OMMS Preparatory Material
As the MSc in Mathematical Sciences is being run parallel to the fourth-year undergraduate course, many of our Part C courses have prerequisites that were courses during the first three years of the undergraduate programme. We have put together extra reading and problem sheets that cover some of the prerequisites. In preparation for the dissertation component of the Masters course, you may want to take a look at the department's notes on using LaTeX.
You might find it beneficial to look at our Guide to Mathematics Options at Part C document which provides a good introduction to the courses on offer in the 2020-21 year (please note this document is reflective of courses offered in 2020-21 and some information may not be accurate for the current academic year).
Logic
Courses: C1.1 Model Theory, C1.2 Godel's Incompleteness Theorem, C1.3 Analytic Topology, C1.4 Axiomatic Set Theory
C1.1 Model Theory and C1.2 Godel's Incompleteness Therorem - Students will be required to have a familiarity with first-order logic. You could take a look over the lecture slides for the B1.1 Logic Lecture Notes. Any book on first-order logic covering the completeness theorem and basic consequences, including compactness, would be sufficient. Enderton [1] for instance. For stronger students, Chapter 1 and 2.l of Chang and Keisler [2], which include a wealth of examples, would be excellent preparation.
C1.3 Analytic Topology - It is recommended that you take a look over the lecture notes for the second year Topology course (up to and including chapter 3, Quotient Spaces) and the first three problem sheets of that course. You could also take a look at Sutherland's "Introduction to Metric and Topological Spaces" [3] or launch straight into the course's reading list with Willard's book on General Topology [4].
C1.4 Axiomatic Set Theory - A familiarity with set theory would be a pre-requisite for the course. Students wishing to take the course could take a look over the Part B Set Theory course lecture notes and the problem sheets below.
Problem Sheet 1
Problem Sheet 2
Problem Sheet 3
Problem Sheet 4
The "Classical Set Theory" by Goldrei [5] would also provide a good understanding. In addition, you could also take a look at "Set Theory" by Kenneth Kunen [6], though, please note that this does cover the material (and more) of the Part C course
[1] W. B. Enderton, A Mathematical Introduction to Logic (Academic Press, 1972).
[2] C.C. Chang and H. Jerome Keisler, Model Theory (Third Edition (Dover Books on Mathematics) Paperback).
[3] W. A. Sutherland, Introduction to Metric and Topological Spaces (Oxford University Press, 1975).
[4] S. Willard, General Topology (Addison-Wesley, 1970), Chs. 1-8.
[5] D. Goldrei, Classic Set Theory (Chapman and Hall, 1996).
[6] K. Kunen, Set Theory (College Publications, 2011) Chapters (I and II).
Algebra
Courses: C2.2 Homological Algebra, C2.3 Representation Theory of Semisimple Lie Algebras, C2.4 Infinite Groups, C2.5 Non-Commutative Rings, C2.6 Introduction to Schemes, C2.7 Category Theory
For many of the Part C courses overseen by the Algebra Subject Panel, the Part B Introduction to Representation Theory and Commutative Algebra courses are a pre-requisite. Below are these course's lecture notes and problem sheets for the 2017-2018 academic year.
Introduction to Representation Theory
Lecture Notes 2017-2018
Sheet 0
Sheet 0 solutions
Sheet 1
Sheet 2
Sheet 3
Sheet 4
Commutative Algebra
Lecture Notes 2017-2018
Sheet 0
Sheet 1
Sheet 2
Sheet 3
Sheet 4
C2.7 Category Theory - There are no essential prerequisites but familiarity with the basic theory of groups, rings, vector spaces, modules and topological spaces would be very useful, and other topics such as Algebraic Geometry, Algebraic Topology, Homological Algebra and Representation Theory are relevant. Category Theory also has links with Logic and Set Theory, but this course will not stress those links.
