The direction of algebra and number theory can be roughly divided into: arithmetic geometry (integrating number theory and algebraic geometry), representation theory and traditional algebra and number theory.
The geometric directions are: low-dimensional topology and curvature flow, mirror symmetry, symplectic geometry and affine structure, noncompact bounded manifold, algebraic geometry.
The analysis direction can be roughly divided into four categories: classical analysis, functional analysis, harmonic analysis, nonlinear analysis and convex analysis. Classical analysis includes inequality theory, summability theory, approximation theory, special function theory and complex variable function theory. The active directions of functional analysis are: matrix analysis, operator theory, evolution equation, operator and function algebra. Harmonic analysis focuses on Fourier transform and wavelet transform in Euclidean space.
Differential equations (including ordinary differential and partial differential) have many important and active fields and topics: 1. Geometric analysis II. Parabola and reaction-diffusion equation 3. Elliptic partial differential equation 4. Ginzburg-Landau equation 5. Nonlinear Schrodinger equation 6. Conservation law equation 7. Naville-Stokes Equation 8. Dynamics and Pozmann Equation 9. Ordinary differential equation 10.
The research direction of discrete mathematics covers: 1. Graph coloring related issues, including point coloring, edge coloring, circle coloring, uniform coloring, T coloring, distance binary labeling and so on. 2. Graph decomposition 3. Algebraic graph theory iv. Combinatorial counting problem 5. Finite body and its application.
Probability direction coverage: 1. Related research and application of Markov process and diffusion process II. The related research of probability theory in financial field. Stochastic analysis and application of infinite dimensional space iv. Mathematical physics. Others.
Scientific calculation can be roughly divided into the theory and application of matrix calculation and the numerical theory and method of partial differential equations. Mainly through physical laws or assumptions, a suitable mathematical model is derived for scientific or engineering problems, and the problems are solved through mathematical analysis and numerical calculation or as pre-experiment predictions. The narrow sense of computational science is to design or apply effective numerical methods to solve some specific mathematical equations.
Mathematics employment situation
In the industrial field, IT is mainly the research and development institutions of large companies such as IT, energy, logistics, film and television. Algorithms in IT field, numerical calculation and simulation in energy field, network or optimization in logistics field, image animation modeling in film and television field, etc. There is also a great demand for this piece of high technology, such as the wind tunnel of aircraft, the aerodynamics of missiles and aerospace vehicles, and the simulation calculation of fluids by people who need to learn mathematics. The exploration of laws by human beings is bound to be more and more elaborate, which also provides a better platform for mathematicians to apply mathematics to practice more widely.
Financial engineering is also a very important employment direction. Mathematics plays an important role in this direction. Based on probability theory, it combines statistics, partial differential equation theory, computational mathematics and mathematical optimization theory.
Algebra and number theory, you can focus on computer coding and cryptography. Many large companies, especially IT, need a group of people to do research on cryptography and computer algorithms. Geometric direction, if you focus on low-dimensional topology, you can use computer graphics in the future. The analysis is mainly harmonic analysis and nonlinear analysis, which have many requirements in application.
The application of differential equations is the most prominent and the most important direction in applied mathematics. Differential equations are widely used in natural science, engineering and various mathematical problems.