path: root/md/research/oxford-math.md

title:Oxford Math Courses
keywords:math,oxford

# Oxford Math Course

https://www.ox.ac.uk/admissions/undergraduate/courses-listing/mathematics  
http://mmathphys.physics.ox.ac.uk/  
https://courses.maths.ox.ac.uk/course_planner  

https://www.ox.ac.uk/admissions/undergraduate/courses-listing/mathematics-and-statistics  
http://www.stats.ox.ac.uk/student-resources/bammath/course-materials/  

## Stages

### Year 1

Mathematics - Algebra,Analysis,Probability and statistics,Geometry and dynamics,Multivariate calculus and mathematical models

Statistics - Algebra,Analysis,Probability and statistics,Geometry and dynamics,Multivariate calculus and mathematical models

### Year 2

Mathematics - Algebra,Complex analysis,Metric spaces,Differential equations (Algebra; Number theory; Analysis; Applied analysis; Geometry; Topology; Fluid dynamics; Probability; Statistics; Numerical analysis; Graph theory; Special relativity; Quantum theory)

Statistics - Probability,Statistics,Algebra and differential equations,Metric spaces and complex analysis,Simulation and statistical programming
### Year 3 and 4

Mathematics - Algebra; Applied and numerical analysis; Algebraic and differential geometry; Algebraic and analytic topology; Logic and set theory; Number theory; Applied probability; Statistics; Theoretical and statistical mechanics; Mathematical physics; Mathematical biology; Mathematical geoscience; Networks; Combinatorics; Information theory; Deep learning; Mathematical philosophy; Computer Science options; History of mathematics

Statistics - Applied and computational statistics,Statistical inference,Statistical machine learning,Applied probability,Statistical lifetime models,Stochastic models in mathematical genetics,Network analysis,Advanced statistical machine learning,Advanced simulation methods,Graphical models,Bayes methods,Computational biology,Algorithmic foundations of learning

## Topics

https://courses.maths.ox.ac.uk/overview/undergraduate/#50879


### A0: Linear Algebra
Definition of an abstract vector space over an arbitrary field. Examples.  
Linear maps. [1]  

Definition of a ring.   
Examples to include Z, F[x], F[A] (where A is a matrix or linear map), End(V). Division algorithm and Bezout's Lemma in F[x].   
Ring homomorphisms and isomorphisms. Examples. [2]  

Characteristic polynomials and minimal polynomials.   
Coincidence of roots. [1]  

Quotient vector spaces.   
The first isomorphism theorem for vector spaces and rank-nullity.  
Induced linear maps. Applications: Triangular form for matrices over C.  
Cayley-Hamilton Theorem. [2]  

Primary Decomposition Theorem.  
Diagonalizability and Triangularizability in terms of minimal polynomials.  
Proof of existence of Jordan canonical form over C (using primary decomposition and inductive proof of form for nilpotent linear maps). [3]  

Dual spaces of finite-dimensional vector spaces.   
Dual bases.   
Dual of a linear map and description of matrix with respect to dual basis.   
Natural isomorphism between a finite-dimensional vector space and its second dual.   
Annihilators of subspaces, dimension formula.   
Isomorphism between U0 and (V/U)'  . [3]  

Recap on real inner product spaces.   
Definition of non-degenerate symmetric bilinear forms and description as isomorphism between V and V'.   
Hermitian forms on complex vector spaces.   
Review of Gram-Schmidt. Orthogonal Complements. [1]

Adjoints for linear maps of inner product spaces.  
Uniqueness.  
Concrete construction via matrices [1]  

Definition of orthogonal/unitary maps.   
Definition of the groups On,SOn,Un,SUn.   
Diagonalizability of self-adjoint and unitary maps. [2]  

#### Reading

1) Richard Kaye and Robert Wilson, Linear Algebra (OUP, 1998) ISBN 0-19-850237-0. Chapters 2--13. [Chapters 6, 7 are not entirely relevant to our syllabus, but are interesting.]

Further Reading: 
1) Paul R. Halmos, Finite-dimensional Vector Spaces, (Springer Verlag, Reprint 1993 of the 1956 second edition), ISBN 3-540-90093-4. sections 1--15, 18, 32--51, 54--56, 59--67, 73, 74, 79.
[Now over 50 years old, this idiosyncratic book is somewhat dated but it is a great classic, and well worth reading.]

