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-archive.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-archive.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-archive.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-archive.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-archive.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-archive.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-archive.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-archive.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-archive.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-archive.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-archive.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 (Cambridge)  

### A10: Fluids and Waves

https://courses-archive.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-archive.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

https://courses-archive.maths.ox.ac.uk/node/50748

### ASO: Number Theory

https://courses-archive.maths.ox.ac.uk/node/50752

#### Course Overview: 
Number theory is one of the oldest parts of mathematics. For well over two thousand years it has attracted professional and amateur mathematicians alike. Although notoriously `pure' it has turned out to have more and more applications as new subjects and new technologies have developed. Our aim in this course is to introduce students to some classical and important basic ideas of the subject.

#### Learning Outcomes: 
Students will learn some of the foundational results in the theory of numbers due to mathematicians such as Fermat, Euler and Gauss. They will also study a modern application of this ancient part of mathematics.

#### Course Synopsis: 
The ring of integers;  
congruences;  
ring of integers modulo n;  
the Chinese Remainder Theorem.  
Wilson's Theorem;  
Fermat's Little Theorem for prime modulus;  
Euler's phi-function.  
Euler's generalisation of Fermat's Little Theorem to arbitrary modulus;  
primitive roots.
Quadratic residues modulo primes.  
Quadratic reciprocity.  
Factorisation of large integers;  
basic version of the RSA encryption method.  

#### Reading list

1) H. Davenport, The Higher Arithmetic (Cambridge University Press, 1992) ISBN 0521422272  
2) G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers (OUP, 1980) ISBN 0198531710  
3) P. Erdos and J. Suranyi, Topics in the Theory of Numbers (Springer, 2003) ISBN 0387953205  

### ASO: Group Theory

https://courses-archive.maths.ox.ac.uk/node/50756

#### Course Overview: 
This group theory course develops the theory begun in prelims, and this course will build on that. After recalling basic concepts, the focus will be on two circles of problems.  
1. The concept of free group and its universal property allow to define and describe groups in terms of generators and relations.  
2. The notion of composition series and the Jordan-Holder Theorem explain how to see, for instance, finite groups as being put together from finitely many simple groups. This leads to the problem of finding and classifying finite simple groups. Conversely, it will be explained how to put together two given groups to get new ones.  
Moreover, the concept of symmetry will be formulated in terms of group actions and applied to prove some group theoretic statements.  

#### Learning Outcomes: 
Students will learn to construct and describe groups. They will learn basic properties of groups and get familiar with important classes of groups. They will understand the crucial concept of simple groups. They will get a better understanding of the notion of symmetry by using group actions.  

#### Course Synopsis: 
Free groups.  
Uniqueness of reduced words and universal mapping property.  
Normal subgroups of free groups and generators and relations for groups. Examples. [2]  
Review of the First Isomorphism Theorem and proof of Second and Third Isomorphism Theorems.  
Simple groups, statement that An is simple (proof for n=5).  
Definition and proof of existence of composition series for finite groups.  
Statement of the Jordan-Holder Theorem. Examples.  
The derived subgroup and solvable groups. [3]  
Discussion of semi-direct products and extensions of groups. Examples. [1]   
Sylow's three theorems.  
Applications including classification of groups of small order. [2]  

#### Reading list

1) Humphreys, J. F. A Course in Group Theory, Oxford, 1996  
2) Armstrong, M. A. Groups and Symmetry, Springer-Verlag, 1988  

### ASO: Projective Geometry

https://courses-archive.maths.ox.ac.uk/node/50765

#### Course Overview: 
Projective Geometry might be viewed as the geometry of perspective. Two observers of a painting - one looking obliquely, one straight on - will not agree on angles and distances but will both sees lines as lines and will agree on whether they meet. So projective transformations (such as relate the two observers views) are less rigid than Euclidean, or even affine, transformations. Projective geometry also introduces the idea of points at infinity - points where parallel lines meet. These points fill in the missing gaps/address some special cases of geometry in a similar way to which complex numbers resolve such problems in algebra. From this point of view ellipses, parabolae and hyperbolae are all projectively equivalent and just happen to include no, one or two points at infinity. The study of such conics also has applications to the study of quadratic Diophantine equations.

#### Learning Outcomes: 
Students will be familiar with the idea of projective space and the linear geometry associated to it, including examples of duality and applications to Diophantine equations.