Geometry, Number Theory and Topology
Courses: C3.1 Algebraic Topology, C3.2 Geometric Group Theory, C3.3 Differentiable Manifolds, C3.4 Algebraic Geometry, C3.5 Lie Groups, C3.6 Modular Forms, C3.7 Elliptic Curves, C3.8 Analytic Number Theory, C3.9 Computational Algebraic Topology, C3.10 Additive Combinatorics, C3.11 Riemannian Geometry, C3.12 Low-Dimensional Topology and Knot Theory
C3.1 Algebraic Topology - A solid understanding of groups, rings, fields, modules, homomorphisms of modules, kernels and cokernels, and classification of finitely generated abelian groups (A3 Rings and Modules); a solid understanding of topological spaces, connectedness, compactness, and classification of compact surfaces (A5 Topology); a solid understanding of tensor products of abelian groups (B2.1 Representation Theory); a solid understanding of homotopic maps, homotopy equivalence, and the fundamental group (B3.5 Topology and Groups). The following are links to the lecture notes pdf that were mentioned above:
Part A Rings and Modules Lecture Notes
Part A Topology.pdf
Lecture Notes 2017-2018.pdf
B3.5 Topology and Groups Lecture Notes
Other useful reference material includes: Massey, "Algebraic topology: an introduction"; Sutherland, "Introduction to metric and topological spaces"; Artin, "Algebra"; Dummit and Foote, "Abstract algebra".
C3.2 Geometric Group Theory - Background: some familiarity with group theory, 1st isomorphism theorem, Lagrange’s theorem, group actions, for example, see below the notes from the 1st year undergraduate course in Oxford.
Groups and Group Actions Lecture Notes
It would be helpful, but not necessary, to have seen before fundamental group and covering spaces. A good reference for this is Chapter 1 of Hatcher’s Algebraic Topology book or the below lecture notes from the Part B course Topology and Groups
B3.5 Topology and Groups Lecture Notes
C3.3 Differentiable Manifolds - This is a high-level course that moves quite fast, and assumes a good knowledge of quite a lot of earlier undergraduate pure mathematics; students coming for example from a first degree in physics may find it challenging.
The most important prerequisite is topology and topological spaces (as in the second year A5: Topology course, a good reference is W. A. Sutherland, Introduction to Metric and Topological Spaces, OUP 1975), including common methods of proof in topology, and the ideas of compact, Hausdorff, second countable, and open cover.
The course also assumes familiarity with ideas about smooth functions on R^n, and commutative algebras over R. Previous exposure to more elementary geometry courses such as the second-year course ASO: Introduction to Manifolds and (especially) the third-year course B3.2 Geometry of Surfaces is very helpful. The following pdfs are the lecture notes for the undergraduate courses mentioned above:
Part A Topology Lecture Notes
ASO Introduction to Manifolds Lecture Notes
B3.2 Geometry of Surfaces Lecture Notes
C3.4 Algebraic Geometry -
Classical algebraic geometry is the study of the sets of of simultaneous solutions of collections of polynomial equations in several variables with coefficients in an algebraically closed field.
Such sets are called algebraic varieties. Algebraic varieties appear in almost every area of mathematics. They play a crucial role in number theory, in topology, in differential geometry and complex geometry (ie the theory of complex manifolds). When the base field is the field of complex numbers, an algebraic variety defines a complex manifold provided it has "no kinks".
A basic reference for classical algebraic geometry is chap. I of D. Mumford's book "The Red Book of Varieties and Schemes” (Springer Lecture Notes in Mathematics 1358). Another reference is chap. I of R. Hartshorne's book "Algebraic Geometry" (Springer). One might also consult the book by M. Reid "Undergraduate algebraic geometry" (London Mathematical Society Student Texts 12, Cambridge University Press 1988). An updated free online version of M. Reid's lectures can be found under
https://homepages.warwick.ac.uk/staff/Miles.Reid/MA4A5/UAG.pdf
The prerequisites (both essential) for this course are the part A course Rings and Modules and the part B course Commutative Algebra (or equivalent courses).
C3.5 Lie Groups - Lie groups are groups that are simultaneously manifolds, that is geometric spaces where the notion of differentiability makes sense, and the group operations are differentiable maps. Matrix groups provide an important set of concrete examples, though the notion of Lie group also includes other examples. Lie groups are vital to modern geometry and theoretical physics, and to the analysis of any physical situation involving continuous symmetries. It is recommended that students interested in taking the C3.5 Lie Groups course also take the C3.3 Differentiable Manifolds course. The following are some useful reading that you might want to take a look at:
[1] J. F. Adams, Lectures on Lie Groups (University of Chicago Press, 1982).
