Schedules (pdf), Lecture notes of Mathematics

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UNIVERSITY OF CAMBRIDGE

Faculty of Mathematics

SCHEDULES OF LECTURE COURSES

AND FORM OF EXAMINATIONS

FOR THE MATHEMATICAL TRIPOS 2021/

INTRODUCTION 1

THE MATHEMATICAL TRIPOS 2021–

CONTENTS

This booklet is the formal description of the content and structure of Parts IA, IB and II of the Mathematical Tripos.^1 In particular, it contains the schedules, or syllabus specifications, that define each course in the undergraduate Tripos, and it contains detailed information about the structure and marking of examinations, and the classification criteria. In addition, the booklet contains many useful pieces of advice and information for students regarding the Mathematical Tripos. It is updated every year to reflect changes approved by the Faculty Board.

Adjustment for Covid-

The sudden arrival of Covid-19 meant that the 2020 examinations differed significantly from the ar- rangements described in the 2019/20 Schedules booklet. While lecturing was online last academic year, it was nevertheless possible to hold ‘normal’ examinations, predominantly in-person, in June 2021. This edition of the Schedules reflects the Faculty’s working assumption that it should again be possible to hold ‘normal’ examinations in June 2022.

The academic content of the lecture courses for 2021/22 is described by the schedules below. Detailed arrangements for the time, location and delivery of lectures in the Mathematical Tripos, and updates on any changes, will be announced by email to students and Directors of Studies, and posted on the undergraduate course page www.maths.cam.ac.uk/undergrad/undergrad as soon as they are available.

SCHEDULES

Syllabus

The schedule for each lecture course is a list of topics that define the course. The schedule is agreed by the Faculty Board. Some schedules contain topics that are ‘starred’ (listed between asterisks); all the topics must be covered by the lecturer but examiners can only set questions on unstarred topics. The numbers which appear in brackets at the end of subsections or paragraphs in these schedules indicate the approximate number of lectures likely to be devoted to that subsection or paragraph. Lecturers decide upon the amount of time they think appropriate to spend on each topic, and also on the order in which they present topics. There is no requirement for this year’s lectures to match the previous year’s notes. Some topics in Part IA and Part IB courses have to be introduced in a certain order so as to tie in with other courses.

Recommended books

A list of books is given after each schedule. Books marked with † are particularly well suited to the course. Some of the books are out of print; these are retained on the list because they should be available in college libraries (as should all the books on the list) and may be found in second-hand bookshops. There may well be many other suitable books not listed; it is usually worth browsing college libraries.

In most cases, the contents of the book will not be exactly the same as the content of the schedule, and different styles suit different people. Hence you are advised to consult library copies in the first instance to decide which, if any, would be of benefit to you. Up-to-date prices, and the availability of hard- and soft-back versions, can most conveniently be checked online.

(^1) This booklet, full name Schedules of Lecture Courses and Form of Examinations for the Mathematical Tripos but often referred to simply as ‘The Schedules’, can be found online at www.maths.cam.ac.uk/undergrad/course/schedules.pdf

STUDY SKILLS

The Faculty produces a booklet Study Skills in Mathematics which is distributed to all first year students and can be obtained in pdf format from www.maths.cam.ac.uk/undergrad/studyskills. There is also a booklet, Supervision in Mathematics, that gives guidance to supervisors obtainable from www.maths.cam.ac.uk/facultyoffice/supervisorsguide/ which may also be of interest to students.

Aims and objectives

The aims of the Faculty for Parts IA, IB and II of the Mathematical Tripos are:

• to provide a challenging course in mathematics and its applications for a range of students that includes some of the best in the country;
• to provide a course that is suitable both for students aiming to pursue research and for students going into other careers;
• to provide an integrated system of teaching which can be tailored to the needs of individual students;
• to develop in students the capacity for learning and for clear logical thinking, and the ability to solve unseen problems;
• to continue to attract and select students of outstanding quality;
• to produce the high calibre graduates in mathematics sought by employers in universities, the professions and the public services.
• to provide an intellectually stimulating environment in which students have the opportunity to develop their skills and enthusiasms to their full potential;
• to maintain the position of Cambridge as a leading centre, nationally and internationally, for teaching and research in mathematics.

The objectives of Parts IA, IB and II of the Mathematical Tripos are as follows:

After completing Part IA, students should have:

• made the transition in learning style and pace from school mathematics to university mathematics;
• been introduced to basic concepts in higher mathematics and their applications, including (i) the notions of proof, rigour and axiomatic development, (ii) the generalisation of familiar mathematics to unfamiliar contexts, (iii) the application of mathematics to problems outside mathematics;
• laid the foundations, in terms of knowledge and understanding, of tools, facts and techniques, to proceed to Part IB.

After completing Part IB, students should have:

• covered material from a range of pure mathematics, statistics and operations research, applied mathematics, theoretical physics and computational mathematics, and studied some of this material in depth;
• acquired a sufficiently broad and deep mathematical knowledge and understanding to enable them both to make an informed choice of courses in Part II and also to study these courses. After completing Part II, students should have:
• developed the capacity for (i) solving both abstract and concrete unseen problems, (ii) present- ing a concise and logical argument, and (iii) (in most cases) using standard software to tackle mathematical problems;
• studied advanced material in the mathematical sciences, some of it in depth.

INTRODUCTION 3

Classification Criteria

As a result of each examination, each candidate is placed in one of the following categories: first class, upper second class (2.1), lower second class (2.2), third class, fail^2 or ‘other’. ‘Other’ here includes, for example, candidates who were ill for all or part of the examination. The examiners place the candidates into the different classes, with particular attention given to all candidates near each borderline. The primary classification criteria for each borderline, which are determined by the Faculty Board, are as follows:

First / upper second 30 α + 5β + m Upper second / lower second 15 α + 5β + m Lower second / third 15 α + 5β + m

Third/ fail

15 α + 5β + m in Part IB and Part II; 2 α + β together with m in Part IA.

Here, m denotes the number of marks and α and β denote the numbers of quality marks. Other factors besides marks and quality marks may be taken into account.

At the third/fail borderline, examiners may consider if most of the marks have been obtained on only one or two courses.

The Faculty Board recommends that no distinction should be made between marks obtained on the Computational Projects courses in Parts IB and II and marks obtained on the written papers. The Faculty Board recommends approximate percentages of candidates for each class: 30% firsts; 70– 75% upper seconds and above; 90–95% lower seconds and above; and 5–10% thirds and below. (These percentages exclude candidates who did not sit all the written papers.)

