. Originally, it can be divided into three categories: the order structure describing the order relationship, the algebraic structure describing the operation relationship, and the topological structure describing the neighborhood relationship. These structures can be combined with each other to form other complex structures, such as geometric structures and measurement structures. Various sets or spaces formed by these structures are the contents of different branches of mathematics. Algebra studies a set of algebraic structures, such as groups, rings, bodies, fields, modules, lattices, linear spaces and various inner product spaces. These structures were originally abstracted from the study of elementary algebra or elementary number theory and equation theory. Algebra includes: elementary algebra, elementary number theory, advanced (linear) algebra, abstract algebra (group theory, ring theory, field theory, etc. ), representation theory, multilinear algebra, algebraic number theory, analytic number theory, differential algebra, combinatorial theory and so on. Geometry studies sets with several geometric topological structures, such as affine space, topological space, metric space, affine inner product space, projective space, differential manifold and so on. Originally developed from Euclidean geometry. Geometry includes: elementary (Euclidean synthesis) geometry, analytic geometry, affine geometry, projective geometry, classical differential geometry, point set topology, algebraic topology, differential topology, global differential geometry, algebraic geometry and so on. Analytics studies sets with several topological measures and the function spaces defined on these sets, such as measurable measure space, normed space, Banach space, Hilbert space and probability space. , developed from calculus. Analysis includes: mathematical analysis, ordinary differential equation, complex variable function theory, real variable function theory, partial differential equation, variational method, functional analysis, harmonic analysis, probability theory and so on.