[2] T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups (Graduate Texts in Mathematics, Springer, 1985).
[3] B. Hall. Lie groups, Lie algebras and representations, Springer.
C3.6 Modular Forms -
The course introduces students to the beautiful theory of modular forms, one of the cornerstones of modern number theory. This theory is a rich and challenging blend of methods from complex analysis and linear algebra, and an explicit application of group actions. A standard first course in complex analysis and metric spaces, and a course in linear algebra covering inner products spaces and self-adjoint maps, are essential, e.g. the course notes for:
A2 Metric Spaces and Complex Analysis
A standard course in elementary number theory, and some familiarity with Riemann surfaces and the Riemann-Roch theorem, are useful but not essential, e.g.
C3.7 Elliptic Curves - It is helpful, but not essential, if students have already taken a standard introduction to algebraic curves and algebraic number theory. For those students who may have gaps in their background, please refer to the below "Preliminary Reading" file, which gives in detail (about 30 pages) the main prerequisite knowledge for the course
C3.8 Analytic Number Theory - A basic familiarity with manipulating functions and series (see M2 Analysis II ) is important, as will some familiarity with basic concepts of complex analysis (See A2: Metric Spaces and Complex Analysis ). Some familiarity with number theory of Fourier analysis will be helpful but is not required.
C3.9 Computational Algebraic Topology and Knot Theory - This course introduces students to homological approaches to topological data science and contextuality in computer science. There are no pre-requesists as such and students generally come with different background knowledge -- from pure maths, applied maths or computer science. Students will, however, be expected to take a mature approach and fill in any gaps they have as necessary. Some prior familiarity with simplicial complexes and notions of homology will be very helpful. [These topics are covered in detail in the Oxford courses B3.5 Topology and Groups (lecture notes below) and C3.1 Algebraic Topology.]
B3.5 Topology and Groups Lecture Notes
C3.10 Additive and Combinatorial Number Theory - There are no essential prerequisites for this course. The course C3.8 Analytic Number Theory, lectured in Hilary, is complementary but you will be able to follow this course without it. The excellent new book Graph Theory and Additive Combinatorics by Yufei Zhao is good complementary reading for this course.
C3.11 Riemannian Geometry -
Riemannian Geometry is the study of curved spaces and provides an important tool with diverse applications from group theory to general relativity. The surprising power of Riemannian Geometry is that we can use local information to derive global results.
This course will study the key notions in Riemannian Geometry: geodesics and curvature. Building on the theory of surfaces in R^3 in the Geometry of Surfaces course, we will describe the notion of Riemannian submanifolds, and study Jacobi fields, which exhibit the interaction between geodesics and curvature. We will prove the Hopf--Rinow theorem, which shows that various notions of completeness are equivalent on Riemannian manifolds, and classify the spaces with constant curvature.
The highlight of the course will be to see how curvature influences topology. We will see this by proving the Cartan--Hadamard theorem, Bonnet-Myers theorem and Synge's theorem.
It is useful for students taking this course to have familiarity with the basic concepts in the C3.3 Differentiable Manifolds course, such as manifolds, vector fields, differential forms and orientability. Another helpful prerequisite is topology, and particularly an understanding of the theory of covering spaces and the fundamental group, as covered in the B3.5 Topology and Groups course, would be useful. For the part of the course on Riemannian submanifolds, it would be useful to have some knowledge of the theory of surfaces in Euclidean space, such as is discussed in the B3.2 Geometry of Surfaces course.
C3.12 Low-Dimensional Topology and Knot Theory - Low-dimensional topology is the study of 3- and 4-manifolds and knots. The classification of manifolds in higher dimensions can be reduced to algebraic topology. These methods fail in dimensions 3 and 4. Dimension 3 is geometric in nature, and techniques from group theory have also been very successful. In dimension 4, gauge-theoretic techniques dominate.
This course provides an overview of the rich world of low-dimensional topology that draws on many areas of mathematics. We will explain why higher dimensions are in some sense easier to understand, and review some basic results in 3- and 4-manifold topology and knot theory.
General Prerequisites: B3.5 Topology and Groups (MT) and C3.1 Algebraic Topology (MT) are essential. B3.2 Geometry of Surfaces (MT) and C3.3 Differentiable Manifolds (MT) are useful but not essential.