The Faculty Board expects that the classification criteria described above should result in classes that can be broadly characterized as follows (after allowing for the possibility that in Parts IB and II stronger performance on the Computational Projects may compensate for weaker performance on the written papers or vice versa):

First Class Candidates placed in the first class will have demonstrated a good command and secure understanding of examinable material. They will have presented standard arguments accurately, showed skill in applying their knowledge, and generally will have produced substantially correct solutions to a significant number of more challenging questions.

Upper Second Class

Candidates placed in the upper second class will have demonstrated good knowledge and understanding of examinable material. They will have presented standard arguments accurately and will have shown some ability to apply their knowledge to solve problems. A fair number of their answers to both straightforward and more challenging questions will have been substantially correct.

Lower Second Class

Candidates placed in the lower second class will have demonstrated knowledge but sometimes imperfect understanding of examinable material. They will have been aware of relevant mathematical issues, but their presentation of standard arguments will sometimes have been fragmentary or imperfect. They will have produced substantially correct solutions to some straightforward questions, but will have had limited success at tackling more challenging problems.

Third Class

Candidates placed in the third class will have demonstrated some knowledge of the examinable material. They will have made reasonable attempts at a small number of questions, but will not have shown the skills needed to complete many of them.

(^2) Very few candidates are placed in the fail category, but anyone who finds themselves in this position should contact their Tutor or Director of Studies at once. There are no ’re-sits’ and, in order to continue to study at Cambridge, an application (based, for example, on medical evidence) must be made to the University.

Examination Data Retention Policy

To meet the University’s obligations under the data protection legislation, the Faculty deals with data relating to individuals and their examination marks as follows:

• All marks for individual questions and computational projects are released routinely to individual candidates and their Colleges after the examinations. The final examination mark book is kept indefinitely by the Undergraduate Office.
• Scripts and Computational Projects submissions are kept, in line with the University policy, for six months following the examinations (in case of appeals). Scripts are then destroyed; and Computa- tional Projects are anonymised and stored in a form that allows comparison (using anti-plagiarism software) with current projects.
• Neither the GDPR nor the Freedom of Information Act entitle candidates to have access to their scripts. Data appearing on individual examination scripts is technically available on application to the University Information Compliance Officer. However, such data consists only of a copy of the examiner’s ticks, crosses, underlines, etc., and the mark subtotals and totals.

Examiners’ reports

For each part of the Tripos, the examiners (internal and external) write a joint report. In addition, the external examiners each submit a report addressed to the Vice-Chancellor. The reports of the external examiners are scrutinised by the Education Committee of the University’s General Board. All the reports, the examination statistics (number of attempts per question, etc), student feedback on the examinations and lecture courses (via the end of year questionnaire and paper questionnaires), and other relevant material are considered by the Faculty Teaching Committee at the start of the Michaelmas term. The Teaching Committee includes two student representatives, and may include other students (for example, previous members of the Teaching Committee and student representatives of the Faculty Board). The Teaching Committee compiles a lengthy report on examinations including various recommenda- tions for the Faculty Board to consider at its second meeting in the Michaelmas term. This report also forms the basis of the Faculty Board’s response to the reports of the external examiners. Pre- vious Teaching Committee reports and recent examiners’ comments on questions can be found at http://www.maths.cam.ac.uk/facultyboard/teachingcommittee.

Transcripts

University guidelines on examinations require the Faculty to produce, for use in official transcripts, a UMS percentage mark and a rank for each candidate. These are issued via CamSIS, and are calculated from the distribution of ‘merit marks’ as follows. The merit mark M is defined in terms of the numbers of marks, alphas and betas by

M =

30 α + 5β + m − 120 for candidates in the first class, or in the upper second class with α ≥ 8 , 15 α + 5β + m otherwise

The UMS percentage mark is obtained by piecewise linear scaling of the merit marks within each class. The 1/2.1, 2.1/2.2, 2.2/3 and 3/fail boundaries are mapped to 69.5%, 59.5%, 49.5% and 39.5% respectively and the merit mark of the 5th ranked candidate is mapped to 95%. If, after linear mapping of the first class, the percentage mark of any candidate is greater than 100, it is reduced to 100%. The percentage of each candidate is then rounded appropriately to integer values. The rank of the candidate is determined by merit-mark order within each class.

INTRODUCTION 4

MISCELLANEOUS MATTERS

Numbers of supervisions, example sheets and workload

The primary responsibility for supervisions rests with colleges, and Directors of Studies are expected to make appropriate arrangements for their students.

Lecturers provide example sheets for each course, which supervisors are generally recommended to use. According to Faculty Board guidelines, the number of example sheets for 24-lecture, 16-lecture and 12-lecture courses should be 4, 3 and 2, respectively, and the content and length of each example sheet should be suitable for discussion (with a typical pair of students) in an hour-long supervision. For a student studying the equivalent of 4 24-lecture courses in each of Michaelmas and Lent Terms, as in Part IA, the 32 example sheets would then be associated with an average of about two supervisions per week, and with revision supervisions in the Easter Term, a norm of about 40 supervisions over the year. Since supervisions on a given course typically begin sometime after the first two weeks of lectures, the fourth supervision of a 24-lecture course is often given at the start of the next term to spread the workload and allow students to catch up.

As described later in this booklet, the structure of Parts IB and II allows considerable flexibility over the selection and number of courses to be studied, which students can use, in consultation with their Directors of Studies, to adjust their workload as appropriate to their interests and to their previous experience in Part IA. Dependent on their course selection, and the corresponding number of example sheets, most students have 35–45 supervisions in Part IB and Part II, with the average across all students being close to 40 supervisions per year. It is impossible to say how long an example sheet ‘should’ take. If a student is concerned that they are regularly studying for significantly more than 48 hours per week in total then they should seek advice from their Director of Studies.

Past papers

Past Tripos papers for the last 8 or more years can be found on the Faculty web site http://www.maths.cam.ac.uk/undergrad/pastpapers/. Some examples of solutions and mark schemes for the 2011 Part IA examination can be found with an explanatory comment at http://www.maths.cam.ac.uk/examples-solutions-part-ia. Otherwise, solutions and mark schemes are not available except in rough draft form for supervisors.

