Below are comments about Part II courses, intended to expand on the prerequisites as given in the schedules. The majority have been sent to me by Part II students, describing what they felt the course needed. There are also comments (in italics) from lecturers and supervisors.
I'll update this page as more comments get sent to me. Feel free to
send comments!
 Algebraic Geometry
 The rings part of GRM is essential. Topological spaces from Met&Top/Analysis II are needed, but not at the same level of detail as for Algebraic Topology. This is often considered a difficult course and any course exposing students to formalizing geometric ideas is useful preparation even if the results are not directly needed (Part IB Geometry, Part II Algebraic Topology, Part II Riemann Surfaces).
 Algebraic Topology

Any material from Geometry will be treated from scratch, but previous
exposure is certainly useful.

EVERYTHING in Met&Top.

Analysis II is really not important (just helps to have played around
with compactness a bit more).

First half of groups IA (i.e. be able to do group theory).
 Met & Top is INCREDIBLY important, being generally fluent with the ideas of the course is key. Analysis II is scarcely relevant, other than building some more familiarity with compactness. IA Groups is key, mainly the parts about groups in the abstract rather than specific examples like Mobius Groups.

Any material from Geometry will be treated from scratch, but previous
exposure is certainly useful.
 Analysis of Functions
 It's essential to have taken Part II Probability & Measure and Part II Linear Analysis.
 Applications of Quantum Mechanics
 You should know Part IB Quantum Mechanics. From Part II Principles of Quantum Mechanics, you will need Dirac bra/ket notation, multiparticle systems, the idea of spin (although not the full angular momentum algebra), and perturbation theory.
 Applied Probability

Markov Chains revisited with some extra spice, nothing else really
needed.
 "Continuoustime Markov Chains" may have been a better course name. It builds on Part IB Markov Chains so most of the materials there are needed, but a lot of concepts are revisited during the course.

Markov Chains revisited with some extra spice, nothing else really
needed.
 Asymptotic Methods

Contour integration when doing method of steepest descent  Complex
analysis/methods useful
 Overlap of the course with Classical Dynamics when talking about Hamiltonians and Poisson Brackets  useful courses to do together

Contour integration when doing method of steepest descent  Complex
analysis/methods useful
 Automata and Formal Languages

Nothing (as far as I remember).
 No official prerequisite, but there are things that are quite similar in flavour to the set theory part of Numbers & Sets. Being comfortable with that will certainly help.

Nothing (as far as I remember).
 Classical Dynamics

This is a prettier version of Part IA Dynamics & Relativity, but
without the relativity. The focus is more on the structure of
classical mechanics, rather than solving problems. It is based on the
variational principle, and Part IB Variational Principles will be
useful, although the relevant material will be reviewed.
 IA D&R very useful: rigid bodies, rotating frames. Very similar to IB Variational Principles so if you enjoyed that you'll love this, e.g. Lagrangians, EulerLagrange equations.

This is a prettier version of Part IA Dynamics & Relativity, but
without the relativity. The focus is more on the structure of
classical mechanics, rather than solving problems. It is based on the
variational principle, and Part IB Variational Principles will be
useful, although the relevant material will be reviewed.
 Coding and Cryptography

Course is not too difficult and is mostly self contained. Basic
knowledge from Probability and Linear algebra assumed, so knowing them
helps.
 Mostly selfcontained, but useful to remember cyclic groups, ideals and polynomial rings from GRM in order to understand cyclic codes.

Course is not too difficult and is mostly self contained. Basic
knowledge from Probability and Linear algebra assumed, so knowing them
helps.
 Cosmology
 This is a straightforward course, that uses some basic facts from IA Dynamics and Relativity and IA Differential Equations, but is otherwise selfcontained.
 Differential Geometry
 The first part of the course is differential topology so Met & Top is essential. I heard people saying that the rest of the materials have a similar flavour with IB Geometry, but as someone who didn't take the course, I don't feel that it's particularly important.
 Dynamical Systems

There are no crucial requirements from Part IB. From Part IA, you
should know most of Differential Equations, and up to continuity in
Analysis I (for example, the Intermediate Value Theorem).

IA DEs  namely phase portraits and systems of ODEs.
 IA DEs  perturbation and bifurcation.

There are no crucial requirements from Part IB. From Part IA, you
should know most of Differential Equations, and up to continuity in
Analysis I (for example, the Intermediate Value Theorem).
 Electrodynamics

You should be comfortable with Part IB Electromagnetism. The
relativistic formalism of electromagnetism is reviewed, but rather
quickly. There will be Green's functions (from Part IB Methods)
and contour integrals (from Part IB Complex Methods).
 IA VC and IB EM very useful, especially the last part in EM about the fourvector representation.