Analysis
Courses: C4.1 Further Functional Analysis, C4.3 Functional Analytic Methods for PDEs, C4.4 Hyperbolic Equations, C4.6 Fixed Point Methods for Nonlinear PDEs, C4.9 Optimal Transport and PDEs
C4.1 Further Functional Analysis - The main prerequisite for this course is familiarity with Hilbert and Banach spaces and their bounded operators (as covered in B4.1 and B4.2, in the notes below) together with the basics of metric and topological spaces (the metric spaces part of the metric spaces and complex analysis course, and the first half of A5 topology). A detailed description of the prerequisites is given in the pdf below, together with the B4.1 and B4.2 notes.
C4.3 Functional Analytic Methods for PDEs - The main prerequisite for this course is a strong working knowledge of Lebesgue measure and integrals (A4 Integration). Familiarity with functional analysis (B4.1-4.2 Functional analysis I, II) and/or distribution theory (B4.3 Distribution Theory and Fourier Analysis) would also be useful, though not essential as the relevant materials will be recalled along the way.
B4.1 Functional Analysis I Lecture Notes
B4.2 Functional Analysis II Lecture Notes
B4.3 Distribution Theory and Fourier Analysis: An Introduction Lecture Notes
C4.6 Fixed Point Methods for Nonlinear PDEs - The only necessary prerequisites for the course are a good understanding of Lebesgue Integration (A4 Integrationintegration.pdf) and some basic results on weak derivatives and functional analysis (the relevant results will also be recalled in the lecture). Ideal preparation for the course is to take the MT course C4.3 Functional Analytic Methods for PDEs.
C4.9 Optimal Transport & Partial Differential Equations - The only necessary prerequisites for the course are a good understanding of Lebesgue Integration (A4 Integration integration.pdf) and the metric spaces part of A2, and some basic results on weak derivatives and functional analysis (the relevant results will also be recalled in the lecture). Ideal preparation for the course is to take the MT course C4.3 Functional Analytic Methods for PDEs.
Mathematical Methods and Applications
Courses: C5.2 Elasticity and Plasticity, C5.4 Networks, C5.5 Perturbation Methods, C5.6 Applied Complex Variables, C5.7 Topics in Fluid Mechanics, C5.9 Mathematical Mechanical Biology, C5.11 Mathematical Geoscience, C5.12 Mathematical Physiology
C5.2 Elasticity and Plasticity -
This course assumes a familiarity with basic Partial Differential Equations, Complex Analysis and Calculus of Variations up to Part A (second year) level. Useful background material on each of these topics may be found in sections 3 and 4 of the A1 Differential Equations I lecture notes, section 1 of the C5.6 Applied Complex Variables lecture notes, as well as the ASO Calculus of Variations lecture notes. Some basic understanding of the Cauchy stress tensor is also helpful and may be found in section 1 of the B5.3 Viscous Flow lecture notes. The course fits particularly well with C5.5 Perturbation Methods and C5.6 Applied Complex Variables.
https://courses.maths.ox.ac.uk/pluginfile.php/30048/mod_resource/content/16/DE1notes22-11-09.pdf
https://courses.maths.ox.ac.uk/pluginfile.php/27181/mod_resource/content/1/C5_6LectureNotes.pdf
https://courses.maths.ox.ac.uk/pluginfile.php/29848/mod_resource/content/1/Notes%202022.pdf
https://courses.maths.ox.ac.uk/pluginfile.php/26002/mod_resource/content/2/B6aLectureNotes.pdf
C5.4 Networks - Students will be required to have a familiarity with linear algebra and programming. Students are invited to read the first section of the lecture notes (partly covered during the first lesson, see below), dedicated to the basic tools required for the course, including on Markov chains and stochastic processes. Problems will be solved in Python in the tutorials, thus learning the basics of Python programming is advised.
C5.5 Perturbation Methods - Knowledge of core complex analysis and of core differential equations will be assumed, at the level of the complex analysis in the Part A (Second Year) course Metric Spaces and Complex Analysis and the phase plane section in Part A Differential Equations I. The optional part A course, Differential Equations II, is useful though not a prerequisite. The final section on approximation techniques in this module is highly recommended reading if it has not already been covered. These lecture notes may be found here.