Student support: colleges and the wider university

An extensive support network is available through colleges and the wider university, to help students get the most from their time in Cambridge and to assist with any issues of a more personal nature that may arise. The first points of contact for any student should be their College Director of Studies (for academic matters) and their College Tutor (for both academic and pastoral concerns). They will be able to offer help and advice directly, or to guide students to others with appropriate expertise, either within their College or elsewhere. While pastoral matters do not usually fall within the remit of the Mathematics Faculty, we strongly encourage our students to seek help and get support if they are experiencing any difficulties, and a summary of some relevant resources can be found by following the Student Support links on the undergraduate course webpages https://www.maths.cam.ac.uk/undergrad/undergrad

Faculty committees and student representatives

The Faculty Board is responsible for setting policies governing arrangements for lecturing and examining in the Mathematical Tripos (https://www.maths.cam.ac.uk/internal/faculty/facultyboard) such as, for example, those described in these Schedules. It meets formally, and also considers other matters. There are two committees that deal exclusively with matters relating to the undergraduate Tripos: the Teaching Committee (http://www.maths.cam.ac.uk/facultyboard/teachingcommittee/) and the Curriculum Committee (http://www.maths.cam.ac.uk/facultyboard/curriculumcommittee/). The role of the Teaching Committee is mainly to monitor feedback (questionnaires, examiners’ reports, etc.) and make recommendations to the Faculty Board on the basis of this feedback. It also formulates policy recommendations at the request of the Faculty Board. The Curriculum Committee is responsible for recommending (to the Faculty Board) changes to the undergraduate Tripos and to the schedules for individual lecture courses. Student representatives have a very important role to play on each of these committees: to advise on the student point of view and to collect opinion and liaise with the wider student body. There are two student representatives on both the Teaching and Curriculum Committees (others may be co-opted). There are also three student representatives on the Faculty Board, two undergraduate and one graduate, elected each year in November. Further details regarding the student representatives, their roles and contributions, can be found at https://www.maths.cam.ac.uk/undergrad/student-representation. They can be contacted by email: [email protected]

Feedback

Constructive feedback of all sorts and from all sources is welcomed by everyone concerned in providing courses for the Mathematical Tripos. There are many different feedback routes.

• Each lecturer hands out a paper questionnaire towards the end of the course.
• There are brief web-based questionnaires after roughly six lectures of each course.
• Students are sent a combined online questionnaire at the end of each year.
• Students (or supervisors) can e-mail [email protected] at any time. Such e-mails are received by the Director of Undergraduate Education and the Chair of the Teaching Committee, who will either deal with your comment, or pass your e-mail (after stripping out any clue to your identity) to the relevant person (a lecturer, for example). Students will receive a rapid response.
• If a student wishes to be entirely anonymous and does not want any response, the web-based comment form at www.maths.cam.ac.uk/feedback.html can be used. (Anonymity also means we can’t ask you for clarifying information to help us deal with the comment.)
• Feedback on college-provided teaching (supervisions, classes) can be given to Directors of Studies or Tutors at any time.

The questionnaires are particularly important in shaping the future of the Tripos and the Faculty Board urges all students to respond.

Formal complaints

The formal complaints procedure to be followed within the University can be found at http://www.studentcomplaints.admin.cam.ac.uk/student-complaints. The Responsible Officer in Step 1 of this procedure for the Faculty of Mathematics is the Chair of the Faculty Board — see http://www.maths.cam.ac.uk/facultyboard for the name of the current Chair.

PART IA 6

GROUPS 24 lectures, Michaelmas Term

Examples of groups Axioms for groups. Examples from geometry: symmetry groups of regular polygons, cube, tetrahedron. Permutations on a set; the symmetric group. Subgroups and homomorphisms. Symmetry groups as subgroups of general permutation groups. The M¨obius group; cross-ratios, preservation of circles, the point at infinity. Conjugation. Fixed points of M¨obius maps and iteration. [4]

Lagrange’s theorem Cosets. Lagrange’s theorem. Groups of small order (up to order 8). Quaternions. Fermat-Euler theorem from the group-theoretic point of view. [5]

Group actions Group actions; orbits and stabilizers. Orbit-stabilizer theorem. Cayley’s theorem (every group is isomorphic to a subgroup of a permutation group). Conjugacy classes. Cauchy’s theorem. [4]

Quotient groups Normal subgroups, quotient groups and the isomorphism theorem. [4]

Matrix groups The general and special linear groups; relation with the M¨obius group. The orthogonal and special orthogonal groups. Proof (in R^3 ) that every element of the orthogonal group is the product of reflections and every rotation in R^3 has an axis. Basis change as an example of conjugation. [3]

Permutations Permutations, cycles and transpositions. The sign of a permutation. Conjugacy in Sn and in An. Simple groups; simplicity of A 5. [4]

Appropriate books M.A. Armstrong Groups and Symmetry. Springer–Verlag 1988 † (^) Alan F Beardon Algebra and Geometry. CUP 2005

R.P. Burn Groups, a Path to Geometry. Cambridge University Press 1987 J.A. Green Sets and Groups: a first course in Algebra. Chapman and Hall/CRC 1988 W. Lederman Introduction to Group Theory. Longman 1976 Nathan Carter Visual Group Theory. Mathematical Association of America Textbooks

VECTORS AND MATRICES 24 lectures, Michaelmas Term

Complex numbers Review of complex numbers, including complex conjugate, inverse, modulus, argument and Argand diagram. Informal treatment of complex logarithm, n-th roots and complex powers. de Moivre’s theorem. [2]

Vectors Review of elementary algebra of vectors in R^3 , including scalar product. Brief discussion of vectors in Rn^ and Cn; scalar product and the Cauchy–Schwarz inequality. Concepts of linear span, linear independence, subspaces, basis and dimension. Suffix notation: including summation convention, δij and ijk. Vector product and triple product: definition and geometrical interpretation. Solution of linear vector equations. Applications of vectors to geometry, including equations of lines, planes and spheres. [5]

Matrices Elementary algebra of 3 × 3 matrices, including determinants. Extension to n × n complex matrices. Trace, determinant, non-singular matrices and inverses. Matrices as linear transformations; examples of geometrical actions including rotations, reflections, dilations, shears; kernel and image, rank–nullity theorem (statement only). [4] Simultaneous linear equations: matrix formulation; existence and uniqueness of solutions, geometric interpretation; Gaussian elimination. [3] Symmetric, anti-symmetric, orthogonal, hermitian and unitary matrices. Decomposition of a general matrix into isotropic, symmetric trace-free and antisymmetric parts. [1]

Eigenvalues and Eigenvectors Eigenvalues and eigenvectors; geometric significance. [2] Proof that eigenvalues of hermitian matrix are real, and that distinct eigenvalues give an orthogonal basis of eigenvectors. The effect of a general change of basis (similarity transformations). Diagonalization of general matrices: sufficient conditions; examples of matrices that cannot be diagonalized. Canonical forms for 2 × 2 matrices. [5] Discussion of quadratic forms, including change of basis. Classification of conics, cartesian and polar forms. [1] Rotation matrices and Lorentz transformations as transformation groups. [1]

Appropriate books Alan F Beardon Algebra and Geometry. CUP 2005 Gilbert Strang Linear Algebra and Its Applications. Thomson Brooks/Cole, 2006 Richard Kaye and Robert Wilson Linear Algebra. Oxford science publications, 1998 D.E. Bourne and P.C. Kendall Vector Analysis and Cartesian Tensors. Nelson Thornes 1992 E. Sernesi Linear Algebra: A Geometric Approach. CRC Press 1993 James J. Callahan The Geometry of Spacetime: An Introduction to Special and General Relativity. Springer 2000

PART IA 7

NUMBERS AND SETS 24 lectures, Michaelmas Term

[Note that this course is omitted from Option (b) of Part IA.]

