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.
-
"Continuous-time 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.
- IB Markov Chains is essential, as most of the time what we do in the course is to reduce the continuous time Markov Chain to a discrete time Markov Chain, and hit it with results we have obtained in IB. It is helpful to revise the concepts of recurrence and transience, simple random walks in R^d and time reversibility as they appear in this course again. The last quarter of the course concerns spatial Poisson processes, and the proofs to some theorems have an extremely slight (just a little, really) measure-theoretic flavour to them. However, with that being said, P&M is definitely NOT necessary for this course (as this course just quotes and uses Fubini and Monotone Convergence without proof, and you can pretty much exchange integrals/sums/expectations as you wish in this 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
-
A surprisingly useful course. Has overlap with FCM and Stat. Phys! No
mention of poisson brackets or hamiltonian at all!!! (I think the
existing comment about hamiltonian and poisson brackets should be with
Integrable Systems)
- The courses studies how we can approximate integrals and solutions to differential equations with very large/small parameters by seeking an asymptotic expansion. Very basic vector calculus on saddles and contour integration suffice (plus knowing how to Taylor expand!). The bit on differential equations overlap with FCM, so it might be beneficial to do both courses together (absolutely not necessary though).
-
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, Euler-Lagrange 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 self-contained, but useful to remember cyclic groups, ideals and
polynomial rings from GRM in order to understand cyclic codes.
- Unlike other courses which focus on developing one or two main ideas, this course comes in short chapters and covers a lot of different topics, though some ideas (e.g. entropy) are recurrent. I agree with what has already been said; would just like to add that the last bit ties in well with Number Theory, though what is really needed is at the level of IA N&S (really just Chinese Remainder Theorem and Fermat-Euler). The course is a nice counterpart to II Quantum Information & Computation.
-
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 self-contained.
- 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.
- The course studies geometric objects like curves and surfaces in R^3. Would say the entirety of A&T is essentia - the first quarter of Differential Geometry, on differential topology, assumes familiarity and fluency with the concepts of compactness, closed & open sets etc. (but in a familiar setting, because everything happens in Euclidean space so nice things like compact iff closed & bounded hold). The rest of the course uses the chain rule, Picard-Lindelof and the Inverse Function Theorem extensively, so it would be good to revise the final quarter of A&T. IB Variational Principles provides good background for the section on geodesics and minimal surfaces. IB Geometry is desirable but absolutely not necessary, as material is treated from scratch. (Somehow the treatment in IB Geometry can be even more abstract and general than in II Diff Geo)
-
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
four-vector representation.
- Algebra heavy and a large section on dipole approximations, be wary if you didn't enjoy the dipole section in the IB course.
-
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.
- A lot better than IB Fluids, not vital to understand IB that well (just how to derive continuity equation, Euler eqn. etc. ) because you cover slightly different topics in the II course in a better way.
-
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.
- A lot of repetition of IB CM, some useful sections on the Gamma and Beta functions which crop up in Asym. Meth.
-
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.
- This course is essentially the study of fields and their extensions. It requires a solid understanding of basic group theory (at the level of IA, and maybe the classification theorem for finite abelian groups from IB GRM), especially familiarity towards symmetric groups (when studying Galois groups of polynomials), abelian groups and dihedral groups (which are common examples of Galois groups). I would say the course assumes general fluency with working with abstract algebraic objects and familiarity with results in the rings bit of the GRM course - it is never a bad idea to review the concepts of the isomorphism theorems, prime ideals, irreducible polynomials, Eisenstein and Gauss etc. Linear Algebra is tangentially relevant in defining field extensions and the trace & the norm, but a basic understanding would suffice.
-
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.
- Cosmology is useful in the last section of the course because you actually find some solutions to Einstein's equations (albeit simplified) whereas the lecturer does this extremely quickly in GR and doesn't explain the physical significance of the solutions Maybe useful to have done IB/II Electromag. because you play around with moving indices up and down.
-
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 1-D 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 1-D 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 ℓp 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 ℓp 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 pre-requisite 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, phase-plane diagrams). Indeed,
'Ordinary Differential Equations' by Robinson (see the schedules for
Differential Equations) chapters 32 and 33 ('coupled non-linear
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.