2) Seymour Lipschutz and Marc Lipson, Schaum's Outline of Linear Algebra (3rd edition, McGraw Hill, 2000), ISBN 0-07-136200-2. [Many worked examples.]

3) C. W. Curtis, Linear Algebra - an Introductory Approach, (4th edition, Springer, reprinted 1994).

4) D. T. Finkbeiner, Elements of Linear Algebra (Freeman, 1972). [Out of print, but available in many libraries.]

### A1: Differential Equations 1

https://courses.maths.ox.ac.uk/node/50806


Picard's Existence Theorem: Picard's Theorem for first-order scalar ODEs with proof.   
Gronwall's inequality leading to uniqueness and continuous dependence on the initial data.  
Examples of blow-up and nonuniqueness, discussion of continuation and global existence.   
Proof of Picard's Theorem via Contraction Mapping (Theorem CMT to be covered in Metric Spaces course) and relationship between the two proofs; extension to systems.  
Application to scalar second order ODEs, with particular reference to linear equations. (5 lectures)  

Phase plane analysis: Phase planes, critical points, definition of stability, classification of critical points and linearisation, Bendixson-Dulac criterion. (4 lectures)  

PDEs in two independent variables: First order semi-linear PDEs (using parameterisation). 
Classification of second order, semilinear PDEs; 
Normal form;  
Ideas of uniqueness and wellposedness.  
Illustration of suitable boundary conditions by example. 
 Poisson's Equation and the Heat Equation: Maximum Principle leading to uniqueness and continuous dependence on the initial data.  
(7 lectures)

P. J. Collins, Differential and Integral Equations (O.U.P., 2006), Chapters 1-7, 14,15.  
Further Reading: 
W. E. Boyce & R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems (7th edition, Wiley, 2000).  

Erwin Kreyszig, Advanced Engineering Mathematics (8th Edition, Wiley, 1999).  

W. A. Strauss, Partial Differential Equations: an Introduction (Wiley, 1992).  

G. F. Carrier & C E Pearson, Partial Differential Equations - Theory and Technique (Academic, 1988).  

J. Ockendon, S. Howison, A. Lacey & A. Movchan, Applied Partial Differential Equations (Oxford, 1999). [More advanced.]  

### A2: Metric Spaces and Complex Analysis

https://courses.maths.ox.ac.uk/node/50681


Basic definitions: metric spaces, isometries, continuous functions (epsilon   gamma definition), homeomorphisms, open sets, closed sets. 
Examples of metric spaces, including metrics derived from a norm on a real vector space, particularly l1,l2,l_infinite norms on Rn, the sup norm on the bounded real-valued functions on a set, and on the bounded continuous real-valued functions on a metric space.
 The characterisation of continuity in terms of the pre-image of open sets or closed sets. The limit of a sequence of points in a metric space. 
 A subset of a metric space inherits a metric. 
 Discussion of open and closed sets in subspaces. 
 The closure of a subset of a metric space. [3]

Completeness (but not completion).
 Completeness of the space of bounded real-valued functions on a set, equipped with the norm, and the completeness of the space of bounded continuous real-valued functions on a metric space, equipped with the metric. 
 Lipschitz maps and contractions. 
 Contraction Mapping Theorem. [2.5]

Connected metric spaces, path-connectedness. 
Closure of a connected space is connected, union of connected sets is connected if there is a non-empty intersection, continuous image of a connected space is connected. 
Path-connectedness implies connectedness. 
Connected open subset of a normed vector space is path-connected. [2]

Definition of sequential compactness and proof of basic properties of compact sets. 
Preservation of compactness under continuous maps, equivalence of continuity and uniform continuity for functions on a compact set. 
Equivalence of sequential compactness with being complete and totally bounded. 
The Arzela-Ascoli theorem (proof non-examinable). 
Open cover definition of compactness. 
Heine-Borel (for the interval only) and proof that compactness implies sequential compactness (statement of the converse only). [2.5]

Complex Analysis (22 lectures)

Basic geometry and topology of the complex plane, including the equations of lines and circles. 
Extended complex plane, Riemann sphere, stereographic projection. 
Mobius transformations acting on the extended complex plane. 
Mobius transformations take circlines to circlines. [3]

Complex differentiation. Holomorphic functions. 
Cauchy-Riemann equations (including z,z- version). 
Real and imaginary parts of a holomorphic function are harmonic. [2]