Introduction to number systems and logic Overview of the natural numbers, integers, real numbers, rational and irrational numbers, algebraic and transcendental numbers. Brief discussion of complex numbers; statement of the Fundamental Theorem of Algebra.

Ideas of axiomatic systems and proof within mathematics; the need for proof; the role of counter- examples in mathematics. Elementary logic; implication and negation; examples of negation of com- pound statements. Proof by contradiction. [2]

Sets, relations and functions Union, intersection and equality of sets. Indicator (characteristic) functions; their use in establishing set identities. Functions; injections, surjections and bijections. Relations, and equivalence relations. Counting the combinations or permutations of a set. The Inclusion-Exclusion Principle. [4]

The integers The natural numbers: mathematical induction and the well-ordering principle. Examples, including the Binomial Theorem. [2]

Elementary number theory Prime numbers: existence and uniqueness of prime factorisation into primes; highest common factors and least common multiples. Euclid’s proof of the infinity of primes. Euclid’s algorithm. Solution in integers of ax+by = c.

Modular arithmetic (congruences). Units modulo n. Chinese Remainder Theorem. Wilson’s Theorem; the Fermat-Euler Theorem. Public key cryptography and the RSA algorithm. [8]

The real numbers Least upper bounds; simple examples. Least upper bound axiom. Sequences and series; convergence of bounded monotonic sequences. Irrationality of

2 and e. Decimal expansions. Construction of a transcendental number. [4]

Countability and uncountability Definitions of finite, infinite, countable and uncountable sets. A countable union of countable sets is countable. Uncountability of R. Non-existence of a bijection from a set to its power set. Indirect proof of existence of transcendental numbers. [4]

Appropriate books

R.B.J.T. Allenby Numbers and Proofs. Butterworth-Heinemann 1997 R.P. Burn Numbers and Functions: steps into analysis. Cambridge University Press 2000 H. Davenport The Higher Arithmetic. Cambridge University Press 1999 A.G. Hamilton Numbers, sets and axioms: the apparatus of mathematics. Cambridge University Press 1983 C. Schumacher Chapter Zero: Fundamental Notions of Abstract Mathematics. Addison-Wesley 2001 I. Stewart and D. Tall The Foundations of Mathematics. Oxford University Press 1977

DIFFERENTIAL EQUATIONS 24 lectures, Michaelmas Term

Basic calculus Informal treatment of differentiation as a limit, the chain rule, Leibnitz’s rule, Taylor series, informal treatment of O and o notation and l’Hˆopital’s rule; integration as an area, fundamental theorem of calculus, integration by substitution and parts. [3] Informal treatment of partial derivatives, geometrical interpretation, statement (only) of symmetry of mixed partial derivatives, chain rule, implicit differentiation. Informal treatment of differentials, including exact differentials. Differentiation of an integral with respect to a parameter. [2]

First-order linear differential equations Equations with constant coefficients: exponential growth, comparison with discrete equations, series solution; modelling examples including radioactive decay. Equations with non-constant coefficients: solution by integrating factor. [2]

Nonlinear first-order equations Separable equations. Exact equations. Sketching solution trajectories. Equilibrium solutions, stability by perturbation; examples, including logistic equation and chemical kinetics. Discrete equations: equi- librium solutions, stability; examples including the logistic map. [4]

Higher-order linear differential equations Complementary function and particular integral, linear independence, Wronskian (for second-order equations), Abel’s theorem. Equations with constant coefficients and examples including radioactive sequences, comparison in simple cases with difference equations, reduction of order, resonance, tran- sients, damping. Homogeneous equations. Response to step and impulse function inputs; introduction to the notions of the Heaviside step-function and the Dirac delta-function. Series solutions including statement only of the need for the logarithmic solution. [8]

Multivariate functions: applications Directional derivatives and the gradient vector. Statement of Taylor series for functions on Rn. Local extrema of real functions, classification using the Hessian matrix. Coupled first order systems: equiv- alence to single higher order equations; solution by matrix methods. Non-degenerate phase portraits local to equilibrium points; stability. Simple examples of first- and second-order partial differential equations, solution of the wave equation in the form f (x + ct) + g(x − ct). [5]

Appropriate books J. Robinson An introduction to Differential Equations. Cambridge University Press, 2004 W.E. Boyce and R.C. DiPrima Elementary Differential Equations and Boundary-Value Problems (and associated web site: google Boyce DiPrima). Wiley, 2004 G.F.Simmons Differential Equations (with applications and historical notes). McGraw-Hill 1991 D.G. Zill and M.R. Cullen Differential Equations with Boundary Value Problems. Brooks/Cole 2001

PART IA 9

VECTOR CALCULUS 24 lectures, Lent Term

Curves in R^3 Parameterised curves and arc length, tangents and normals to curves in R^3 ; curvature and torsion. [1]

Integration in R^2 and R^3 Line integrals. Surface and volume integrals: definitions, examples using Cartesian, cylindrical and spherical coordinates; change of variables. [4]

Vector operators Directional derivatives. The gradient of a real-valued function: definition; interpretation as normal to level surfaces; examples including the use of cylindrical, spherical ∗and general orthogonal curvilinear∗ coordinates. Divergence, curl and ∇^2 in Cartesian coordinates, examples; formulae for these operators (statement only) in cylindrical, spherical ∗and general orthogonal curvilinear∗^ coordinates. Solenoidal fields, irro- tational fields and conservative fields; scalar potentials. Vector derivative identities. [5]

