# Mathematics Options

Choosing your options in mathematics is an important part of the Maths/Physics degree in the second, third and fourth years. Whereas it is possible in the first year to cover all the key areas of mathematics, this is very difficult in the second and later years - even for students studying maths on its own.

The maths modules can be loosely grouped into the four categories shown below. The natural areas for maths/physics students to concentrate on are **Applied Mathematics** and **Analysis**. The categories are rough and ready. Within the Applied Mathematics section, good modules to consider are the modules which are about physics or the mathematics directly used in physics. These are listed in the dropdown menu **Mathematical Physics** on the right-hand side. You should note that not all modules are taught every year and that from time to time new modules may be introduced and others discontinued.

Choices in pure mathematics in the second year are particularly important. If you want to continue with analysis beyond the second year, you should take *Metric Spaces*. If you want to continue to study algebra in later years, you should take * Algebra I: Advanced Linear Algebra* and *Algebra II: Groups and Rings*.

You can 'sample' more options than you are required to take before committing yourself. However, lecture modules are taught on the basis that you are keeping up with the material and completing the problem sheets.

Only the modules which would normally be taken by a significant number of Maths/Physics students are included in the course regulations and these are predominantly from the categories Applied Mathematics and Analysis. However, you are free to take almost any other module available to mathematics students as unusual options. Please note that you may not take Mathematical Methods for Physicists nor Mathematical Methods for Physicists III (which cover material you meet in the modules *Multivariable Calculus*, *Partial Differential Equations* and *Variational Principles*), nor may you take MA117 Programming for Scientists after year 1. To take an unusual option, you should obtain the necessary permission from the lecturer offering the module. You should then complete an unusual option form.

In addition to the options listed, it is possible to take certain modules, offered by other departments, as unusual options. You must consult your Personal Tutor, **in advance**, in order to obtain the necessary permission (you will also need to obtain approval from the department offering the module).

The maths modules are listed below by area. However, you should note that the provision does change from year to year. Fourth year (Maths List C) modules are in italics. Modules marked with an asterisk are those which are not listed explicitly in the Maths/Physics course regulations. The module Historical Challenges in Maths (homepage to appear) is due to run 19/20 for the first time.

**APPLIED MATHEMATICS**

**Second Year**

Partial Differential Equations (core); Multivariable Calculus (core); Methods of Mathematical Physics (core); Physics of Fluids (core);
Theory of ODEs;
Variational Principles (core); Introduction to Systems Biology; Differential Equations: Modelling and Numerics;

**Third and Fourth Years**

Fluid Dynamics (core for MMathsPhys students);

Topics in Mathematical Biology; Problem Solving^{*}; Theory of PDEs;
Numerical Analysis and PDEs^{*}; Markov Processes and Percolation Theory^{*};
Control Theory^{*};
Bifurcations, Catastrophes and Symmetry^{*}; Mathematical Modelling and PDEs^{*}; Approximation Theory and Applications^{*}; High-dimensional Probability^{*}; Maths of Machine Learning^{*}; *Dynamical Systems*;
*Quantum Mechanics: Basic Principles and Probabilistic Methods*;
*Population Dynamics: Ecology and Epidemiology*^{*};
*Applied Dynamical Systems*^{*};
*Atmospheric Dynamics*^{*};
*Mathematics and Biophysics of Cell Dynamics*;
*Statistical Mechanics*;
*Mathematical Acoustics*

**ANALYSIS**

**Second Year**

Analysis III (1-10, core); Metric Spaces

**Third and Fourth Years**

Complex Analysis; Measure Theory; Functional Analysis I: Applied Analysis; Functional Analysis II: Linear Analysis; Manifolds^{*}; *Ergodic Theory*^{*}; *Fourier Analysis*;
*Riemann Surfaces*^{*}; *Stochastic Analysis*^{*}; *Advanced PDEs*; *Brownian Motion*^{*};
*Advanced Real Analysis*^{*};

**ALGEBRA, COMBINATORICS and NUMBER THEORY**

**Second Year**

Combinatorics^{*}; Algebra I: Advanced Linear Algebra; Algebra II: Groups and Rings;
Combinatorial Optimization^{*};
Introduction to Number Theory^{*}

**Third and Fourth Years**

Rings and Modules^{*}; Matrix Analysis and Algorithms^{*}; Algebraic Number Theory^{*}; Galois Theory^{*}; Groups and Representations^{*};
Commutative Algebra^{*};
Set Theory^{*};
Combinatorics II^{*}; *Group Theory*^{*};
*Lie Groups*^{*};
*Geometric Group Theory*^{*};
*Ring Theory*^{*}; *Graph Theory*^{*};
*Analytical Number Theory*^{*};

**GEOMETRY and TOPOLOGY**

**Second Year**

Geometry

**Third and Fourth Years**

Fractal Geometry^{*}; Geometry of Curves and Surfaces^{*}; Introduction to Topology^{*};
Algebraic Topology^{*};
*Elliptic Curves*^{*};
*Algebraic Geometry*^{*}; *Differential Geometry*^{*}; *Cohomology and Poincare Duality*^{*}; *Algebraic Curves*^{*};