Mathematics has always been a subject which fascinates great minds. Mathematicians have been trying to solve all different kinds of problems. In the past, mathematics was a subject which is purely theoretical. You will see lots of lemmas, theorems, corollary etc. This is the essence of PURE MATHEMATICS. Great mathematicians like Hilbert, Banach, Galois etc have established results in different areas of pure mathematics. Some of these lie in abstract algebra and analysis.
Now we are in the age of the 21st century and mathematics has evolved into a subject where applications in other branches of sciences are very important. Some of these areas of APPLIED MATHEMATICS are mathematical physics, operations research, financial mathematics, wavelets and signal processing etc. Scientists and engineers have to work with mathematicians because the underlying theory in all branches of science is still mathematics. Newton and Einstein were physicists but yet their most fascinating results are in mathematics. Advancement in science and technology rely very much on applied mathematics. Nevertheless,we should not forget that mathematics should be taught as a theoretical subject.
The following are the various main topics in Mathematics(Pure and Applied Mathematics) at university level
Algebra: Group Theory, Number Theory
Analysis: Differential and Integral Calculus of single variable and multi-variables functions, Functional Analysis, Complex Analysis
Geometry: Differential Geometry,
Differential Equations: Ordinary Differential Equations, Partial Differential Equations
Linear Algebra: Systems of Linear Equations, Linear Spaces, Eigenvalues and Eigenvectors, Inner Product Spaces
Mathematical Methods: Method of Separation of Variables in Solving PDE, Fourier Series, Sturm Liouville Problems, Laplace and Fourier Transform, Calculus of Variations
Operations Research: Linear Programming, Nonlinear Programming, Optimisation and Game Theory
Discrete Mathematics: Combinatories and Graph Theory
Mathematical Physics: Newtonian Mechanics, Classical Mechanics, Special Relativity, General Relativity, Nonlinear Dynamics