You should be comfortable with Part IB Electromagnetism. The
relativistic formalism of electromagnetism is reviewed, but rather
quickly. There will be Green's functions (from Part IB Methods)
and contour integrals (from Part IB Complex Methods).
 Fluid Dynamics

Part IB Methods and Fluid Dynamics are essential. Pretty
much all of Methods except characteristics is used in this course in
one way or another (e.g., Fourier series, Green's Functions, methods
for Laplace/heat equations such as separation of variables, Dirac
delta function, PDEs). While one could find parts of Part IB Fluids
that are not strictly required (e.g. the last two lectures on rotating
flows), even that material contains further experience with stream
functions and the use of vector calculus that is valuable in Part II.
Regarding Part IA, Differential Equations and Vector Calculus are
essential in order to do IB Methods, and so are also required here.
 IB Fluids, although can be grasped with background knowledge and not necessarily deep understanding of this.

Part IB Methods and Fluid Dynamics are essential. Pretty
much all of Methods except characteristics is used in this course in
one way or another (e.g., Fourier series, Green's Functions, methods
for Laplace/heat equations such as separation of variables, Dirac
delta function, PDEs). While one could find parts of Part IB Fluids
that are not strictly required (e.g. the last two lectures on rotating
flows), even that material contains further experience with stream
functions and the use of vector calculus that is valuable in Part II.
Regarding Part IA, Differential Equations and Vector Calculus are
essential in order to do IB Methods, and so are also required here.
 Further Complex Methods
 Either of Part IB Complex Methods or Complex Analysis is sufficient for this course. A good knowledge of parts of Part IB Methods (solving PDEs using transforms) and Part IA Differential Equations (series solutions) would also be helpful.
 Galois Theory

Galois theory provides a connection between field theory and group
theory. Using Galois theory, certain problems in field theory can be
'reduced' to group theory, which in some sense is simpler and better
understood. Motivation for the subject lay in understanding the
symmetries in the roots of a polynomial and the results were used to
show the solubility or otherwise of a given polynomial equation of
some fixed degree. Thus the required prerequisites include Part IA
Groups, modular congruences from Part IA Numbers & Sets, together with
the groups section of Part IB Groups, Rings & Modules, basic ideas
from Parts IB Linear Algebra (basis and dimension), and some
elementary properties of rings (essentially the rings part of GRM).

GRM: None of G, ALL of R, none of M.
 So far it really has been mostly the ring section that is essential, although as we have only been dealing with fields we haven't really done much with ideals and overall it seems to me that the course is separate from GRM, but still requires good understanding of what was happening in GRM. The reducible/irreducible polynomial arguments are as in GRM.

Galois theory provides a connection between field theory and group
theory. Using Galois theory, certain problems in field theory can be
'reduced' to group theory, which in some sense is simpler and better
understood. Motivation for the subject lay in understanding the
symmetries in the roots of a polynomial and the results were used to
show the solubility or otherwise of a given polynomial equation of
some fixed degree. Thus the required prerequisites include Part IA
Groups, modular congruences from Part IA Numbers & Sets, together with
the groups section of Part IB Groups, Rings & Modules, basic ideas
from Parts IB Linear Algebra (basis and dimension), and some
elementary properties of rings (essentially the rings part of GRM).
 General Relativity

Should be familiar with tensor algebra (in particular how up and down
indices work  they will teach it again, but very quickly).
 Should know how special relativity is built up from just the Minkowski metric (i.e. last bit of D&R)  they will teach it again, but very quickly.

Should be familiar with tensor algebra (in particular how up and down
indices work  they will teach it again, but very quickly).
 Graph Theory

One must like combinatorics and playing around with stuff.
 No prerequisite. Some elementary knowledge from IA Probability will be revisited, specifically property of expectation, Markov's inequality and Chebyshev's inequality, but that is a very very small part of the course.

One must like combinatorics and playing around with stuff.
 Integrable Systems

In the early part of the course which covers integrability of finite
dimensional Hamiltonian systems there is a definite advantage to
attending the Part II Classical Dynamics course, and a small
benefit to attending Part II Dynamical Systems. For the material
on KdV it's very helpful to have been to Part IB Quantum Mechanics,
in particular the part of the course dealing with scattering and bound
states for a 1D potential. Of the courses listed as essential in
the schedules, Part IB Methods is probably more important than Part IB
Complex Methods and Complex Analysis – some contour integration
may be required, and either of CM/CA will be sufficient.
 IB QM (Scattering) incredibly useful, but retaught. Groups and group actions are talked about at the end of the course so IA Groups is useful, although again, this is retaught.

