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INTRODUCTION 1
THE MATHEMATICAL TRIPOS 2014-2015
CONTENTS
This booklet contains the schedule, or syllabus specification,for each course of the undergraduateTripos together withinformation about the examinations. It is updated every year.Suggestions andcorrections should be e-mailed to[emailprotected].
SCHEDULES
Syllabus
The schedule for each lecture course is a list of topics thatdefine the course. The schedule is agreed bythe Faculty Board. Someschedules contain topics that are ‘starred’ (listed betweenasterisks); all thetopics must be covered by the lecturer butexaminers can only set questions on unstarred topics.
The numbers which appear in brackets at the end of subsectionsor paragraphs in these schedules indicatethe approximate number oflectures likely to be devoted to that subsection or paragraph.Lecturersdecide upon the amount of time they think appropriate tospend on each topic, and also on the order
in which they present to topics. Some topics in Part IA and PartIB courses have to be introduced ina certain order so as to tie inwith other courses.
Recommended books
A list of books is given after each schedule. Books markedwith† are particularly well suited to thecourse. Someof the books are out of print; these are retained on the listbecause they should be availablein college libraries (as should allthe books on the list) and may be found in second-handbookshops.There may well be many other suitable books not listed;it is usually worth browsing college libraries.
Most books on the list have a price attached: this is based onthe most up to date information availableat the time of productionof the schedule, but it may not be accurate.
STUDY SKILLS
The Faculty produces a bookletStudy Skills inMathematicswhich is distributed to all first yearstudents
and can be obtained in pdf format fromwww.maths.cam.ac.uk/undergrad/studyskills.
There is also a booklet, Supervision in Mathematics,that gives guidance to supervisors obtainablefromwww.maths.cam.ac.uk/facultyoffice/supervisorsguide/whichmay also be of interest to students.
Aims and objectives
Theaimsof the Faculty for Parts IA, IB and II of theMathematical Tripos are:
• to provide a challenging course in mathematics and itsapplications for a range of students thatincludes some of the bestin the country;
• to provide a course that is suitable both for studentsaiming to pursue research and for studentsgoing into othercareers;
• to provide an integrated system of teaching which canbe tailored to the needs of individual students;• to developin students the capacity for learning and for clear logicalthinking, and the ability to
solve unseen problems;• to continue to attract and selectstudents of outstanding quality;• to produce the high calibregraduates in mathematics sought by employers in universities,the
professions and the public services.• to provide anintellectually stimulating environment in which students have theopportunity 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.
Theobjectivesof Parts IA, IB and II of theMathematical Tripos are as follows:
After completing Part IA, students should have:
• made the transition in learning style and pace fromschool mathematics to university mathematics;• beenintroduced to basic concepts in higher mathematics and theirapplications, including (i) the
notions of proof, rigour and axiomatic development, (ii) thegeneralisation of familiar mathematicsto unfamiliar contexts, (iii)the application of mathematics to problems outside mathematics;
• laid the foundations, in terms of knowledge andunderstanding, of tools, facts and techniques, toproceed to PartIB.
After completing Part IB, students should have:
• covered material from a range of pure mathematics,statistics and operations research, appliedmathematics, theoreticalphysics and computational mathematics, and studied some of thismaterialin depth;
• acquired a sufficiently broad and deep mathematicalknowledge and understanding to enable themboth to make an informedchoice 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 abstractand concrete unseen problems, (ii) present-ing a concise andlogical argument, and (iii) (in most cases) using standard softwareto tacklemathematical problems;
• studied advanced material in the mathematical sciences,some of it in depth.
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INTRODUCTION 2
EXAMINATIONS
Overview
There are three examinations for the undergraduate MathematicalTripos: Parts IA, IB and II. Theyare normally taken in consecutiveyears. This page contains information that is common to allthreeexaminations. Information that is specific to individualexaminations is given later in this booklet in
the General Arrangementssections for theappropriate part of the Tripos.The form of each examination (numberof papers, numbers of questions on each lecture course,distri-bution of questions in the papers and in the sections ofeach paper, number of questions which may beattempted) isdetermined by the Faculty Board. The main structure has to beagreed by Universitycommittees and is published as a Regulation inthe Statutes and Ordinances of the University of Cam-bridge(http://www.admin.cam.ac.uk/univ/so). (Any significant change tothe format is announcedin theReporteras a Formand Conduct notice.) The actual questions and marking schemes, andpreciseborderlines (following general classing criteria agreed bythe Faculty Board — see below) are determinedby the examiners.
The examiners for each part of the Tripos are appointed by theGeneral Board of the University. Theinternal examiners are normallyteaching staff of the two mathematics departments and they arejoinedby one or more external examiners from other universities(one for Part IA, two for Part IB and threefor Part II).
For all three parts of the Tripos, the examiners arecollectively responsible for the examination questions,though forPart II the questions are proposed by the individual lecturers. Allquestions have to be
signed off by the relevant lecturer; no question can be usedunless the lecturer agrees that it is fair andappropriate to thecourse he or she lectured.
