1. abstract algebra:
A branch of modern algebra, which was formed in the 1920s, expanded the research object from numbers in primitive algebra to more general elements, and studied their algebraic operation rules and properties as well as various algebraic structures. Abstract algebra has penetrated into different branches of mathematics and combined with other mathematical disciplines, resulting in new mathematical theories such as algebraic geometry and algebraic theory.
2. Topological structure:
A branch of modern algebra, formed in the 1920s and 1930s, studies the invariant global properties of geometric figures under continuous deformation. Modern mathematics has obtained many profound results from topology, which cannot be obtained by other methods. It is the most colorful branch of mathematics in the 20th century, and it has more and more important applications in natural science and engineering technology, such as physics, molecular biology, biochemistry and other disciplines.
3. Functional analysis:
A branch of modern algebra was formed in the 1930s. It is developed from the study of variational problems, integral equations and theoretical physics. It comprehensively uses the viewpoints of function theory, geometry and algebra to study functions (also called functionals), operators and limit theory in infinite dimensional vector space, which can be regarded as analytical geometry and mathematical analysis of infinite dimensional vector space. Functional analysis has applications in mathematical physics equations, probability theory, computational mathematics, quantum physics and other branches, and is also a mathematical tool for studying infinite freedom physical systems. It can be said that it is the most comprehensive basic subject of modern mathematics in the 20th century.