In the early part of the course which covers integrability of finite
dimensional Hamiltonian systems there is a definite advantage to
attending the Part II Classical Dynamics course, and a small
benefit to attending Part II Dynamical Systems. For the material
on KdV it's very helpful to have been to Part IB Quantum Mechanics,
in particular the part of the course dealing with scattering and bound
states for a 1D potential. Of the courses listed as essential in
the schedules, Part IB Methods is probably more important than Part IB
Complex Methods and Complex Analysis – some contour integration
may be required, and either of CM/CA will be sufficient.
 Linear Analysis

From Part IB Analysis II, you need some elementary notions of
completeness and knowledge of ℓ_{p} spaces, but overall
very little material is required. From Part IB Metric & Topological
Spaces, you need the notion of a topological space, the Hausdorff
property, and topological and sequential compactness.

You definitely need to revise the operator norm and it's
properties however (which is technically the start of the
differentiation chapter in Analysis II).
 Awareness of Linear Algebra has been useful especially dimension arguments, picking linearly independent sets, extending functions linearly, bilinear forms, but most useful for the course is Analysis II, so far without the differentiation bit. A lot of the arguments have Analysis II style and in fact even some example sheet questions are a little bit repeated, so I think good command of it is expected. Awareness of Met/Top is a good idea as although most of the terms are defined again one has to be confident in doing stuff with open, closed sets, interiors, closures etc.

From Part IB Analysis II, you need some elementary notions of
completeness and knowledge of ℓ_{p} spaces, but overall
very little material is required. From Part IB Metric & Topological
Spaces, you need the notion of a topological space, the Hausdorff
property, and topological and sequential compactness.
 Logic and Set Theory

Part IA Numbers & Sets is essential. There are questions and
examples that assume that you know what an abelian group is, what a
field is, and what the characteristic of a field is. So, while Part
IB Groups, Rings & Modules isn't officially a prerequisite, it would
help you appreciate such examples. Really, the main
prerequisite is a good brain and a refusal to be intimidated!
 The SET part of Numbers & Sets is essential, the actual number theory is irrelevant. Otherwise pretty prerequisite free.

Part IA Numbers & Sets is essential. There are questions and
examples that assume that you know what an abelian group is, what a
field is, and what the characteristic of a field is. So, while Part
IB Groups, Rings & Modules isn't officially a prerequisite, it would
help you appreciate such examples. Really, the main
prerequisite is a good brain and a refusal to be intimidated!
 Mathematical Biology

Part II Dynamical Systems is helpful for parts of this course but
certainly not essential. If you did not do Dynamical Systems, then it
might be wise to do a little revision of some of Part IA Differential
Equations: stability of equilibria of discrete and continuous time
systems (Jacobians, saddles/focus/nodes, phaseplane diagrams). Indeed,
'Ordinary Differential Equations' by Robinson (see the schedules for
Differential Equations) chapters 32 and 33 ('coupled nonlinear
equations' and 'ecological models') will put you right on track for
this course. The middle part of the course on stochastic systems will
use some knowledge from Part IA Probability, including generating
functions. It would be a good idea to revise separable solutions from
Part IB Methods for the last part of the course on diffusion.
 Not as scary in terms of needing a lot of probability, you can get by learning it in lectures and understanding it. I know applied probability has similar things covered in their lectures also.

Part II Dynamical Systems is helpful for parts of this course but
certainly not essential. If you did not do Dynamical Systems, then it
might be wise to do a little revision of some of Part IA Differential
Equations: stability of equilibria of discrete and continuous time
systems (Jacobians, saddles/focus/nodes, phaseplane diagrams). Indeed,
'Ordinary Differential Equations' by Robinson (see the schedules for
Differential Equations) chapters 32 and 33 ('coupled nonlinear
equations' and 'ecological models') will put you right on track for
this course. The middle part of the course on stochastic systems will
use some knowledge from Part IA Probability, including generating
functions. It would be a good idea to revise separable solutions from
Part IB Methods for the last part of the course on diffusion.
 Number Fields
 Depends VERY heavily on the rings bit of GRM, it's all about doing number theory in more interesting rings than just Z. The rest of GRM isn't super relevant.
 Number Theory

Nothing is really essential except Part IA Groups and Numbers
& Sets. It would be helpful to have had more experience of group
theory and analysis than Part IA provides, so some knowledge from Part
IB Analysis II and Groups, Rings & Modules would be useful but not
essential.