#### Course Synopsis: 
1-2: Projective Spaces (as P(V) of a vector space V).  
Homogeneous Co-ordinates.  
Linear Subspaces. 
3-4: Projective Transformations.  
General Position.  
Desargues Theorem.  
Cross-ratio.  
5: Dual Spaces.  
Duality.  
6-7: Symmetric Bilinear Forms.  
Conics.  
Singular conics, singular points.  
Projective equivalence of non-singular conics.  
7-8: Correspondence between P1 and a non-singular conic.  
Simple applications to Diophantine Equations.  

#### Reading List: 
1) N.J. Hitchin, Maths Institute notes on Projective Geometry (found under 'Teaching')  
2) M. Reid and B. Szendroi, Geometry and topology, Cambridge University Press, 2005 (Chapter 5).  
3) R. Casse, Projective Geometry, An Introduction, Oxford University Press (2006)  

### ASO: Introduction to Manifolds

https://courses-archive.maths.ox.ac.uk/node/50777

Course Overview: 
In this course, the notion of the total derivative for a function f:Rm->Rn is introduced. Roughly speaking, this is an approximation of the function near each point in Rn by a linear transformation. This is a key concept which pervades much of mathematics, both pure and applied. It allows us to transfer results from linear theory locally to nonlinear functions. For example, the Inverse Function Theorem tells us that if the derivative is an invertible linear mapping at a point then the function is invertible in a neighbourhood of this point. Another example is the tangent space at a point of a surface in R3, which is the plane that locally approximates the surface best.

Learning Outcomes: 
Students will understand the concept of derivative in n dimensions and the implict and inverse function theorems which give a bridge between suitably nondegenerate infinitesimal information about mappings and local information. They will understand the concept of manifold and see some examples such as matrix groups.

Course Synopsis: 
Definition of a derivative of a function from Rm to Rn; examples; elementary properties; partial derivatives; the chain rule; the gradient of a function from Rn to R; Jacobian. Continuous partial derivatives imply differentiability, Mean Value Theorems. [3 lectures]

The Inverse Function Theorem and the Implicit Function Theorem (proofs non-examinable). [2 lectures]

The definition of a submanifold of Rm. Its tangent and normal space at a point, examples, including two-dimensional surfaces in R3. [2 lectures]

Lagrange multipliers. [1 lecture]

Reading List: 
Access ORLO reading list
 
Theodore Shifrin, Multivariable Mathematics (Wiley, 2005). Chapters 3-6.

T. M. Apostol, Mathematical Analysis: Modern Approach to Advanced Calculus (World Students) (Addison Wesley, 1975). Chapters 6 and 7.

S. Dineen, Multivariate Calculus and Geometry (Springer, 2001). Chapters 1-4.

J. J. Duistermaat and J A C Kolk, Multidimensional Real Analysis I, Differentiation (Cambridge University Press, 2004).

M. Spivak, Calculus on Manifolds: A modern approach to classical theorems of advanced calculus, W. A. Benjamin, Inc., New York-Amsterdam, 1965.

Further Reading: 
William R. Wade, An Introduction to Analysis (Second Edition, Prentice Hall, 2000). Chapter 11.

M. P. Do Carmo, Differential Geometry of Curves and Surfaces (Prentice Hall, 1976).

Stephen G. Krantz and Harold R. Parks, The Implicit Function Theorem: History, Theory and Applications (Birkhaeuser, 2002).

### ASO: Calculus of Variations

### ASO: Graph Theory

https://courses-archive.maths.ox.ac.uk/node/50793

#### Course Overview: 
This course introduces some central topics in graph theory.

#### Learning Outcomes: 
By the end of the course, students should have an appreciation of the methods and results of graph theory. They should have a good understanding of the basic objects in graph theory, such as trees, Euler circuits and matchings, and they should be able to reason effectively about graphs.

#### Course Synopsis: 
Introduction. Paths, walks, cycles and trees.   
Euler circuits.   
Hamiltonian cycles.  
Hall's theorem.  
Application and analysis of algorithms for minimum cost spanning trees, shortest paths, bipartite matching and the Chinese Postman Problem.  

#### Reading List:
Access ORLO reading list  
R. J. Wilson, Introduction to Graph Theory, 5th edition, Prentice Hall, 2010.  
D.B. West, Introduction to Graph Theory, 2nd edition, Prentice Hall, 2001.  