Elements of the third years' courses, B5.3 Viscous Flow (Boundary Layers, Lubrication Theory) and B5.5 Further Mathematical Biology (Enzyme kinetics, Fitzhugh Nagumo Equations) are helpful in providing examples from the Natural Sciences of how, after non-dimensionalisation, equations with small parameters emerge and how they are analysed.
C5.6 Complex Analysis
The course requires Part A core complex analysis (link to notes), and is devoted to extensions and applications of that material.
A knowledge of the basic properties of the Fourier transform, as found for example in Part A Integral Transforms, will be assumed (link to notes). Part A Fluid Dynamics and Waves is helpful but not absolutely essential: the necessary results from inviscid two-dimensional hydrodynamics will be quoted as required.
The course fits well with C5.5 Perturbation Methods, which is helpful in the analysis of certain contour integrals.
C5.7 Topics in Fluid Mechanics - The course develops various topics in fluid mechanics which lie beyond the remit of the introductory fluid mechanics courses, which deal variously with wave motion in inviscid flows, viscous boundary layers at high Reynolds number, inertia-free flows such as Stokes flow or those of lubrication theory. As such it will assume a fluency with these subjects, and the problem sheet 0 (available below) provides some exercises on them. Having said that, all that is really required as a prerequisite is a knowledge of the form and origin of the Navier-Stokes equations, and the topics to be studied can be thought of as variations on an underlying theme.
Four topics will be dealt with: these are thin films, convection, rotating flows and two-phase flows. As the course has a new lecturer this year, be aware that some of the online material will be updated. The problem sheets 1 to 4 will be revised, and I have left them there only for guidance. Equally, there are online lecture notes, and these also will be left, although it is likely that an embryo version of my own notes may appear at some point during the term. The problem sheet 0 should have answers posted in week 0; there is no tutorial assistance available for this, however.
C5.11 Mathematical Geoscience - An outline of the background material useful for this course is included at the start of the online lecture notes, which will be made available at the start of term. The course will use a variety of techniques of applied mathematics that may have been come across in other contexts, but most will be introduced briefly during the course. Students will be expected to have some knowledge of Non-dimensionalisation and scaling analysis, skills to solve ordinary differential equations and standard partial differential equations such as the heat equation, phase-plane analysis and the method of characteristics for first-order hyperbolic equations (both of these are covered in the Oxford Part A course Differential Equations I Lecture Notes). Some knowledge of fluid dynamics is useful (though not essential), in particular, the ideas of lubrication theory (covered in the Oxford Part B course B5.3 Viscous FlowB5.3 Viscous Flow Lecture Notes).
C5.12 Mathematical Physiology - This course develops the subject of mathematical biology, such as is treated in J.D. Murray’s book, Mathematical biology [1]. The basic topics covered in such a book revolve around the mathematical subjects of oscillations, travelling waves, spatial pattern, singular perturbation theory, and reaction-diffusion systems, as exemplified in the study of predator-prey systems, enzyme kinetics, the Fisher equation, the Belousov-Zhabotinskii reaction and diffusion-driven instability. This course assumes a familiarity with these topics, and also with some of the biological background, and extends it specifically to the study of physiological systems. In particular, there is an attempt to treat a sequence of topics at a range of scales, from the intra-cellular to the systemic. Most of the techniques should be familiar and are assumed (phase plane analysis, stability theory, relaxation oscillations, but there will be some new material introduced, such as spatio-temporal chaos and differential-delay equations.
[1] J. D. Murray, Mathematical Biology (Springer-Verlag, 2nd ed., 1993). [Third edition, Vols I and II, (Springer-Verlag, 2003).]
Numerical Analysis
Courses: C6.1 Numerical Linear Algebra, C6.2 Continuous Optimisation, C6.5 Theories of Deep Learning
C6.1 Numerical Linear Algebra - A reasonable knowledge of basic linear algebra is required, though there are no formal prerequisites. There are very many textbooks on this fundamental subject, one of the most popular around the world being that by Gil Strang [1] and Trefethen and Bau [2]. Since this is a course describing and analysing practical numerical methods for problems of linear algebra, emphasis will be on matrices rather than linear transformations.
