The CDT begins with four foundation modules and these are designed to bring all students within the cohort to a similar level of understanding regarding key topics. If you haven't studied some of these topics previously then it may be a good idea to take some time to do so before arriving at the CDT.
Introduction to Partial Differential Equations
Recommended prerequisites include undergraduate-level advanced calculus, linear algebra, and ODE theory and some exposure to complex analysis.
W. Strauss, Partial Differential Equations: An Introduction, John Wiley and Sons, 1992.
H. Weinberger, A First Course in Partial Differential Equations, with Complex Variables and Transform Methods, Dover, 1965.
Measure and Probability Theory
It is expected that students will have studied undergraduate analysis, probability and topology.
L.C. Evans and R.F. Gariepy, Measure theory and Fine properties of functions, CRC Press, Boca Raton, FL, 1992
E.H. Lieb and M. Loss, Analysis, 2nd Edition, (Graduate Studies in Mathematics), American Mathematical Society, 2001
D. Williams, Probability with Martingales, Cambridge University Press, 1995
Function Spaces and Distribution Theory
It is expected that students will have understanding of basic Functional Analysis and Lebesgue Integration.
In addition to basic properties of L^p spaces, the topics of Appendix D.1-D.4 and E.1-E.3 of the book of Evans on PDEs can serve as a rough check list of topics students should be familiar with or might want to read up on before the course.
L.C. Evans, Partial Differential Equations, American Mathematical Society, 2nd edition, 2010
Introduction to Differential Geometry
The Hitchin lecture notes on differential manifolds would be good for pre-reading.