Update oxford course list

1 files changed, 322 insertions, 81 deletions

diff --git a/md/research/oxford-math.md b/md/research/oxford-math.md
index 8359867..c2306c7 100644
--- a/md/research/oxford-math.md
+++ b/md/research/oxford-math.md
@@ -410,7 +410,8 @@ F. Daly, D.J. Hand, M.C. Jones, A.D. Lunn and K.J. McConway, Elements of Statist
 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
+A. C. Davison, Statistical Models (Cambridge)  
+
 ### A10: Fluids and Waves
 
 https://courses.maths.ox.ac.uk/node/50862
@@ -469,24 +470,159 @@ https://courses.maths.ox.ac.uk/node/50748
 
 https://courses.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.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.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.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.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
@@ -583,65 +719,117 @@ D. Eisenbud: Commutative Algebra with a view towards Algebraic Geometry, (Spring
 
 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).
+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.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.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.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: 
+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
@@ -700,41 +888,35 @@ https://courses.maths.ox.ac.uk/node/49027
 
 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.
+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
 
@@ -756,7 +938,35 @@ https://courses.maths.ox.ac.uk/node/49119
 
 https://courses.maths.ox.ac.uk/node/49135
 
-### 8.5 Graph Theory
+### 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
 
@@ -919,6 +1129,37 @@ https://courses.maths.ox.ac.uk/node/49375
 
 https://courses.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