Integration theorems Divergence theorem, Green’s theorem, Stokes’s theorem, Green’s second theorem: statements; infor- mal proofs; examples; application to fluid dynamics, and to electromagnetism including statement of Maxwell’s equations. [5]

Laplace’s equation Laplace’s equation in R^2 and R^3 : uniqueness theorem and maximum principle. Solution of Poisson’s equation by Gauss’s method (for spherical and cylindrical symmetry) and as an integral. [4]

Cartesian tensors in R^3 Tensor transformation laws, addition, multiplication, contraction, with emphasis on tensors of second rank. Isotropic second and third rank tensors. Symmetric and antisymmetric tensors. Revision of principal axes and diagonalization. Quotient theorem. Examples including inertia and conductivity. [5]

Appropriate books H. Anton Calculus. Wiley Student Edition 2000 T.M. Apostol Calculus. Wiley Student Edition 1975 M.L. Boas Mathematical Methods in the Physical Sciences. Wiley 1983 † (^) D.E. Bourne and P.C. Kendall Vector Analysis and Cartesian Tensors. 3rd edition, Nelson Thornes

1999 E. Kreyszig Advanced Engineering Mathematics. Wiley International Edition 1999 J.E. Marsden and A.J.Tromba Vector Calculus. Freeman 1996 P.C. Matthews Vector Calculus. SUMS (Springer Undergraduate Mathematics Series) 1998 † (^) K. F. Riley, M.P. Hobson, and S.J. Bence Mathematical Methods for Physics and Engineering. Cam-

bridge University Press 2002 H.M. Schey Div, grad, curl and all that: an informal text on vector calculus. Norton 1996 M.R. Spiegel Schaum’s outline of Vector Analysis. McGraw Hill 1974

DYNAMICS AND RELATIVITY 24 lectures, Lent Term [Note that this course is omitted from Option (b) of Part IA.] Familarity with the topics covered in the non-examinable Mechanics course is assumed.

Basic concepts Space and time, frames of reference, Galilean transformations. Newton’s laws. Dimensional analysis. Examples of forces, including gravity, friction and Lorentz. [4] Newtonian dynamics of a single particle Equation of motion in Cartesian and plane polar coordinates. Work, conservative forces and potential energy, motion and the shape of the potential energy function; stable equilibria and small oscillations; effect of damping. Angular velocity, angular momentum, torque. Orbits: the u(θ) equation; escape velocity; Kepler’s laws; stability of orbits; motion in a repulsive potential (Rutherford scattering). Rotating frames: centrifugal and Coriolis forces. Brief discussion of Foucault pendulum. [8]

Newtonian dynamics of systems of particles Momentum, angular momentum, energy. Motion relative to the centre of mass; the two body problem. Variable mass problems; the rocket equation. [2]

Rigid bodies Moments of inertia, angular momentum and energy of a rigid body. Parallel axis theorem. Simple examples of motion involving both rotation and translation (e.g. rolling). [3]

Special relativity The principle of relativity. Relativity and simultaneity. The invariant interval. Lorentz transformations in (1 + 1)-dimensional spacetime. Time dilation and length contraction. The Minkowski metric for (1 + 1)-dimensional spacetime. Lorentz transformations in (3 + 1) dimensions. 4–vectors and Lorentz invariants. Proper time. 4– velocity and 4–momentum. Conservation of 4–momentum in particle decay. Collisions. The Newtonian limit. [7]

Appropriate books † (^) D. Gregory Classical Mechanics. Cambridge University Press 2006 G.F.R. Ellis and R.M. Williams Flat and Curved Space-times. Oxford University Press 2000 A.P. French and M.G. Ebison Introduction to Classical Mechanics. Kluwer 1986 T.W.B. Kibble and F.H. Berkshire Introduction to Classical Mechanics. Kluwer 1986 M.A. Lunn A First Course in Mechanics. Oxford University Press 1991 P.J. O’Donnell Essential Dynamics and Relativity. CRC Press 2015 † (^) W. Rindler Introduction to Special Relativity. Oxford University Press 1991 E.F. Taylor and J.A. Wheeler Spacetime Physics: introduction to special relativity. Freeman 1992

PART IA 10

COMPUTATIONAL PROJECTS 8 lectures, Easter Term of Part IA

The Computational Projects course is examined in Part IB. However introductory practical sessions are offered at the end of Lent Full Term and the beginning of Easter Full Term of the Part IA year (students are advised by email how to register for a session), and lectures are given in the Easter Full Term of the Part IA year. The lectures cover an introduction to algorithms and aspects of the MATLAB programming language. The projects that need to be completed for credit are published by the Faculty in a manual usually by the end of July at the end of the Part IA year. The manual contains details of the projects and information about course administration. The manual is available on the Faculty website at http://www.maths.cam.ac.uk/undergrad/catam/. Full credit may obtained from the submission of the two core projects and a further two additional projects. Once the manual is available, these projects may be undertaken at any time up to the submission deadlines, which are near the start of the Full Lent Term in the IB year for the two core projects, and near the start of the Full Easter Term in the IB year for the two additional projects. A list of suitable books can be found in the manual.

MECHANICS (non-examinable) 10 lectures, Michaelmas Term This course is intended for students who have taken only a limited amount of Mechanics at A-level (or the equivalent). The material is prerequisite for Dynamics and Relativity in the Lent Term.

Lecture 1 Brief introduction

Lecture 2: Kinematics of a single particle Position, velocity, speed, acceleration. Constant acceleration in one-dimension. Projectile motion in two-dimensions.

Lecture 3: Equilibrium of a single particle The vector nature of forces, addition of forces, examples including gravity, tension in a string, normal reaction (Newton’s third law), friction. Conditions for equilibrium.

Lecture 4: Equilibrium of a rigid body Resultant of several forces, couple, moment of a force. Conditions for equilibrium.

Lecture 5: Dynamics of particles Newton’s second law. Examples of pulleys, motion on an inclined plane.

Lecture 6: Dynamics of particles Further examples, including motion of a projectile with air-resistance.

Lecture 7: Energy Definition of energy and work. Kinetic energy, potential energy of a particle in a uniform gravitational field. Conservation of energy. Lecture 8: Momentum Definition of momentum (as a vector), conservation of momentum, collisions, coefficient of restitution, impulse.

Lecture 9: Springs, strings and SHM Force exerted by elastic springs and strings (Hooke’s law). Oscillations of a particle attached to a spring, and of a particle hanging on a string. Simple harmonic motion of a particle for small displacement from equilibrium.

Lecture 10: Motion in a circle Derivation of the central acceleration of a particle constrained to move on a circle. Simple pendulum; motion of a particle sliding on a cylinder.