-
The probability parts are very simple and taught in lectures (IA Prob.
is fine), more important to be comfortable with IB Methods (separable
solutions, finding coefficients in Fourier series), II Fluids also
very helpful with similarity solutions and linearisation
- This course studies the mathematics behind various biological phenomena using the language of differential equations. Pretty much pre-requisite free, apart from basic knowledge from IA DE's (on phase portraits, stability of fixed points, difference equations etc, but will be retaught), dimensionality arguments from IA D&R (which are revisited in detail), Fourier series and separable solutions from IB Methods and very basic IA Probability. (The continuity equation gets a mention somewhere in II MB too, if I recall correctly?) The middle bit of the course, on master equations, pairs surprisingly well with Continuous Time Markov Chains in II Applied Probability. II Dynamical Systems is highly relevant but not essential, though the courses pair well and the former allows a better understanding of the latter according to my own experience.
-
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, phase-plane diagrams). Indeed,
'Ordinary Differential Equations' by Robinson (see the schedules for
Differential Equations) chapters 32 and 33 ('coupled non-linear
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.
- Mathematics of Machine Learning
- 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.
- This course studies the ring of algebraic integers of different number fields, not just Q. Would strongly recommend doing Galois Theory in Michaelmas, as this course is a natural continuation of Galois Theory and quotes definitions, facts and results from Galois Theory quite frequently (without proof, as they were done in Galois Theory). Relies on quite a lot of rings and a bit of modules from GRM; in particular about quotient rings, maximal/prime/principal ideals from rings and finitely generated modules and the structure theorem for modules. It might be a very good idea to review GRM and Galois Theory before taking this course.
-
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.
- This course does what it says on the tin - classical number theory is introduced in the first half, and the distribution of primes, continued fractions and primality testing methods are covered in the second half. Basically no prerequisites, although remembering the most basic stuff from N&S is helpful. An elementary understanding towards groups and rings will allow one to appreciate certain theorems phrased in alternative (e.g. group- or ring-theoretic) notations, e.g. the Chinese Remainder Theorem, or the structure of (Z/nZ)*. Last bit goes well with Coding & Cryptography.
-
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 non-exhaustive 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 Rd. 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 Rd. 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 self-contained 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.
- This course studies the novel properties of quantum protocols and algorithms. Any previous exposure to Part II Quantum courses might be beneficial, but is definitely not necessary for the course (I would even argue one could pick up the course without IB Quantum Mechanics, with just a bit of extra effort) - in particular, it is of different flavour to IB Quantum Mechanics with much less focus on solving the Schrodinger equation. Bra-ket notation is taught at a comfortable pace at the beginning of the course, so it wouldn't be necessary to have met them before in Part II QM courses. The course is really just linear algebra mostly, and only requires a few recurring ideas from Quantum Mechanics (e.g. collapse of the wavefunction, Born rule). Pairs well with II Coding & Cryptography, as one can readily compare classical and quantum information theory (in fact, part of the QIC course was one in CC when QIC wasn't a separate course yet!).
-
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
finite-dimensional 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.
- Fun crossover between Linear Algebra and Group Theory that enables cool stuff about groups to be proved, like Burnside's p^a q^b theorem! Requires basic familiarity with working with endomorphisms of finite-dimensional vector spaces, ideas about inner products, basic facts about eigenvalues, eigenvectors and the trace of a linear map from Linear Algebra, while familiarity with IA Groups (excluding the section on Mobius maps) and the Sylow Theorems would suffice for the bit on Group Theory. The last quarter of the course is about topological groups; a very basic understanding of topology would suffice as the course doesn't go into that much depth anyway.
-
The course studies linear actions of groups acting on
finite-dimensional 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
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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.
- I was worried about taking this course since I hadn't done PQM. However, Cosmology gave me a sufficient understanding of bosons/fermions and I didn't feel at a disadvantage at all. If you have done Cosmology, I don't think you should worry about not having done PQM. Tong's online Stat Phys notes approach the content from more of a PQM angle, but this isn't that important and can be ignored.
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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 "one-sentence" 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.
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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 self-contained, 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.