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 (Addison Wesley, 1995) Chapters 7-10 (and Chapters 1-6 for background).  
J. A. Rice, Mathematical Statistics and Data Analysis (2nd edition, Wadsworth, 1995) Sections 8.5, 8.6, 9.1-9.7, 9.9, 10.3-10.6, 11.2, 11.3, 12.2.1, 13.3, 13.4.  
T Leonard and J.S.J. Hsu, Bayesian Methods (CUP, 1999), Chapters 2 and 3.  
G. Casella and R. L. Berger, Statistical Inference (2nd edition, Wadsworth, 2001).  
A. C. Davison, Statistical Models (Cambri
### A10: Fluids and Waves

https://courses.maths.ox.ac.uk/node/50862

Incompressible flow. Convective derivative, streamlines and particle paths. Euler's equations of motion for an inviscid fluid. Bernoulli's Theorem. Vorticity, circulation and Kelvin's Theorem. The vorticity equation and vortex motion.

Irrotational incompressible flow; velocity potential. Two-dimensional flow, stream function and complex potential. Line sources and vortices. Method of images, circle theorem and Blasius's Theorem.

Uniform flow past a circular cylinder. Circulation, lift. Use of conformal mapping to determine flow past a flat wing. Water waves, including effects of finite depth and surface tension. Dispersion, simple introduction to group velocity.

D. J. Acheson, Elementary Fluid Dynamics (OUP, 1997). Chapters 1, 3.1-3.5, 4.1-4.8, 4.10-4.12, 5.1, 5.2, 5.6, 5.7.

R. P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics, volume II, (Addison Wesley 1964) Chapter 40 (http://www.feynmanlectures.caltech.edu/II_40.html)

M. van Dyke, An Album of Fluid Motion (Parabolic Press, 1982).


### A11: Quantum Theory

https://courses.maths.ox.ac.uk/node/50872

Wave-particle duality; Schrodinger's equation; stationary states; quantum states of a particle in a box (infinite squarewell potential).

Interpretation of the wave function; boundary conditions; probability density and conservation of current; parity.

The one-dimensional harmonic oscillator; higher-dimensional oscillators and normal modes; degeneracy. The rotationally symmetric states of the hydrogen atom with fixed nucleus.

The mathematical structure of quantum mechanics and the postulates of quantum mechanics.

Commutation relations. Heisenberg's uncertainty principle.

Creation and annihilation operators for the harmonic oscillator. Measurements and the collapse of the wave function.

Schrodinger's cat. Angular momentum in quantum mechanics. The particular case of spin-1/2. Particle in a central potential. General states of the hydrogen atom.

B.H. Bransden and C.J Joachain Quantum Mechanics (Second edition, Pearson Education Limited, 2000). Chapters 1-4.

P.C.W. Davies and D.S. Betts, Quantum Mechanics (Physics and its Applications) (2nd edition, Taylor & Francis Ltd, 1994). Chapters 1,2,4.

R.P Feynman, R.B Leighton, M. Sands The Feynman Lectures on Physics, Volume 3, (Addison-Wesley, 1998). Chapters 1,2 (for physical background).

K.C Hannabuss, An Introduction to Quantum Theory (Oxford University Press 1997). Chapters 1-4.

A.I.M. Rae, Quantum Mechanics (4th Edition, Taylor & Francis Ltd, 2002). Chapters 1-3.


### A12 Simulation and Statistical Programming



### ASO: Integral Transforms

### ASO: Number Theory

### ASO: Group Theory

### ASO: Projective Geometry

### ASO: Introduction to Manifolds

### ASO: Calculus of Variations

### ASO: Graph Theory

### ASO: Special Relativity

### ASO: Mathematical Modelling in Biology



### B1.1 Logic

https://courses.maths.ox.ac.uk/node/48789

The notation, meaning and use of propositional and predicate calculus. The formal language of propositional calculus: truth functions; conjunctive and disjunctive normal form; tautologies and logical consequence. The formal language of predicate calculus: satisfaction, truth, validity, logical consequence.

Deductive system for propositional calculus: proofs and theorems, proofs from hypotheses, the Deduction Theorem; Soundness Theorem. Maximal consistent sets of formulae; completeness; constructive proof of completeness.

