“Part III Mathematics is most simply described as a taught Master’s course in mathematics; but this simple description is inadequate for the culminating part of the oldest and most famous mathematics examination in the world.”
— T. W. Körner, An Unofficial Guide To Part III

Part III Guide to Courses

Pure Mathematics (DPMMS):

- Algebra:
-- Commutative Algebra
-- Finite Dimensional Lie and Associative Algebras
-- Profinite Groups
-- Representation Theory of Symmetric Groups
-- Lie Algebras, Vertex Algebras and Shtukas
-- Sofic Groups

- Algebraic Geometry:
-- Algebraic Geometry
-- Geometric Aspects of p-adic Hodge Theory
-- Algebraic Surfaces

- Analysis and PDEs:
-- Introduction to Nonlinear Analysis
-- Analysis of Partial Differential Equations
-- Metric Embeddings
-- Fractal Geometry
-- The Minimal Surface Equation and Related Topics

- Combinatorics:
-- Topics in Combinatorics
-- Additive Combinatorics
-- Ramsey Theory
-- Higher-order uniformity and applications

- Differential Geometry and Topology:
-- Algebraic Topology
-- Differential Geometry
-- Complex Dynamics
-- Homotopy Theory
-- Mapping Class Groups
-- Symplectic Topology
-- Random Groups and related topics

- Foundations:
-- Model Theory
-- Category Theory
-- Infinite Games

- Number Theory:
-- Local Fields
-- Elliptic Curves
-- Multiplicative Functions
-- Modular Forms
-- Algebraic Number Theory
-- Analytic Number Theory
-- Diophantine Approximation

Statistical Laboratory (DPMMS):

- Probability:
-- Advanced Probability
-- Percolation and Related Topics
-- Mixing Times of Markov Chains
-- Stochastic Calculus
-- Information Theory
-- Random Planar Geometry

- Statistics:
-- Modern Statistical Methods
-- Causal Inference
-- Statistics in Medical Practice (Statistics in Medicine)
-- Topics in Statistical Theory
-- Robust Statistics
-- Astrostatistics
-- Analysis of Survival Data (Statistics in Medicine)
-- Statistical Learning in Practice
-- Functional Data Analysis

Applied Mathematics and Theoretical Physics (DAMTP):

- Applied and Computational Analysis:
-- Inverse Problems
-- Distribution Theory and Applications
-- Numerical Solution of Differential Equations

- Astrophysics:
-- Structure and Evolution of Stars
-- Galaxy Formation
-- Planetary System Dynamics
-- Extra Solar Planets
-- The Life and Death of Galaxies
-- Astrophysical Fluid Dynamics
-- Astrophysical Black Holes
-- Astrostatistics
-- Dynamics of Astrophysical Discs
-- Binary Stars

- Continuum Mechanics:
-- Slow Viscous Flow
-- Fluid Dynamics of Climate
-- Perturbation Methods
-- Non-Newtonian Fluid Mechanics
-- Fluid Dynamics of the Environment
-- Theoretical Physics of Soft Condensed Matter
-- Computational Methods in Fluid Dynamics
-- Hydrodynamic Stability
-- Fluid Dynamics of the Solid Earth

- Math Biology:
-- Mathematical Phyllotaxis

- Particle Physics, Quantum Fields and Strings:
-- Quantum Field Theory
-- Symmetries, Fields and Particles
-- Statistical Field Theory
-- Non-Equilibrium Statistical Field Theory
-- Advanced Quantum Field Theory
-- Standard Model
-- String Theory
-- Supersymmetry
-- Classical and Quantum Solitons
-- Physics Beyond the Standard Model

- Philosophy of Physics:
-- Philosophical Aspects of Quantum Fields
-- Further Topics in Philosophy of Quantum Field Theory

- Quantum Computation, Information and Foundations:
-- Quantum Computation
-- Quantum Information Theory

- Relativity and Cosmology:
-- General Relativity
-- Cosmology
-- Black Holes
-- Field Theory in Cosmology
-- Applications of Differential Geometry to Physics
-- Applications of Analysis in Physics
-- Gauge / Gravity Duality

https://www.maths.cam.ac.uk/postgrad/...