Fairly sure you only need to remember Euclid's algorithm and
modular arithmetic from IA N&S.
 Numbers and sets is useful, but everything is done from scratch so nothing seems essential to me.

Nothing is really essential except Part IA Groups and Numbers
& Sets. It would be helpful to have had more experience of group
theory and analysis than Part IA provides, so some knowledge from Part
IB Analysis II and Groups, Rings & Modules would be useful but not
essential.
 Numerical Analysis

Taking Numerical Analysis in 2nd year gives a good sense of the things
you will do over and over again in this course, but in terms of
contents, it's not really essential as concepts will be reintroduced.
 This extends IB NA, which focuses mostly on 1D analysis, to look at 2D things. Here is a nonexhaustive list of things that will be revisited from IB courses. IB NA: Chebyshev expansion, QR decomposition; IB Methods: PDEs (wave, advection, diffusion, Laplace  because different numerical schemes will be applied to them, it is useful to recall their properties), Fourier Series, Fourier Transform.

Taking Numerical Analysis in 2nd year gives a good sense of the things
you will do over and over again in this course, but in terms of
contents, it's not really essential as concepts will be reintroduced.
 Principles of Quantum Mechanics

Linear Algebra is needed throughout the course. From IB Methods, you need
Fourier series, Fourier transforms, Legendre polynomials, Dirac delta,
and from IB Quantum Mechanics, Schrodinger's eqn, Harmonic Oscillator,
Angular Momentum, Hydrogen are also needed.

IB Linear Algebra!

Not very closely related to IB QM (I think), it's more like
reconstruct everything in a more algebraic (lots of linear algebra)
rather than physical way.
 IB QM isn't super relevant, mainly the bits about the deeper mathematical framework and the postulates, eg "observables are eigenvalues". Being good at Linear Algebra is ESSENTIAL.

Linear Algebra is needed throughout the course. From IB Methods, you need
Fourier series, Fourier transforms, Legendre polynomials, Dirac delta,
and from IB Quantum Mechanics, Schrodinger's eqn, Harmonic Oscillator,
Angular Momentum, Hydrogen are also needed.
 Principles of Statistics

Many concepts from IB Stats would be reintroduced but it
doesn't hurt if one is familiar with the basic concepts, so when
things get complicated it wouldn't too be bad.

Analysis I would be helpful in understanding some of the proofs
and Probability and Measure is useful to understand concepts such as
"converge almost surely", "in probability", "in distribution", etc.
This is definitely more pure focused than I expected.
 I agree with everything that has been said.

Many concepts from IB Stats would be reintroduced but it
doesn't hurt if one is familiar with the basic concepts, so when
things get complicated it wouldn't too be bad.
 Probability and Measure

There are very few requirements from Part IB. One definition requires
the notion of a topological space, but in examples this is almost
always R or R^{d}. Occasionally, examples or
proofs will use standard facts from Part IA Analysis I or Part IB
Analysis II (for instance, that a uniform limit of continuous
functions is continuous). The course is completely independent of
Part IA Probability. Indeed, the title is rather misleading since
it's a completely different flavour to the Part IA course.

Metric spaces and compactness are definitely important, but
beyond that Analysis I is everything you need (I think).

IA probability... it's probably a good idea to revise random
variables and continuous probability distributions (but it all gets
retaught).

It would be helpful if one is familiar with Analysis I and II
as certain results would be quoted without proof. Metrics and
compactness are very useful in understanding the course. Familiarity
with techniques used in analysis would also be helpful in
understanding the intuition behind proofs and ideas.
 To me, the exposure to both Analysis II and Met & Top in IB helps in developing the kind of intuition needed, but no technical concepts beyond metrics, compactness and Analysis I are required.

There are very few requirements from Part IB. One definition requires
the notion of a topological space, but in examples this is almost
always R or R^{d}. Occasionally, examples or
proofs will use standard facts from Part IA Analysis I or Part IB
Analysis II (for instance, that a uniform limit of continuous
functions is continuous). The course is completely independent of
Part IA Probability. Indeed, the title is rather misleading since
it's a completely different flavour to the Part IA course.
 Quantum Information and Computation

Some familiarity with bra ket notation and inner products from
Principles Of QM or even Part IB QM would be helpful to make the
course more comfortable  especially to begin with but really is a
self contained course and everything you need is covered from scratch.
 Pretty much selfcontained except that you may need to familiarise yourself with bra ket notation. Nice coverage on information theory which is not found in any other course (except for perhaps one part in Coding and Cryptography?). I would argue that IB Quantum Mechanics is not really needed: it's helpful for you to have an understanding of how quantum states work (either from QM or from somewhere else), but the course is more about putting algorithms into quantum settings rather than solving Schrödinger's equations again.