Statement of Soundness and Completeness Theorems for a deductive system for predicate calculus; derivation of the Compactness Theorem; simple applications of the Compactness Theorem.

A deductive system for predicate calculus; proofs and theorems; prenex form. Proof of Completeness Theorem. Existence of countable models, the downward Lowenheim-Skolem Theorem.

R. Cori and D. Lascar, Mathematical Logic: A Course with Exercises (Part I) (Oxford University Press, 2001), sections 1, 3, 4.
A. G. Hamilton, Logic for Mathematicians (2nd edition, Cambridge University Press, 1988), pp.1-69, pp.73-76 (for statement of Completeness (Adequacy) Theorem), pp.99-103 (for the Compactness Theorem).
W. B. Enderton, A Mathematical Introduction to Logic (Academic Press, 1972), pp.101-144.
D. Goldrei, Propositional and Predicate Calculus: A model of argument (Springer, 2005).
A. Prestel and C. N. Delzell, Mathematical Logic and Model Theory (Springer, 2010).
Further Reading: 
R. Cori and D. Lascar, Mathematical Logic: A Course with Exercises (Part II) (Oxford University Press, 2001), section 8.

### B1.2 Set Theory

https://courses.maths.ox.ac.uk/node/48811

What is a set? Introduction to the basic axioms of set theory. Ordered pairs, cartesian products, relations and functions. Axiom of Infinity and the construction of the natural numbers; induction and the Recursion Theorem.

Cardinality; the notions of finite and countable and uncountable sets; Cantor's Theorem on power sets. The Tarski Fixed Point Theorem. The Schroder-Bernstein Theorem. Basic cardinal arithmetic.

Well-orders. Comparability of well-orders. Ordinal numbers. Transfinite induction; transfinite recursion [informal treatment only]. Ordinal arithmetic.

The Axiom of Choice, Zorn's Lemma, the Well-ordering Principle; comparability of cardinals. Equivalence of WO, CC, AC and ZL. Cardinal numbers.

D. Goldrei, Classic Set Theory (Chapman and Hall, 1996).
H. B. Enderton, Elements of Set Theory (Academic Press, 1978).
Further Reading: 
R. Cori and D. Lascar, Mathematical Logic: A Course with Exercises (Part II) (Oxford University Press, 2001), section 7.1-7.5.
R. Rucker, Infinity and the Mind: The Science and Philosophy of the Infinite (Princeton University Press, 1995). An accessible introduction to set theory.
J. W. Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Princeton University Press, 1990). For some background, you may find JW Dauben's biography of Cantor interesting.
M. D. Potter, Set Theory and its Philosophy: A Critical Introduction (Oxford University Press, 2004). An interestingly different way of establishing Set Theory, together with some discussion of the history and philosophy of the subject.
W. Sierpinski, Cardinal and Ordinal Numbers (Polish Scientific Publishers, 1965). More about the arithmetic of transfinite numbers.
J. Stillwell, Roads to Infinity (CRC Press, 2010).

### B2.1 Introduction to Representation Theory

https://courses.maths.ox.ac.uk/node/48817

Noncommutative rings, one- and two-sided ideals. Associative algebras (over fields). Main examples: matrix algebras, polynomial rings and quotients of polynomial rings. Group algebras, representations of groups.

Modules and their relationship with representations. Simple and semisimple modules, composition series of a module, Jordan-Holder Theorem. Semisimple algebras. Schur's Lemma, the Wedderburn Theorem, Maschke's Theorem. Characters of complex representations. Orthogonality relations, finding character tables. Tensor product of modules. Induction and restriction of representations. Application: Burnside's paqb Theorem.

K. Erdmann, T. Holm Algebras and Representation Theory, Springer Undergraduate Mathematical Series (2018), ISSN 1615-2085
G. D. James and M. Liebeck, Representations and Characters of Finite Groups (2nd edition, Cambridge University Press, 2001).
Further Reading: 
J. L. Alperin and R. B. Bell, Groups and Representations, Graduate Texts in Mathematics 162 (Springer-Verlag, 1995).
P. M. Cohn, Classic Algebra (Wiley & Sons, 2000). (Several books by this author available.)
C. W. Curtis, and I. Reiner, Representation Theory of Finite Groups and Associative Algebras (Wiley & Sons, 1962).
L. Dornhoff, Group Representation Theory (Marcel Dekker Inc., New York, 1972).
I. M. Isaacs, Character Theory of Finite Groups (AMS Chelsea Publishing, American Mathematical Society, Providence, Rhode Island, 2006).
J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics 42 (Springer-Verlag, 1977).
P. Etingof, Introduction to representation theory (Online course notes, MIT 2011).