Recap on power series and differentiation of power series. 
Exponential function and logarithm function. 
Fractional powers     examples of multifunctions. 
The use of cuts as method of defining a branch of a multifunction. [3]

Path integration. 
Cauchy's Theorem. (Sketch of proof only     students referred to various texts for proof.) Fundamental Theorem of Calculus in the path integral/holomorphic situation. [2]

Cauchy's Integral formulae. 
Taylor expansion. 
Liouville's Theorem.
 Identity Theorem. 
 Morera's Theorem. [4]

Laurent's expansion. 
Classification of isolated singularities. 
Calculation of principal parts, particularly residues. [2]

Residue Theorem. 
Evaluation of integrals by the method of residues (straightforward examples only but to include the use of Jordan's Lemma and simple poles on contour of integration). [3]

Conformal mappings. 
Riemann mapping theorem (no proof), Mobius transformations, exponential functions, fractional powers; mapping regions (not Christoffel transformations or Joukowski's transformation). [3]



W. A. Sutherland, Introduction to Metric and Topological Spaces (Second Edition, OUP, 2009).  
H. A. Priestley, Introduction to Complex Analysis (Second edition, OUP, 2003).  
Further Reading:  
L. Ahlfors, Complex Analysis (McGraw-Hill, 1979).  
Reinhold Remmert, Theory of Complex Functions (Springer, 1989) (Graduate Texts in Mathematics 122).  

### A3: Rings and Modules

https://courses.maths.ox.ac.uk/node/50723

Recap on rings (not necessarily commutative) and examples: Z, fields, polynomial rings (in more than one variable), matrix rings. 
Zero-divisors, integral domains. 
Units. 
The characteristic of a ring. 
Discussion of fields of fractions and their characterization (proofs non-examinable) [2]

Homomorphisms of rings. 
Quotient rings, ideals and the first isomorphism theorem and consequences, e.g. 
Chinese remainder theorem. 
Relation between ideals in R and R/I. 
Prime ideals and maximal ideals, relation to fields and integral domains. 
Examples of ideals. 
Application of quotients to constructing fields by adjunction of elements; examples to include C=R[x]/(x2+1) and some finite fields. 
Degree of a field extension, the tower law. [4]

Euclidean Domains. Examples. 
Principal Ideal Domains. 
EDs are PIDs. 
Unique factorisation for PIDs. 
Gauss's Lemma and Eisenstein's Criterion for irreducibility. [3]

Modules: Definition and examples: vector spaces, abelian groups, vector spaces with an endomorphism. Submodules and quotient modules and direct sums. 
The first isomorphism theorem. [2]

Row and column operations on matrices over a ring. 
Equivalence of matrices. Smith Normal form of matrices over a Euclidean Domain. [1.5]

Free modules and presentations of finitely generated modules. 
Structure of finitely generated modules of a Euclidean domain. [2]

Application to rational canonical form and Jordan normal form for matrices, and structure of finitely generated Abelian groups. [1.5]

1) M. E. Keating, A First Course in Module Theory, Imperial College Press (1998)
Covers almost all material of the course. Out of print but many libraries should have it and second hand copies readily available.  

2) Joseph Gallian, Contemporary Abstract Algebra (9th edition, CENGAGE 2016) (Excellent text covering material on groups, rings and fields).  

3) B. Hartley, T. O. Hawkes, Chapman and Hall, Rings, Modules and Linear Algebra. (Out of print, but many libraries should have it. Relatively concise and covers all the material in the course).  

4) Neils Lauritzen, Concrete Abstract Algebra, CUP (2003) (Excellent on groups, rings and fields, and covers topics in the Number Theory course also. Does not cover material on modules).  

5) Michael Artin, Algebra (2nd ed. Pearson, (2010). (Excellent but highly abstract text covering everything in this course and much more besides).  

### A4: Integration

https://courses.maths.ox.ac.uk/node/50730

Measure spaces. 
Outer measure, null set, measurable set. 
The Cantor set. Lebesgue measure on the real line. 
Counting measure. 
Probability measures. 
Construction of a non-measurable set (non-examinable). 
Simple function, measurable function, integrable function. 
Reconciliation with the integral introduced in Prelims.