Some familiarity with bra ket notation and inner products from
Principles Of QM or even Part IB QM would be helpful to make the
course more comfortable  especially to begin with but really is a
self contained course and everything you need is covered from scratch.
 Representation Theory
 The course studies linear actions of groups acting on finitedimensional vector spaces over (usually) the complex numbers. For the 'groups' part, you will need to understand the entire contents of Part IA Groups and the groups section of Part IB Groups, Rings & Modules, with a working familiarity with various families of groups such as symmetric, dihedral and matrix groups, as well as groups of prime power order (including Sylow's theorems). The notions of conjugacy classes, actions of groups, the first isomorphism theorem and the structure of finite abelian groups (from GRM) are also important. For the 'vector spaces' part, you should be familiar with Part IB Linear Algebra, especially the criteria for diagonalisation of linear transformations, vector space duals and inner product spaces. The final quarter of the course focuses on certain infinite groups which are also topological spaces (such as SU(2)), hence some familiarity with the ideas of compactness and connectedness from Part IB Metric & Topological Spaces is also required.
 Riemann Surfaces

All of Part IB Complex Analysis is necessary (Complex Methods alone is
not sufficient). Most of Part IB Metric & Topological Spaces is also
needed, particularly connectedness, basis of a topology, and quotient
and product topologies. A small amount of content overlaps with Part
II Algebraic Topology (homotopy, simply connected spaces, monodromy
without proof), and taking Algebraic Topology concurrently may be
helpful for those wishing a deeper understanding of these concepts.

I have been told by friends that you HAVE to do Alg Top alongside it.
 I think familiarity with the embedded surfaces part of IB Geometry is useful, at least the idea of charts and atlases (though it doesn't directly depend on this)

All of Part IB Complex Analysis is necessary (Complex Methods alone is
not sufficient). Most of Part IB Metric & Topological Spaces is also
needed, particularly connectedness, basis of a topology, and quotient
and product topologies. A small amount of content overlaps with Part
II Algebraic Topology (homotopy, simply connected spaces, monodromy
without proof), and taking Algebraic Topology concurrently may be
helpful for those wishing a deeper understanding of these concepts.
 Statistical Modelling

Feels more of an extension from Part IB Statistics more than it does
from Part II Principles of Statistics, though having done both
definitely helps. It is a course that involves using R so knowing how
to code helps..? (Being able to do CATAM should be good enough)
 IB Statistics is essential. The course ties in really well with Part II Principles of Statistics and touches upon a lot of statistical techniques. I personally would recommend taking them together but it doesn't hurt too much if you leave PoS out.

Feels more of an extension from Part IB Statistics more than it does
from Part II Principles of Statistics, though having done both
definitely helps. It is a course that involves using R so knowing how
to code helps..? (Being able to do CATAM should be good enough)
 Statistical Physics
 Most importantly, this course doesn't need any statistics. From Part II Principles of Quantum Mechanics, you will need to be comfortable with Dirac bra/ket notation and the concept of multiparticle systems of bosons and fermions.
 Stochastic Financial Models

Familiarity in probability and Markov chains help build some intuition
but is not crucial. This course is more pure than it sounds and builds
off measure theory, so knowing measure theory (from prob and measure
presumably) is rather important.
 Part II Probability & Measure is important. Lagrangians from IB Optimisation / Variational Principles are needed but it's not too difficult to just pick it up along the way. Many people have dropped the course because they first thought it's a finance course until they realised the course is actually very pure. Other than that, it introduces the notion of martingales which are helpful to know if you are a probability person, and also explores Brownian motion with one entire chapter if you feel that the "onesentence" explanation in Principle of Statistics is not enough. The course also touches on several optimisation techniques in the context of finance, including dynamic programming, which should be quite straightforward.

Familiarity in probability and Markov chains help build some intuition
but is not crucial. This course is more pure than it sounds and builds
off measure theory, so knowing measure theory (from prob and measure
presumably) is rather important.
 Topics in Analysis
 Waves
 The course is selfcontained, but there is considerable synergy with Fluids II (e.g. stress tensors are used in both courses), and the two courses naturally fit together. Familiarity with material from Asymptotic Methods is also useful.