Appropriate books Peter J O’Donnell Essential Dynamics and Relativity. CRC Press, 2014 J. Hebborn and J. Littlewood Mechanics 1, Mechanics 2 and Mechanics 3 (Edexel). Heinemann, 2000 Anything similar to the above, for the other A-level examination boards

PART IB 12

Approximate class boundaries

The following tables, based on information supplied by the examiners, show approximate borderlines in recent years. For convenience, we define M 1 and M 2 by

M 1 = 30α + 5β + m − 120 , M 2 = 15α + 5β + m.

M 1 is related to the primary classification criterion for the first class and M 2 is related to the primary classification criterion for the upper and lower second and third classes. The second column of each table shows a sufficient criterion for each class (in terms of M 1 for the first class and M 2 for the other classes). The third and fourth columns show M 1 (for the first class) or M 2 (for the other classes), raw mark, number of alphas and number of betas of two representative candidates placed just above the borderline.

The sufficient condition for each class is not prescriptive: it is just intended to be helpful for interpreting the data. Each candidate near a borderline is scrutinised individually. The data given below are relevant to one year only; borderlines may go up or down in future years. (Both 2020 and 2021 were not typical years.)

Part IB 2018 Class Sufficient condition Borderline candidates 1 M 1 > 721 722/452,11,12 723/428,13, 5 2.1 M 2 > 505 506/386, 5, 9 508/373, 6, 9 2.2 M 2 > 373 374/239, 7, 6 379/274, 4, 9 3 M 2 > 200 201/166, 1, 4 262/217, 2, 3

Part IB 2019 Class Sufficient condition Borderline candidates 1 M 1 > 753 754/449,13, 7 761/466,12, 2.1 M 2 > 492 493/358, 6, 9 497/367, 4, 2.2 M 2 > 330 336/286, 0,10 337/262, 3, 6 3 M 2 > 200 219/189, 0, 6 259/204, 1, 8

Part II dependencies

The relationships between Part IB courses and Part II courses are shown in the following tables. A blank in the table means that the material in the Part IB course is not directly relevant to the Part II course. The terminology is as follows:

Essential: (E) a good understanding of the methods and results of the Part IB course is essential; Desirable: (D) knowledge of some of the results of the Part IB course is required; Background: (B) some knowledge of the Part IB course would provide a useful background.

Linear AlgebraGroups, Rings and ModulesAnalysis and TopologyComplex AnalysisComplex MethodsGeometryVariational PrinciplesMethodsQuantum MechanicsElectromagnetismFluid DynamicsNumerical AnalysisStatisticsOptimisationMarkov Chains Number Theory Topics in Analysis B Coding and Cryptography D E Automata and Form. Lang. Statistical Modelling E Mathematical Biology Further Complex Methods E Classical Dynamics E Cosmology Quantum Inf. and Comp. D Logic and Set Theory Graph Theory Galois Theory D E Representation Theory E E Number Fields E D Algebraic Topology E Linear Analysis E E Analysis of Functions E E Riemann Surfaces D E Algebraic Geometry E Differential Geometry E D Prob. and Measure E Applied Prob. E Princ. of Stats E Stochastic FM’s D D D Maths. Machine Learning D D Asymptotic Methods E D Dynamical Systems Integrable Systems D E Principles of QM D E Applications of QM B E Statistical Physics E Electrodynamics D E General Relativity D D Fluid Dynamics II E E Waves E D Numerical Analysis D D D E

PART IB 13

LINEAR ALGEBRA 24 lectures, Michaelmas Term

Definition of a vector space (over R or C), subspaces, the space spanned by a subset. Linear indepen- dence, bases, dimension. Direct sums and complementary subspaces. Quotient spaces. [3]

Linear maps, isomorphisms. Relation between rank and nullity. The space of linear maps from U to V , representation by matrices. Change of basis. Row rank and column rank. [4]

Determinant and trace of a square matrix. Determinant of a product of two matrices and of the inverse matrix. Determinant of an endomorphism. The adjugate matrix. [3]

Eigenvalues and eigenvectors. Diagonal and triangular forms. Characteristic and minimal polynomials. Cayley–Hamilton Theorem over C. Algebraic and geometric multiplicity of eigenvalues. Statement and illustration of Jordan normal form. [4]

Dual of a finite-dimensional vector space, dual bases and maps. Matrix representation, rank and determinant of dual map [2]

Bilinear forms. Matrix representation, change of basis. Symmetric forms and their link with quadratic forms. Diagonalisation of quadratic forms. Law of inertia, classification by rank and signature. Complex Hermitian forms. [4]

Inner product spaces, orthonormal sets, orthogonal projection, V = W ⊕ W ⊥. Gram–Schmidt or- thogonalisation. Adjoints. Diagonalisation of Hermitian matrices. Orthogonality of eigenvectors and properties of eigenvalues. [4]

Appropriate books

C.W. Curtis Linear Algebra: an introductory approach. Springer 1984 P.R. Halmos Finite-dimensional vector spaces. Springer 1974 K. Hoffman and R. Kunze Linear Algebra. Prentice-Hall 1971

GROUPS, RINGS AND MODULES 24 lectures, Lent Term

Groups Basic concepts of group theory recalled from Part IA Groups. Normal subgroups, quotient groups and isomorphism theorems. Permutation groups. Groups acting on sets, permutation representations. Conjugacy classes, centralizers and normalizers. The centre of a group. Elementary properties of finite p-groups. Examples of finite linear groups and groups arising from geometry. Simplicity of An. Sylow subgroups and Sylow theorems. Applications, groups of small order. [8]

Rings Definition and examples of rings (commutative, with 1). Ideals, homomorphisms, quotient rings, iso- morphism theorems. Prime and maximal ideals. Fields. The characteristic of a field. Field of fractions of an integral domain. Factorization in rings; units, primes and irreducibles. Unique factorization in principal ideal domains, and in polynomial rings. Gauss’ Lemma and Eisenstein’s irreducibility criterion. Rings Z[α] of algebraic integers as subsets of C and quotients of Z[x]. Examples of Euclidean domains and uniqueness and non-uniqueness of factorization. Factorization in the ring of Gaussian integers; representation of integers as sums of two squares. Ideals in polynomial rings. Hilbert basis theorem. [10]

Modules Definitions, examples of vector spaces, abelian groups and vector spaces with an endomorphism. Sub- modules, homomorphisms, quotient modules and direct sums. Equivalence of matrices, canonical form. Structure of finitely generated modules over Euclidean domains, applications to abelian groups and Jordan normal form. [6]