A simple comparison theorem. 
Integrability of polynomial and exponential functions over suitable intervals. Monotone Convergence Theorem. 
Fatou's Lemma. 
Dominated Convergence Theorem. 
Corollaries and applications of the Convergence Theorems (including term-by-term integration of series).

Theorems of Fubini and Tonelli (proofs not examinable). 
Differentiation under the integral sign. 
Change of variables.

Brief introduction to Lp spaces. 
H  lder and Minkowski inequalities.

M. Capinski & E. Kopp, Measure, Integral and Probability (Second Edition, Springer, 2004).  
F. Jones, Lebesgue Integration on Euclidean Space (Second Edition, Jones & Bartlett, 2000).  
Further Reading:   
D. S. Kurtz & C. W. Swartz, Theories of Integration (Series in Real Analysis Vol.9, World Scientific, 2004).  
H. A. Priestley, Introduction to Integration (OUP, 1997). [Useful for worked examples, although adopts a different approach to construction of the integral].  
H. L. Royden, Real Analysis (various editions; 4th edition has P. Fitzpatrick as co author).  
E. M. Stein & R. Shakarchi, Real Analysis: Measure Theory, Integration and Hilbert Spaces (Princeton Lectures in Analysis III, Princeton University Press, 2005).  
R. L. Schilling, Measures, Integrals and Martingales (CUP first ed. 2005, or second ed. 2017).  

### A5: Topology

https://courses.maths.ox.ac.uk/node/50742

Axiomatic definition of an abstract topological space in terms of open sets. Basic definitions: closed sets, continuity, homeomorphism, convergent sequences, connectedness and comparison with the corresponding definitions for metric spaces. Examples to include metric spaces (definition of topological equivalence of metric spaces), discrete and indiscrete topologies, cofinite topology. The Hausdorff condition. Subspace topology. [2 lectures]

Accumulation points of sets. Closure of a set. Interior of a set. Continuity if and only if f(A        )   f(A)                        [2 lectures]

Basis of a topology. Product topology on a product of two spaces and continuity of projections. [2 lectures]

Compact topological spaces, closed subset of a compact set is compact, compact subset of a Hausdorff space is closed. Product of two compact spaces is compact. A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. Equivalence of sequential compactness and abstract compactness in metric spaces. [2 lectures]

Quotient topology. Quotient maps. Characterisation of when quotient spaces are Hausdorff in terms of saturated sets. Examples, including the torus, Klein bottle and real projective plane. [2 lectures]

Abstract simplicial complexes and their topological realisation. A triangulation of a space. Any compact triangulated surface is homeomorphic to the sphere with g handles (g   0) or the sphere with h cross-caps (h   1). (No proof that these surfaces are not homeomorphic, but a brief informal discussion of Euler characteristic.) [6 lectures]

W. A. Sutherland, Introduction to Metric and Topological Spaces (Oxford University Press, 1975). Chapters 2-6, 8, 9.1-9.4. (New edition to appear shortly.)

J. R. Munkres, Topology, A First Course (Prentice Hall, 1974), chapters 2, 3, 7.

Further Reading: 
B. Mendelson, Introduction to Topology (Allyn and Bacon, 1975). (cheap paperback edition available).

G. Buskes, A. Van Rooij, Topological Spaces (Springer, 1997).

N. Bourbaki, General Topology (Springer, 1998).

J. Dugundji, Topology (Allyn and Bacon, 1966), chapters 3, 4, 5, 6, 7, 9, 11. [Although out of print, available in some libraries.]

### A6: Differential Equations 2

https://courses.maths.ox.ac.uk/node/50812

Models leading to two point boundary value problems for second order ODEs

Inhomogeneous two point boundary value problems (Ly=f); Wronskian and variation of parameters. Green's functions.

Adjoints. Self-adjoint operators. Eigenfunction expansions (issues of convergence and completeness noted but full treatment deferred to later courses). Sturm-Liouville theory. Fredholm alternative.

Series solutions. Method of Frobenius. Special functions.

Asymptotic sequences. Approximate roots of algebraic equations. Regular perturbations in ODE's. Introduction to boundary layer theory.

K. F. Riley, M. P. Hobson and S. J. Bence, Mathematical Methods for Physics and Engineering, (3rd Ed. Cambridge University Press, 2006).

W. E. Boyce & R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems (7th edition, Wiley, 2000).

P. J. Collins, Differential and Integral Equations (O.U.P., 2006).

Erwin Kreyszig, Advanced Engineering Mathematics (8th Edition, Wiley, 1999).