### ASO: Special Relativity

### ASO: Mathematical Modelling in Biology



### B1.1 Logic

https://courses-archive.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-archive.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-archive.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-archive.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-archive.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.  

#### Reading list
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

https://courses-archive.maths.ox.ac.uk/node/48839

#### Course Overview: 
Different ways of thinking about surfaces (also called two-dimensional manifolds) are introduced in this course: first topological surfaces and then surfaces with extra structures which allow us to make sense of differentiable functions (`smooth surfaces'), holomorphic functions (`Riemann surfaces') and the measurement of lengths and areas ('Riemannian 2-manifolds').

These geometric structures interact in a fundamental way with the topology of the surfaces. A striking example of this is given by the Euler number, which is a manifestly topological quantity, but can be related to the total curvature, which at first glance depends on the geometry of the surface.

The course ends with an introduction to hyperbolic surfaces modelled on the hyperbolic plane, which gives us an example of a non-Euclidean geometry (that is, a geometry which meets all of Euclid's axioms except the axiom of parallels).

#### Learning Outcomes: 
Students will be able to implement the classification of surfaces for simple constructions of topological surfaces such as planar models and connected sums; be able to relate the Euler characteristic to branching data for simple maps of Riemann surfaces; be able to describe the definition and use of Gaussian curvature; know the geodesics and isometries of the hyperbolic plane and their use in geometrical constructions.

#### Course Synopsis: 
The concept of a topological surface (or 2-manifold); examples, including polygons with pairs of sides identified.  
Orientation and the Euler characteristic.  
Classification theorem for compact surfaces (the proof will not be examined).  
Riemann surfaces; examples, including the Riemann sphere, the quotient of the complex numbers by a lattice, and double coverings of the Riemann sphere.  
Holomorphic maps of Riemann surfaces and the Riemann-Hurwitz formula.  
Elliptic functions.  
Smooth surfaces in Euclidean three-space and their first fundamental forms.  
The concept of a Riemannian 2-manifold; isometries; Gaussian curvature.    
Geodesics.   
The Gauss-Bonnet Theorem (statement of local version and deduction of global version).  
Critical points of real-valued functions on compact surfaces.  
The hyperbolic plane, its isometries and geodesics.  
Compact hyperbolic surfaces as Riemann surfaces and as surfaces of constant negative curvature.  

#### Reading List: 
A. Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series (Springer-Verlag, 2001). (Chapters 4-8 and 10-11.)  
G. B. Segal, Geometry of Surfaces, Mathematical Institute Notes (1989).  
R. Earl, The Local Theory of Curves and Surfaces, Mathematical Institute Notes (1999).  
J. McCleary, Geometry from a Differentiable Viewpoint, (Cambridge, 1997).  
#### Further Reading: 
P. A. Firby and C. E. Gardiner, Surface Topology (Ellis Horwood, 1991) (Chapters 1-4 and 7).  
F. Kirwan, Complex Algebraic Curves, Student Texts 23 (London Mathematical Society, Cambridge, 1992) (Chapter 5.2 only).  
B. O'Neill, Elementary Differential Geometry (Academic Press, 1997).  
M. P. do Carmo, Differential Geometry of Curves and Surfaces (Dover, 2016)  

### B3.3 Algebraic Curves

https://courses-archive.maths.ox.ac.uk/node/48854

#### Course Overview: 
A real algebraic curve is a subset of the plane defined by a polynomial equation p(x,y)=0. The intersection properties of a pair of curves are much better behaved if we extend this picture in two ways: the first is to use polynomials with complex coefficients, the second to extend the curve into the projective plane. In this course projective algebraic curves are studied, using ideas from algebra, from the geometry of surfaces and from complex analysis.

#### Learning Outcomes: 
Students will know the concepts of projective space and curves in the projective plane. They will appreciate the notion of nonsingularity and know some basic features of intersection theory. They will view nonsingular algebraic curves as examples of Riemann surfaces, and be familiar with divisors, meromorphic functions and differentials.

#### Course Synopsis: 
Projective spaces, homogeneous coordinates, projective transformations.  
Algebraic curves in the complex projective plane.  
Irreducibility, singular and nonsingular points, tangent lines.   
Bezout's Theorem (the proof will not be examined).  
Points of inflection, and normal form of a nonsingular cubic.  
Nonsingular algebraic curves as Riemann surfaces.  
Meromorphic functions, divisors, linear equivalence.  
Differentials and canonical divisors.  
The group law on a nonsingular cubic.  
The Riemann-Roch Theorem (the proof will not be examined).  
The geometric genus. Applications.  

