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Stage 2 Probability and Statistics

Probability and Statistics:
Well, being a well educated mathematician, you should have basic knowledge of statistics. Things to learn: basic probability theory (independent events, conditional probability, Bayes' Theorem), random variables (r.v.), expectation, convergence of r.v., maximum likelihood estimator, basic hypothesis testing, p-values are the basic. If you want to go a bit further, check out linear regression, linear model, residual, categorical predictors, logistic regression, ANOVA (analysis of variance) etc.

Probability:

Statistics:

Second course in Statistics:

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Stage 2 Differential Equations

Differential Equation:
Isaac Newton originally studies behaviour of dynamical systems using differential equations. Mathematical models are often described with differential equations and they are widely used in many fields, including physics, chemistry, biology, economics etc. Mathematical methods used in solving differential equation also play an important in advanced studies. Key topics to learn are: Wronskians, series solutions, reduction of order, variation of parameters, Frobenius normal form, Bessel's equation, Legendre's equation, two points boundary problem, Fredholm alternative, Green's function, complete orthogonal system, Sturm-Liouville problems (inhomogeneous, singular), basic partial differential equation, elliptic eigen-problems, heat equation and wave equation.
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Stage 2 Complex Variable

Complex Variables (Introductory Complex Analysis):
Another beautiful branch of mathematics. While you're in high school, you may wonder what's the point of introducing imaginary number i, does it really exists? Is it really useful? In an introductory complex analysis course, you will see the beauty of this construction. Gauss, Riemann, Weierstrass and Cauchy are key figures in this area. Key concepts to learn are: analytic functions, Cauchy-Riemann equations, contour integral, Cauchy-Goursat theorem, residues and poles.
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Stage 2 Calculus

Calculus (Introductory Real Analysis, Several Variables Calculus, Vector Calculus, etc):
 Learn the very basic concept of real analysis, like open set, close set, boundary point, closure, limit point, bounded set, connected set, compact set, Bolzano-Weierstrass theorem. To get yourself ready for the stage 3 analysis, make sure you understand those stuff. Next, the Lagrange multiplier, inverse and implicit function theorems can also be studied. For the several variables section, gradients, double, triple and surface integrals, cylindrical coordinates, Green's theorem and divergence theorem are the key things. Also learn basic Fourier series.
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Stage 2 Introductory Higher Algebra

Introductory Higher Algebra:
I've taken this course entitled "Finite Mathematics", a course for computer scientists, software engineers and pure mathematicians (optional but useful). Assuming you have learnt the very basic of number theory in the discrete mathematics section, you are ready to get a taste of higher algebra here. Things like prime numbers, tests for primality, Fundamental Theorem of Arithmetic, Fermat's little theorem, Gauss' lemma, Euler's theorem, Chinese remainder theorem and their applications (coding, RSA etc.) are concerned in this section. This is a bridge that connects first stage basic algebra, discrete mathematics and the third stage abstract algebra.
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Stage 2 Linear Algebra

Linear Algebra:
Vectors, vector spaces, linear transformations, multilinear map, inner product spaces, norms, orthogonality, Gram-Schmidt algorithm, QR-factorisation, least square, Householder algorithm, normal matrices, Jordan canonical forms, Cayley-Hamilton theorem, minimal and characteristic polynomials, direct sum decompositions, generalised eigenspaces, functions of matrices, exponentials of matrices, etc will be studied in this course. Such material can be applied to linear programming, computer graphics, fractals, and many areas in natural sciences and social sciences.
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Stage 2

Mathematics program in the University are constructed in the way that courses are offered in favor of other schools (Physics, Chemistry, Biology, Engineering, Economic, etc). For example, Physics students have to take linear algebra, several variable calculus, mathematical analysis and differential equation. As a consequence, textbooks are written and catalog like the following way. Several authors focus on the application and often do calculation without justifying, beware of it.
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