Appropriate books P.M.Cohn Classic Algebra. Wiley, 2000 P.J. Cameron Introduction to Algebra. OUP J.B. Fraleigh A First Course in Abstract Algebra. Addison Wesley, 2003 B. Hartley and T.O. Hawkes Rings, Modules and Linear Algebra: a further course in algebra. Chapman and Hall, 1970 I. Herstein Topics in Algebra. John Wiley and Sons, 1975 P.M. Neumann, G.A. Stoy and E.C. Thomson Groups and Geometry. OUP 1994 M. Artin Algebra. Prentice Hall, 1991

PART IB 15

COMPLEX ANALYSIS 16 lectures, Lent Term

Analytic functions Complex differentiation and the Cauchy-Riemann equations. Examples. Conformal mappings. Informal discussion of branch points, examples of log z and zc. [3]

Contour integration and Cauchy’s theorem Contour integration (for piecewise continuously differentiable curves). Statement and proof of Cauchy’s theorem for star domains. Cauchy’s integral formula, maximum modulus theorem, Liouville’s theorem, fundamental theorem of algebra. Morera’s theorem. [5]

Expansions and singularities Uniform convergence of analytic functions; local uniform convergence. Differentiability of a power series. Taylor and Laurent expansions. Principle of isolated zeros. Residue at an isolated singularity. Classification of isolated singularities. [4]

The residue theorem Winding numbers. Residue theorem. Jordan’s lemma. Evaluation of definite integrals by contour integration. Rouch´e’s theorem, principle of the argument. Open mapping theorem. [4]

Appropriate books L.V. Ahlfors Complex Analysis. McGraw–Hill 1978 † (^) A.F. Beardon Complex Analysis. Wiley

D.J.H. Garling A Course in Mathematical Analysis (Vol 3). Cambridge University Press 2014 † (^) H.A. Priestley Introduction to Complex Analysis. Oxford University Press 2003

I. Stewart and D. Tall Complex Analysis. Cambridge University Press 1983

COMPLEX METHODS 16 lectures, Lent Term

Analytic functions Definition of an analytic function. Cauchy-Riemann equations. Analytic functions as conformal map- pings; examples. Application to the solutions of Laplace’s equation in various domains. Discussion of log z and za. [6]

Contour integration and Cauchy’s Theorem [Proofs of theorems in this section will not be examined in this course.] Contours, contour integrals. Cauchy’s theorem and Cauchy’s integral formula. Taylor and Laurent series. Zeros, poles and essential singularities. [4]

Residue calculus Residue theorem, calculus of residues. Jordan’s lemma. Evaluation of definite integrals by contour integration. [3]

Fourier and Laplace transforms Laplace transform: definition and basic properties; inversion theorem (proof not required); convolution theorem. Examples of inversion of Fourier and Laplace transforms by contour integration. Applications to differential equations. [3]

Appropriate books M.J. Ablowitz and A.S. Fokas Complex Variables: Introduction and Applications. CUP 2003 G.B. Arfken, H.J. Weber & F.E. Harris Mathematical Methods for Physicists. Elsevier 2013 G.J.O. Jameson A First Course in Complex Functions. Chapman and Hall 1970 T. Needham Visual Complex Analysis. Clarendon 1998 † (^) H.A. Priestley Introduction to Complex Analysis. Clarendon 1990 † (^) K. F. Riley, M. P. Hobson, and S.J. Bence Mathematical Methods for Physics and Engineering: a Comprehensive Guide. Cambridge University Press 2002

PART IB 16

VARIATIONAL PRINCIPLES 12 lectures, Easter Term

Stationary points for functions on Rn. Necessary and sufficient conditions for minima and maxima. Importance of convexity. Variational problems with constraints; method of Lagrange multipliers. The Legendre Transform; need for convexity to ensure invertibility; illustrations from thermodynamics. [4]

The idea of a functional and a functional derivative. First variation for functionals, Euler-Lagrange equations, for both ordinary and partial differential equations. Use of Lagrange multipliers and multi- plier functions. [3]

Fermat’s principle; geodesics; least action principles, Lagrange’s and Hamilton’s equations for particles and fields. Noether theorems and first integrals, including two forms of Noether’s theorem for ordinary differential equations (energy and momentum, for example). Interpretation in terms of conservation laws. [3]

Second variation for functionals; associated eigenvalue problem. [2]

Appropriate books

D.S. Lemons Perfect Form. Princeton Unversity Press 1997 C. Lanczos The Variational Principles of Mechanics. Dover 1986 R. Weinstock Calculus of Variations with applications to physics and engineering. Dover 1974 I.M. Gelfand and S.V. Fomin Calculus of Variations. Dover 2000 W. Yourgrau and S. Mandelstam Variational Principles in Dynamics and Quantum Theory. Dover 2007 S. Hildebrandt and A. Tromba Mathematics and Optimal Form. Scientific American Library 1985

METHODS 24 lectures, Michaelmas Term

Self-adjoint ODEs Periodic functions. Fourier series: definition and simple properties; Parseval’s theorem. Equations of second order. Self-adjoint differential operators. The Sturm–Liouville equation; eigenfunctions and eigenvalues; reality of eigenvalues and orthogonality of eigenfunctions; eigenfunction expansions (Fourier series as prototype), approximation in mean square, statement of completeness. [5]

PDEs on bounded domains: separation of variables Physical basis of Laplace’s equation, the wave equation and the diffusion equation. General method of separation of variables in Cartesian, cylindrical and spherical coordinates. Legendre’s equation: derivation, solutions including explicit forms of P 0 , P 1 and P 2 , orthogonality. Bessel’s equation of integer order as an example of a self-adjoint eigenvalue problem with non-trivial weight. Examples including potentials on rectangular and circular domains and on a spherical domain (axisym- metric case only), waves on a finite string and heat flow down a semi-infinite rod. [6]

Inhomogeneous ODEs: Green’s functions Properties of the Dirac delta function. Initial value problems and forced problems with two fixed end points; solution using Green’s functions. Eigenfunction expansions of the delta function and Green’s functions. [3]

Fourier transforms Fourier transforms: definition and simple properties; inversion and convolution theorems. The discrete Fourier transform. Examples of application to linear systems. Relationship of transfer function to Green’s function for initial value problems. [4]

PDEs on unbounded domains Classification of PDEs in two independent variables. Well posedness. Solution by the method of characteristics. Green’s functions for PDEs in 1, 2 and 3 independent variables; fundamental solutions of the wave equation, Laplace’s equation and the diffusion equation. The method of images. Application to the forced wave equation, Poisson’s equation and forced diffusion equation. Transient solutions of diffusion problems: the error function. [6]