#### Reading List: 
F. Kirwan, Complex Algebraic Curves, Student Texts 23 (London Mathematical Society, Cambridge, 1992), Chapters 2-6.  
W. Fulton, Algebraic Curves, 3rd ed., downloadable at http://www.math.lsa.umich.edu/~wfulton  

### B3.4 Algebraic Number Theory

https://courses-archive.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  

#### Reading list
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-archive.maths.ox.ac.uk/node/48869

### B4.1 Functional Analysis I

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

### B4.2 Functional Analysis II

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

### B4.3 Distribution Theory

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

### B4.4 Fourier Analysis

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

### B5.1 Stochastic Modelling of Biological Processes

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

### B5.2 Applied Partial Differential Equations

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

### B5.3 Viscous Flow

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

### B5.4 Waves and Compressible Flow

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

### B5.5 Further Mathematical Biology

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

### B5.6 Nonlinear Systems

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

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

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

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

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

### B6.3 Integer Programming

https://courses-archive.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.  

#### Reading list

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-archive.maths.ox.ac.uk/node/49105

### B8.2 Continuous Martingales and Stochastic Calculus

### B8.3 Mathematical Models of Financial Derivatives

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

### B8.4 Information Theory

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

### B8.5 Graph Theory

https://courses-archive.maths.ox.ac.uk/node/49141

#### Course Overview: 
Graphs (abstract networks) are among the simplest mathematical structures, but nevertheless have a very rich and well-developed structural theory. Since graphs arise naturally in many contexts within and outside mathematics, Graph Theory is an important area of mathematics, and also has many applications in other fields such as computer science.

The main aim of the course is to introduce the fundamental ideas of Graph Theory, and some of the basic techniques of combinatorics.

#### Learning Outcomes: 
The student will have developed a basic understanding of the properties of graphs, and an appreciation of the combinatorial methods used to analyze discrete structures.

#### Course Synopsis: 
Introduction: basic definitions and examples.  
Trees and their characterization.  
Euler circuits; long paths and cycles.  
Vertex colourings: Brooks' theorem, chromatic polynomial.  
Edge colourings: Vizing's theorem.  
Planar graphs, including Euler's formula, dual graphs.  
Maximum flow - minimum cut theorem: applications including Menger's theorem and Hall's theorem.  
Tutte's theorem on matchings.  
Extremal Problems: Turan's theorem, Zarankiewicz problem, Erdos-Stone theorem.  

#### Reading List: 
B. Bollobas, Modern Graph Theory, Graduate Texts in Mathematics 184 (Springer-Verlag, 1998)  
#### Further Reading: 
J. A. Bondy and U. S. R. Murty, Graph Theory: An Advanced Course, Graduate Texts in Mathematics 244 (Springer-Verlag, 2007).  
R. Diestel, Graph Theory, Graduate Texts in Mathematics 173 (third edition, Springer-Verlag, 2005).  
D. West, Introduction to Graph Theory, Second edition, (Prentice-Hall, 2001).  

### BEE Mathematical Extended Essay

### BSP Structured Projects

### BO1.1 History of Mathematics

https://courses-archive.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-archive.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