### B2.2 Commutative Algebra

https://courses.maths.ox.ac.uk/node/48825

Modules, ideals, prime ideals, maximal ideals.

Noetherian rings; Hilbert basis theorem. Minimal primes.

Localization.

Polynomial rings and algebraic sets. Weak Nullstellensatz.

Nilradical and Jacobson radical; strong Nullstellensatz.

Integral extensions. Prime ideals in integral extensions.

Noether Normalization Lemma.

Krull dimension; dimension of an affine algebra.

Noetherian rings of small dimension, Dedekind domains.

M. F. Atiyah and I. G. MacDonald: Introduction to Commutative Algebra, (Addison-Wesley, 1969).
D. Eisenbud: Commutative Algebra with a view towards Algebraic Geometry, (Springer GTM, 1995).

### B3.1 Galois Theory

https://courses.maths.ox.ac.uk/node/48832

Review of polynomial rings, factorisation, integral domains. Reminder that any nonzero homomorphism of fields is injective. Fields of fractions.

Review of group actions on sets, Gauss' Lemma and Eisenstein's criterion for irreducibility of polynomials, field extensions, degrees, the tower law. Symmetric polynomials.

Separable extensions. Splitting fields and normal extensions. The theorem of the primitive element. The existence and uniqueness of algebraic closure (proofs not examinable).

Groups of automorphisms, fixed fields. The fundamental theorem of Galois theory.

Examples: Kummer extensions, cyclotomic extensions, finite fields and the Frobenius automorphism. Techniques for calculating Galois groups.

Soluble groups. Solubility by radicals, solubility of polynomials of degree at most 4, insolubility of the general quintic, impossibility of some ruler and compass constructions.

J. Rotman, Galois Theory (Springer-Verlag, NY Inc, 2001/1990).
I. Stewart, Galois Theory (Chapman and Hall, 2003/1989).
D.J.H. Garling, A Course in Galois Theory (Cambridge University Press I.N., 1987).
Herstein, Topics in Algebra (Wiley, 1975).


### B3.2 Geometry of Surfaces

### B3.3 Algebraic Curves

### B3.4 Algebraic Number Theory

https://courses.maths.ox.ac.uk/node/48862

Field extensions, minimum polynomial, algebraic numbers, conjugates, discriminants, Gaussian integers, algebraic integers, integral basis

Examples: quadratic fields

Norm of an algebraic number

Existence of factorisation

Factorisation in Q(d         )
Ideals, Z-basis, maximal ideals, prime ideals

Unique factorisation theorem of ideals

Relationship between factorisation of number and of ideals

Norm of an ideal

Ideal classes

Statement of Minkowski convex body theorem

Finiteness of class number

Computations of class number to go on example sheets


I. Stewart and D. Tall, Algebraic Number Theory and Fermat's Last Theorem (Third Edition, Peters, 2002).

Further Reading: 
D. Marcus, Number Fields (Springer-Verlag, New York-Heidelberg, 1977). ISBN 0-387-90279-1.

### B3.5 Topology and Groups

https://courses.maths.ox.ac.uk/node/48869

### B4.1 Functional Analysis I

https://courses.maths.ox.ac.uk/node/48881

### B4.2 Functional Analysis II

https://courses.maths.ox.ac.uk/node/48890

### B4.3 Distribution Theory

https://courses.maths.ox.ac.uk/node/48897

### B4.4 Fourier Analysis

https://courses.maths.ox.ac.uk/node/48912

### B5.1 Stochastic Modelling of Biological Processes

https://courses.maths.ox.ac.uk/node/48922

### B5.2 Applied Partial Differential Equations

https://courses.maths.ox.ac.uk/node/48944

### B5.3 Viscous Flow

https://courses.maths.ox.ac.uk/node/48954

### B5.4 Waves and Compressible Flow

https://courses.maths.ox.ac.uk/node/48960

### B5.5 Further Mathematical Biology

https://courses.maths.ox.ac.uk/node/48968

### B5.6 Nonlinear Systems

https://courses.maths.ox.ac.uk/node/48977

### B6.1 Numerical Solution of Differential Equations I

https://courses.maths.ox.ac.uk/node/48986

### B6.2 Numerical Solution of Differential Equations II

https://courses.maths.ox.ac.uk/node/49027

### B6.3 Integer Programming

https://courses.maths.ox.ac.uk/node/49036

Week 1:

