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Stage 4 Biostatistics

Biostatistics (optional):
When mathematics is applied to biology or medical science, a subject with significant importance. For more free online material, see also John Hopkins Bloomberg School of Public Health's biostatistics section.

In general:

Statistical Methods in Epidemiology:

Design and Analysis of Clinical Trials:

Longitudinal Data Analysis:

Survival Analysis:

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Stage 4 Statistics

Statistics (optional):
May I put a quotation here:
If the results disagree with informed opinion, do not admit a simple logical interpretation, and do not show up clearly in a graphical presentation, they are probably wrong. There is no magic about numerical methods, and many ways in which they can break down. They are a valuable aid to the interpretation of data, not sausage machines automatically transforming bodies of numbers into packets of scientific fact.
(by F.H.C. Marriott, cited in Johnson and Wichern)

In general (mainly inference):

Statistical Models and Regression:

Multivariate Analysis:

Bayesian Statistics:
The BUGS Project web resources directs you to many other material.

Simulation and the Monte Carlo Method:

Nonparametric Statistics:

Categorical Data Analysis:

Data Mining:

Time Series:

Further Reading and Reference:

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Stage 4 Probability

Probability (optional):
These days, studying probability without measure is like studying physics without calculus. If you have done some baby measure theory in stage 3, you are probably ready for the followings.

Check out Probability Theory As Extended Logic for collected probability papers.

Probaility built upon Measure Theory:

Stochastic Processes:

Stochastic Analysis:

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Stage 4 Mathematical Physics

Mathematical Physics (optional):
Advanced studies of modern physics rely heavily on mathematics. In fact, throughout history, these two subjects are highly related.
Introductory level:
Relatively advanced level:
Mathematical methods in physics:
Relativity:
Quantum Mechanics:
Quantum Field Theory:
String Theory:
Chaos:
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Stage 4 Topology

Topology:
See Geometry & Topology Monographs for further study. Also consult Allen's list of recommended books on topology [.pdf].

In general:

Geometric Topology:

Algebraic Topology:

K-theory:

Differential Topology:

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Stage 4 Further Calculus

Further Calculus (optional):

Ordinary Differential Equations:

Partial Differential Equation:
Differential equation with several variables, i.e. equations with partial derivatives. Particularly useful in physics, e.g. heat, electrostatics, electrodynamics, etc.

Calculus of Variations:

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Stage 4 Number Theory

Number Theory:
There are several branch of Number Theory and we focus in algebraic and analytic here. As the title indicated, algebraic number theory employs algebra result to number theory, analytic one applies analysis theory. It is recommended you have a solid background of algebra before you begin to study algebraic number theory. Riemann hypothesis is one of the Millennium Prize Problems. Fermat's last theorem and the Poincaré conjecture have been solved, this is probably the biggest challenge to date.

This page consists links to online number lecture notes. Algebraic Number Theory Archives provide papers for further study.

In general:

Algebraic Number Theory:

Class Field Theory:

Analytic Number Theory:

Riemann Zeta Function and the Hypothesis:

Further readings (including Modular Forms, Elliptic Curves, etc.):
See Franz Lemmermeyer's page for further study.

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