https://courses-archive.maths.ox.ac.uk/node/49212



### C1.3 Analytic Topology

### C2.1 Lie Algebras

### C2.2 Homological Algebra

### C2.4 Infinite Groups

https://courses-archive.maths.ox.ac.uk/node/49282

### C2.7 Category Theory

https://courses-archive.maths.ox.ac.uk/node/49315

### C3.1 Algebraic Topology

https://courses-archive.maths.ox.ac.uk/node/49323

### C3.3 Differentiable Manifolds

### C3.4 Algebraic Geometry

### C3.8 Analytic Number Theory

https://courses-archive.maths.ox.ac.uk/node/49389

### 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

https://courses-archive.maths.ox.ac.uk/node/49542

### C6.1 Numerical Linear Algebra

https://courses-archive.maths.ox.ac.uk/node/49550

### C6.3 Approximation of Functions

https://courses-archive.maths.ox.ac.uk/node/49601

### C6.5 Theories of Deep Learning

https://courses-archive.maths.ox.ac.uk/node/49612

### C7.1 Theoretical Physics (C6)

### C7.5 General Relativity I

### C8.1 Stochastic Differential Equations

### C8.3 Combinatorics

https://courses-archive.maths.ox.ac.uk/node/49710

### CCD Dissertations on a Mathematical Topic

### COD Dissertations on the History of Mathematics

### C1.2 Goodel's Incompleteness Theorems

### C1.4 Axiomatic Set Theory

https://courses-archive.maths.ox.ac.uk/node/49234

### 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

https://courses-archive.maths.ox.ac.uk/node/49375

### C3.7 Elliptic Curves

https://courses-archive.maths.ox.ac.uk/node/49382

#### Course Overview: 
Elliptic curves give the simplest examples of many of the most interesting phenomena which can occur in algebraic curves; they have an incredibly rich structure and have been the testing ground for many developments in algebraic geometry whilst the theory is still full of deep unsolved conjectures, some of which are amongst the oldest unsolved problems in mathematics. The course will concentrate on arithmetic aspects of elliptic curves defined over the rationals, with the study of the group of rational points, and explicit determination of the rank, being the primary focus. Using elliptic curves over the rationals as an example, we will be able to introduce many of the basic tools for studying arithmetic properties of algebraic varieties.

#### Learning Outcomes: 
On completing the course, students should be able to understand and use properties of elliptic curves, such as the group law, the torsion group of rational points, and 2-isogenies between elliptic curves. They should be able to understand and apply the theory of fields with valuations, emphasising the p-adic numbers, and be able to prove and apply Hensel's Lemma in problem solving. They should be able to understand the proof of the Mordell-Weil Theorem for the case when an elliptic curve has a rational point of order 2, and compute ranks in such cases, for examples where all homogeneous spaces for descent-via-2-isogeny satisfy the Hasse principle. They should also be able to apply the elliptic curve method for the factorisation of integers.

#### Course Synopsis: 
Non-singular cubics and the group law; Weierstrass equations.  
Elliptic curves over finite fields; Hasse estimate (stated without proof).  
p-adic fields (basic definitions and properties).  
1-dimensional formal groups (basic definitions and properties).  
Curves over p-adic fields and reduction mod p.  
Computation of torsion groups over Q; the Nagell-Lutz theorem.  
2-isogenies on elliptic curves defined over Q, with a Q-rational point of order 2.  
Weak Mordell-Weil Theorem for elliptic curves defined over Q, with a Q-rational point of order 2.  
Height functions on Abelian groups and basic properties.  
Heights of points on elliptic curves defined over Q; statement (without proof) that this gives a height function on the Mordell-Weil group.  
Mordell-Weil Theorem for elliptic curves defined over Q, with a Q-rational point of order 2.  
Explicit computation of rank using descent via 2-isogeny.  
Public keys in cryptography; Pollard's (p-1) method and the elliptic curve method of factorisation.  

#### Reading List:  
J.W.S. Cassels, Lectures on Elliptic Curves, LMS Student Texts 24 (Cambridge University Press, 1991).   
N. Koblitz, A Course in Number Theory and Cryptography, Graduate Texts in Mathematics 114 (Springer, 1987).  
J.H. Silverman and J. Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics (Springer, 1992).  
J.H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 106 (Springer, 1986).  
#### Further Reading: 
A. Knapp, Elliptic Curves, Mathematical Notes 40 (Princeton University Press, 1992).  
G, Cornell, J.H. Silverman and G. Stevans (editors), Modular Forms and Fermat's Last Theorem (Springer, 1997).  
J.H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 151 (Springer, 1994). 

### 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

https://courses-archive.maths.ox.ac.uk/node/49460

### C5.6 Applied Complex Variables

### C5.9 Mathematical Mechanical Biology

https://courses-archive.maths.ox.ac.uk/node/49511

### C6.2 Continuous Optimisation

https://courses-archive.maths.ox.ac.uk/node/49577

### 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

https://courses-archive.maths.ox.ac.uk/node/50988

### C8.2 Stochastic Analysis and PDEs

### C8.4 Probabilistic Combinatorics

https://courses-archive.maths.ox.ac.uk/node/49719

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

https://courses-archive.maths.ox.ac.uk/node/49728

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

https://courses-archive.maths.ox.ac.uk/node/49733

### 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