Classical examples of Integer Programming problems (IP), modelling and basic terminology.
Linear programming I: the simplex method.
Week 2:

Linear programming II: Duality Theory.
Total unimodularity I: Ideal formulations of IPs and totally unimodular matrices.
Week 3:

Total Unimodularity II: Exact theoretical characterisation, practical sufficient criteria, bipartite matching, the shortest path problem.
Submodularity I: Submodular functions and submodular optimisation problems.
Week 4:

Submodularity II: Submodular rank functions, matroids, the greedy algorithm and the maximum weight independent set problem.
Branch-and-Bound I: LP based branch-and-bound for general integer programming problems.
Week 5:

Branch-and-bound II: general B&B, pre-processing, warm starting of LPs, dual simplex method.
Dantzig-Wolfe decomposition, delayed column generation.
Week 6:

Branch-and-Price, application to the cutting stock problem.
Preprocessing of LPs and IPs, generating valid cuts, cutting plane algorithm.
Week 7:

Chvatal cuts, Gomoroy cuts, branch-and-cut algorithm.
The Generalised Assignment Problem.
Week 8:

Lagrangian relaxation and Lagrangian duality.
The subgradient algorithm.

M. Conforti, G. Cornuejols, G. Zambelli, Integer Programming (Springer 2014), ISBN 978-3-319-11007-3.
L. A. Wolsey, Integer Programming (John Wiley & Sons, 1998), parts of chapters 1-5 and 7.

### B7.1 Classical Mechanics

### B7.2 Electromagnetism

### B7.3 Further Quantum Theory

### B8.1 Probability, Measure and Martingales

https://courses.maths.ox.ac.uk/node/49105

### B8.2 Continuous Martingales and Stochastic Calculus

### B8.3 Mathematical Models of Financial Derivatives

https://courses.maths.ox.ac.uk/node/49119

### B8.4 Information Theory

https://courses.maths.ox.ac.uk/node/49135

### 8.5 Graph Theory

### BEE Mathematical Extended Essay

### BSP Structured Projects

### BO1.1 History of Mathematics

https://courses.maths.ox.ac.uk/node/49179

Introduction: ancient mathematical knowledge and its transmission to early modern Europe; the development of symbolic notation up to the end of the sixteenth century.
Seventeenth century: analytic geometry; the development of calculus; Newton's Principia.
Eighteenth century: from calculus to analysis; functions, limits, continuity; equations and solvability.
Nineteenth century: group theory and abstract algebra; the beginnings of modern analysis; rigorous definitions of real numbers; integration; complex analysis; set theory; linear algebra.

Jacqueline Stedall, Mathematics emerging: a sourcebook 1540-1900 (Oxford University Press, 2008).
Victor Katz, A history of mathematics (brief edition) (Pearson Addison Wesley, 2004), or:
Victor Katz, A history of mathematics: an introduction (third edition) (Pearson Addison Wesley, 2009).
Benjamin Wardhaugh, How to read historical mathematics (Princeton, 2010).
Jacqueline Stedall, The history of mathematics: a very short introduction (Oxford University Press, 2012).
Further Reading: 
John Fauvel and Jeremy Gray (eds), The history of mathematics: a reader, (Macmillan, 1987).
June Barrow-Green, Jeremy Gray and Robin J. Wilson, The history of mathematics : a source-based approach, vol. I (Mathematical Association of America, 2019).