E. J. Hinch, Perturbation Methods (Cambridge University Press, Cambridge, 1991).

J. D. Logan, Applied Mathematics, (3rd Ed. Wiley Interscience, 2006).

### A7: Numerical Analysis

https://courses.maths.ox.ac.uk/node/50826

Lagrange interpolation [1 lecture]

Gaussian elimination, LU, QR factorisations, least-squares problems [3.5 lectures]

Eigenvalues: Gershgori    s Theorem, symmetric QR algorithm, polynomial rootfinding via eigenvalues [3.5 lectures]

SVD and low-rank matrix approximation [2 lectures]

Best approximation in inner product spaces, orthogonal polynomials, Gauss quadrature [3 lectures]

Forward and backward Euler, trapezium rule, leapfrog, Runge-Kutta methods [3 lectures]

Linear multi-step methods and Dahlquist   s theorem [2 lectures]

1) L. N. Trefethen and D. Bau, Numerical Linear Algebra (SIAM, 1997). (For Numerical Linear Algebra)

2) L. N. Trefethen, Approximation Theory and Approximation Practice (SIAM, 2012; extended edition 2020). (Highly recommended for Function Approximation)

3) E. Suli and D. F. Mayers, An Introduction to Numerical Analysis (CUP, 2003). Of which the relevant chapters are: 6, 7, 2, 5, 9, 11. (For ODEs; covers the subject broadly)



### A8 Probability
https://courses.maths.ox.ac.uk/node/50703

Continuous random variables.  
Jointly continuous random variables, independence, conditioning, functions of one or more random variables, change of variables.  
Examples including some with later applications in statistics.  
Moment generating functions and applications.  
Statements of the continuity and uniqueness theorems for moment generating functions. Characteristic functions (definition only).  
Convergence in distribution and convergence in probability.  
Weak law of large numbers and central limit theorem for independent identically distributed random variables.  
Strong law of large numbers (proof not examinable).  
Discrete-time Markov chains: definition, transition matrix, n-step transition probabilities, communicating classes, absorption, irreducibility, periodicity, calculation of hitting probabilities and mean hitting times.  
Recurrence and transience.  
Invariant distributions, mean return time, positive recurrence, convergence to equilibrium (proof not examinable), ergodic theorem (proof not examinable).  
Random walks (including symmetric and asymmetric random walks on Z, and symmetric random walks on Zd).  

Poisson processes in one dimension: exponential spacing   s, Poisson counts, thinning and superposition.
#### Literature
G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes (3rd edition, OUP, 2001). Chapters 4, 6.1-6.5, 6.8.  
G. R. Grimmett and D. R. Stirzaker, One Thousand Exercises in Probability (OUP, 2001).  
G. R. Grimmett and D J A Welsh, Probability: An Introduction (OUP, 1986). Chapters 6, 7.4, 8, 11.1-11.3.  
J. R. Norris, Markov Chains (CUP, 1997). Chapter 1.  
D. R. Stirzaker, Elementary Probability (Second edition, CUP, 2003). Chapters 7-9 excluding 9.9.  

### A9 Statistics

https://courses.maths.ox.ac.uk/node/50850


Order statistics, probability plots.

Estimation: observed and expected information, statement of large sample properties of maximum likelihood estimators in the regular case, methods for calculating maximum likelihood estimates, large sample distribution of sample estimators using the delta method.  

Hypothesis testing: simple and composite hypotheses, size, power and p-values, Neyman-Pearson lemma, distribution theory for testing means and variances in the normal model, generalized likelihood ratio, statement of its large sample distribution under the null hypothesis, analysis of count data.  

Confidence intervals: exact intervals, approximate intervals using large sample theory, relationship to hypothesis testing.  

Probability and Bayesian Inference.  
Posterior and prior probability densities.  
 Constructing priors including conjugate priors, subjective priors, Jeffreys priors. 
 Bayes estimators and credible intervals.  
 Statement of asymptotic normality of the posterior.  
 Model choice via posterior probabilities and Bayes factors.  

Examples: statistical techniques will be illustrated with relevant datasets in the lectures.

#### Reading


F. Daly, D.J. Hand, M.C. Jones, A.D. Lunn and K.J. McConway, Elements of Statistics (Addiso