Appropriate books G.B. Arfken, H.J. Weber & F.E. Harris Mathematical Methods for Physicists. Elsevier 2013 M.L. Boas Mathematical Methods in the Physical Sciences. Wiley 2005 J. Mathews and R.L. Walker Mathematical Methods of Physics. Benjamin/Cummings 1970 K. F. Riley, M. P. Hobson, and S.J. Bence Mathematical Methods for Physics and Engineering: a comprehensive guide. Cambridge University Press 2002 Erwin Kreyszig Advanced Engineering Mathematics. Wiley

PART IB 18

FLUID DYNAMICS 16 lectures, Lent Term

Parallel viscous flow Plane Couette flow, dynamic viscosity. Momentum equation and boundary conditions. Steady flows including Poiseuille flow in a channel. Unsteady flows, kinematic viscosity, brief description of viscous boundary layers (skin depth). [3]

Kinematics Material time derivative. Conservation of mass and the kinematic boundary condition. Incompressibil- ity; streamfunction for two-dimensional flow. Streamlines and path lines. [2]

Dynamics Statement of Navier-Stokes momentum equation. Reynolds number. Stagnation-point flow; discussion of viscous boundary layer and pressure field. Conservation of momentum; Euler momentum equation. Bernoulli’s equation. Vorticity, vorticity equation, vortex line stretching, irrotational flow remains irrotational. [4]

Potential flows Velocity potential; Laplace’s equation, examples of solutions in spherical and cylindrical geometry by separation of variables. Translating sphere. Lift on a cylinder with circulation. Expression for pressure in time-dependent potential flows with potential forces. Oscillations in a manometer and of a bubble. [3]

Geophysical flows Linear water waves: dispersion relation, deep and shallow water, standing waves in a container, Rayleigh-Taylor instability. Euler equations in a rotating frame. Steady geostrophic flow, pressure as streamfunction. Motion in a shallow layer, hydrostatic assumption, modified continuity equation. Conservation of potential vorticity, Rossby radius of deformation. [4]

Appropriate books † (^) D.J. Acheson Elementary Fluid Dynamics. Oxford University Press 1990

G.K. Batchelor An Introduction to Fluid Dynamics. Cambridge University Press 2000 G.M. Homsey et al. Multi-Media Fluid Mechanics. Cambridge University Press 2008 M. van Dyke An Album of Fluid Motion. Parabolic Press M.G. Worster Understanding Fluid Flow. Cambridge University Press 2009

NUMERICAL ANALYSIS 16 lectures, Lent Term

Polynomial approximation Interpolation by polynomials. Divided differences of functions and relations to derivatives. Orthogonal polynomials and their recurrence relations. Least squares approximation by polynomials. Gaussian quadrature formulae. Peano kernel theorem and applications. [6]

Computation of ordinary differential equations Euler’s method and proof of convergence. Multistep methods, including order, the root condition and the concept of convergence. Runge-Kutta schemes. Stiff equations and A-stability. [5]

Systems of equations and least squares calculations LU triangular factorization of matrices. Relation to Gaussian elimination. Column pivoting. Fac- torizations of symmetric and band matrices. The Newton-Raphson method for systems of non-linear algebraic equations. QR factorization of rectangular matrices by Gram–Schmidt, Givens and House- holder techniques. Application to linear least squares calculations. [5]

Appropriate books † (^) S.D. Conte and C. de Boor Elementary Numerical Analysis: an algorithmic approach. McGraw–Hill 1980 G.H. Golub and C. Van Loan Matrix Computations. Johns Hopkins University Press 1996 A Iserles A first course in the Numerical Analysis of Differential Equations. CUP 2009 E. Suli and D.F. Meyers An introduction to numerical analysis. CUP 2003 A. Ralston and P. Rabinowitz A first course in numerical analysis. Dover 2001 M.J.D. Powell Approximation Theory and Methods. CUP 1981 P.J. Davis Interpolation and Approximation. Dover 1975

PART IB 19

STATISTICS 16 lectures, Lent Term

Estimation Review of distribution and density functions, parametric families. Examples: binomial, Poisson, gamma. Sufficiency, minimal sufficiency, the Rao–Blackwell theorem. Maximum likelihood estimation. Confi- dence intervals. Use of prior distributions and Bayesian inference. [6]

Hypothesis testing Simple examples of hypothesis testing, null and alternative hypothesis, critical region, size, power, type I and type II errors, Neyman–Pearson lemma. Significance level of outcome. Uniformly most powerful tests. Likelihood ratio, and use of generalised likelihood ratio to construct test statistics for composite hypotheses. Examples, including t-tests and F -tests. Relationship with confidence intervals. Goodness- of-fit tests and contingency tables. [4]

Linear models Derivation and joint distribution of maximum likelihood estimators, least squares, Gauss-Markov the- orem. Testing hypotheses, geometric interpretation. Examples, including simple linear regression and one-way analysis of variance. ∗Use of software∗. [6]

Appropriate books

D.A. Berry and B.W. Lindgren Statistics, Theory and Methods. Wadsworth 1995 G. Casella and R.L. Berger Statistical Inference. Duxbury 2001 M.H. DeGroot and M.J. Schervish Probability and Statistics. Pearson Education 2001

MARKOV CHAINS 12 lectures, Michaelmas Term

Discrete-time chains Definition and basic properties, the transition matrix. Calculation of n-step transition probabilities. Communicating classes, closed classes, absorption, irreducibility. Calculation of hitting probabilities and mean hitting times; survival probability for birth and death chains. Stopping times and statement of the strong Markov property. [5] Recurrence and transience; equivalence of transience and summability of n-step transition probabilities; equivalence of recurrence and certainty of return. Recurrence as a class property, relation with closed classes. Simple random walks in dimensions one, two and three. [3] Invariant distributions, statement of existence and uniqueness. Mean return time, positive recurrence; equivalence of positive recurrence and the existence of an invariant distribution. Convergence to equilib- rium for irreducible, positive recurrent, aperiodic chains and proof by coupling. Long-run proportion of time spent in given state. [3] Time reversal, detailed balance, reversibility; random walk on a graph. [1]

Appropriate books G.R. Grimmett and D.R. Stirzaker Probability and Random Processes. OUP 2001 G.R. Grimmett and D. Welsh Probability, An Introduction. OUP, 2nd edition, 2014 J.R. Norris Markov Chains. CUP 1997