[1] G. Strang. Introduction to Linear Algebra, Fifth Edition (2016)
[2] L. N. Trefethen and D. Bau, Numerical Linear Algebra (1997)
C6.2 Continuous Optimisation -
This course connects well to the Part B courses on Optimization for Data Science (B6.2) and on Integer Programming (B6.1). You may want to take a look at the lecture notes of these courses, though neither one of these courses is a requirement/pre-requisite for choosing the C6.2 course.
Mathematical Physics
Courses: C7.1 Theoretical Physics, C7.4 Introduction to Quantum Information, C7.5 General Relativity I, C7.6 General Relativity II, C7.7 Random Matrix Theory
C7.4 Introduction to Quantum Information - Please visit https://qubit.guide/ for further information on C7.4
C7.5 General Relativity I - This course introduces a modern perspective of gravitation. Understanding special relativity is a necessary prerequisite for the course, including Lorentz transformations and the Einstein—Summation convention. There will be a one-lecture review of this material but it will be brief and assume prior exposure to the material. For those in need of a refresher see chapter 1 of the book Carroll: Spacetime and Geometry, an introduction to general relativity. By far the hardest part of the course will be the first few lectures on differential geometry. Flicking through either the book Nakaha: geometry, topology and physics or chapters 2 and 3 of Carroll will give you a good overview but is by no means necessary. Full lecture notes will be provided.
C7.6 General Relativity II - This course builds on the Michaelmas course C7.5 General Relativity 1 and is a necessary prerequisite. A deeper understanding of differential geometry will be provided and for those interested in getting a head-start can consult any of: Nakaha; geometry, topology and physics, Wald; General Relativity or Carroll; Spacetime and Geometry, an introduction to general relativity, though this is by no means necessary.
C7.7 Random Matrix Theory - Random Matrix Theory provides generic tools to analyse random linear systems. It plays a central role in a broad range of disciplines and application areas, including complex networks, data science, finance, machine learning, number theory, population dynamics, and quantum physics. Within Mathematics, it connects with asymptotic analysis, combinatorics, integrable systems, numerical analysis, probability, and stochastic analysis. This course aims to provide an introduction to this highly active, interdisciplinary field of research, covering some of the foundational concepts, methods, questions, and results. There are no formal prerequisites, but familiarity with basic concepts and results from linear algebra and probability will be assumed, at the level of A0 (Linear Algebra) and A8 (Probability). Full lecture notes are available on the course website.
Stochastics, Discrete Mathematics and Information
Courses: C4.9 Optimal Transport and PDEs, C8.1 Stochastic Differential Equations, C8.2 Stochastic Analysis and PDEs, C8.3 Combinatorics, C8.4 Probabilistic Combinatorics, C8.6 Limit Theorems and Large Deviations in Probability
C8.1 Stochastic Differential Equations - This course will assume familiarity with martingales and some knowledge of Brownian motion and stochastic calculus. Suitable background can be found in the lecture notes from the part B course: B8.2 Continuous Martingales and Stochastic Calculus.
C8.2 Stochastic Analysis and PDEs - This course builds a little bit on C8.1 in that students will be expected to have a grasp of stochastic calculus and stochastic differential equations. The B8.2 lecture notes would serve as good preparatory reading.
C8.3 Combinatorics - This course is essentially self-contained, and so will not rely crucially on previous material. That said, students wishing to take the course might find it useful to have some prior exposure to some combinatorial arguments, along the lines of Part B Graph Theory. An excellent source for the material covered in the course is Bollobás' book [1].
B8.5 Graph Theory Lecture Notes
[1] Bela Bollobás, Combinatorics, CUP, 1986.
Statistics and Probability
Courses: Stochastic Models in Mathematical Genetics (MT), Probability and Statistics for Network Analysis (MT), Graphical Models (MT), Probability on Graphs and Lattices (MT), Advanced Topics in Statistical Machine Learning (HT), Advanced Simulation Methods (HT), Topics in Computational Biology (HT), Bayes Methods (HT)
For courses on offer at the Department of Statistics, students are advised to take a look at the prerequisite document which has been produced for the MSc in Statistical Science students (available below). Some sections will not be relevant for OMMS and Part C Maths students (e.g. 2.4), though, there are some references and suggested ideas for preparing that students will find useful.