### BOE: Other Mathematical Extended Essay


### SB3.1 Applied Probability

### BEE Mathematical Extended Essay

### BSP Structured Projects

### BO1.1 History of Mathematics

### BOE: Other Mathematical Extended Essay


### SB1.1 Applied Statistics

### SB1.2 Computational Statistics

### SB2.1 Foundations of Statistical Inference

### SB2.2 Statistical Machine Learning

### SB3.1 Applied Probability

https://courses.maths.ox.ac.uk/node/49150

Poisson processes and birth processes. Continuous-time Markov chains. Transition rates, jump chains and holding times. Forward and backward equations. Class structure, hitting times and absorption probabilities. Recurrence and transience. Invariant distributions and limiting behaviour. Time reversal. Renewal theory. Limit theorems: strong law of large numbers, strong law and central limit theorem of renewal theory, elementary renewal theorem, renewal theorem, key renewal theorem. Excess life, inspection paradox.

Applications in areas such as: queues and queueing networks - M/M/s queue, Erlang's formula, queues in tandem and networks of queues, M/G/1 and G/M/1 queues; insurance ruin models; applications in applied sciences.

J. R. Norris, Markov Chains (Cambridge University Press, 1997).
G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes (3rd edition, Oxford University Press, 2001).
G. R. Grimmett and D. R. Stirzaker, One Thousand Exercises in Probability (Oxford University Press, 2001).
S. M. Ross, Introduction to Probability Models (4th edition, Academic Press, 1989).
D. R. Stirzaker, Elementary Probability (2nd edition, Cambridge University Press, 2003).

### SB3.2 Statistical Lifetime Models


### C1.1 Model Theory

### C1.3 Analytic Topology

### C2.1 Lie Algebras

### C2.2 Homological Algebra

### C2.4 Infinite Groups

### C2.7 Category Theory

### C3.1 Algebraic Topology

### C3.3 Differentiable Manifolds

### C3.4 Algebraic Geometry

### C3.8 Analytic Number Theory

### C3.10 Additive and Combinatorial Number Theory

### C4.1 Further Functional Analysis

### C4.3 Functional Analytic Methods for PDEs

### C4.8 Complex Analysis: Conformal Maps and Geometry

### C5.1 Solid Mechanics

### C5.5 Perturbation Methods

### C5.7 Topics in Fluid Mechanics

### C5.11 Mathematical Geoscience

### C5.12 Mathematical Physiology

### C6.1 Numerical Linear Algebra

### C6.3 Approximation of Functions

### C6.5 Theories of Deep Learning

### C7.1 Theoretical Physics (C6)

### C7.5 General Relativity I

### C8.1 Stochastic Differential Equations

### C8.3 Combinatorics

### CCD Dissertations on a Mathematical Topic

### COD Dissertations on the History of Mathematics

### C1.2 Goodel's Incompleteness Theorems

### C1.4 Axiomatic Set Theory

### C2.3 Representation Theory of Semisimple Lie Algebras

### C2.5 Non-Commutative Rings

### C2.6 Introduction to Schemes

### C3.2 Geometric Group Theory

### C3.5 Lie Groups

### C3.7 Elliptic Curves

### C3.9 Computational Algebraic Topology

### C3.11 Riemannian Geometry

### C4.6 Fixed Point Methods for Nonlinear PDEs

### C4.9 Optimal Transport & Partial Differential Equations

### C5.2 Elasticity and Plasticity

### C5.3 Statistical Mechanics

### C5.4 Networks

### C5.6 Applied Complex Variables

### C5.9 Mathematical Mechanical Biology

### C6.2 Continuous Optimisation

### C6.4 Finite Element Method for PDEs

### C7.1 Theoretical Physics (C6)

### C7.4 Introduction to Quantum Information

### C7.6 General Relativity II

### C7.7 Random Matrix Theory

### C8.2 Stochastic Analysis and PDEs

### C8.4 Probabilistic Combinatorics

### C8.5 Introduction to Schramm-Loewner Evolution

### C8.6 Limit Theorems and Large Deviations in Probability

### CCD Dissertations on a Mathematical Topic

### COD Dissertations on the History of Mathematics


### SC1 Stochastic Models in Mathematical Genetics

### SC2 Probability and Statistics for Network Analysis

### SC4 Advanced Topics in Statistical Machine Learning

### SC5 Advanced Simulation Methods

### SC6 Graphical Models

### SC7 Bayes Methods

### SC8 Topics in Computational Biology

### SC9 Probability on Graphs and Lattices

### SC10 Algorithmic Foundations of Learning