Form of the examination
The examination for each part of the Tripos consists of fourwritten papers and candidates take all four.For Parts IB and II,candidates may in addition submit Computational Projects. Eachwritten paperhas two sections: Section I contains questions thatare intended to be accessible to any student whohas studied thematerial conscientiously. They should not contain any significant‘problem’ element.Section II questions are intended to be morechallenging
Calculators are not allowed in any paper of the MathematicalTripos; questions will be set in such away as not to require theuse of calculators. The rules for the use of calculators in thePhysics paper ofthe Mathematics with Physics option of PartIA are set out in the regulations for the NaturalSciencesTripos.
Formula booklets are not permitted, but candidates will not berequired to quote elaborate formulaefrom memory.
Past papers
Past Tripos papers for the last 10 or more years can be found onthe Faculty websitehttp://www.maths.cam.ac.uk/undergrad/pastpapers/. Solutions andmark schemes are not avail-able except in rough draft form forsupervisors.
Marking conventions
Section I questions are marked out of 10 and Section IIquestions are marked out of 20. In additionto a numerical mark,extra credit in the form of a quality mark may be awarded for eachquestiondepending on the completeness and quality of each answer.For a Section I question, a betaqualitymark isawarded for a mark of 8 or more. For a Section II question,analphaquality mark is awardedfor a mark of 15 ormore, and a betaquality mark is awarded for amark between 10 and 14, inclusive.
The Faculty Board recommends that no distinction should be madebetween marks obtained on theComputational Projects courses andmarks obtained on the written papers.
On some papers, there are restrictions on the number ofquestions that may be attempted, indicatedby a rubric of the form ‘You may attempt at mostNquestions in SectionI’. The Faculty policy is thatexaminers mark all attempts,even if the number of these exceeds that specified in the rubric,and thecandidate is assessed on the best attempts consistent withthe rubric. This policy is intended to dealwith candidates whoaccidently violate the rubric: it is clearly not in candidates’best interests to tacklemore questions than is permitted by therubric.
Examinations are ‘single-marked’, but safety checks are made onall scripts to ensure that all work ismarked and that all marks arecorrectly added and transcribed. Scripts are identified only bycandidatenumber until the final class-lists have been drawn up. Indrawing up the class list, examiners makedecisions based only onthe work they see: no account is taken of the candidates’ personalsituation orof supervision reports. Candidates for whom a warningletter has been received may be removed fromthe class list pendingan appeal. All appeals must be made through official channels (viaa college tutorif you are seeking an allowance due to, for example,ill health; either via a college tutor or directly to
the Registrary if you wish to appeal against the mark you weregiven: for further information, a tutoror the CUSU web site shouldbe consulted). Examiners should not be approached either bycandidatesor their directors of studies as this might jeopardiseany formal appeal.
Data Protection Act
To meet the University’s obligations under the Data ProtectionAct (1998), the Faculty deals with datarelating to individuals andtheir examination marks as follows:
• Marks for individual questions and ComputationalProjects are released routinely after the exam-inations.
• Scripts and Computational Projects submissions arekept, in line with the University policy, forsix months followingthe examinations (in case of appeals). Scripts and are thendestroyed; andComputational Projects are anonymised and stored in aform that allows comparison (using anti-plagiarism software) withcurrent projects.
• Neither the Data Protection Act nor the Freedom ofInformation Act entitle candidates to haveaccess to their scripts.However, data app earing on individual examination scripts areavailable onapplication to the University Data Protection Officerand on payment of a fee. Such data wouldconsist of little more thanticks, crosses, underlines, and mark subtotals and totals.
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INTRODUCTION 3
Classification Criteria
As a result of each examination, each candidate is placed in oneof the following categories: first class,upper second class (2.1),lower second class (2.2), third class, fail or ‘other’. ‘Other’here includes, forexample, candidates who were ill for all or partof the examination.
In the exceptionally unlikely event of yourbeing placed in the fail category, you should contact yourtutor ordirector of studies at once: if you wish to continue to study atCambridge an appeal (based, for
example, on medical evidence) must be made to the Council of theUniversity. There are no ‘re-sits’ inthe usual sense; inexceptional circ*mstances the regulations permit you to re-take aTripos examinationthe following year.
The examiners place the candidates into the different classes;they do not rank the candidates withinthe classes. The primaryclassification criteria for each borderline, which are determinedby the FacultyBoard, are as follows:
First / upp er second 30α + 5β+ m
Upper second / lower second 15α + 5β+ mLower second/ thir d 15α + 5β+ m
Third/ fail
15α + 5β+ m in Part IB and Part II;
2α + β together withm in Part IA.
Here,m denotes the number of marks and αand βdenote the numbers of quality marks.Other factorsbesides marks and quality marks may be taken intoaccount.
At the third/fail borderline, individual considerations arealways paramount.
The Faculty Board recommends approximate percentages ofcandidates for each class: 30% firsts; 40-45% upper seconds; 20-25%lower seconds; and up to 10% thirds. These percentages shouldexcludecandidates who did not sit the examination.
The Faculty Board intends that the classification criteriadescribed above should result in classes thatcan be characterizedas follows:
First Class
Candidates placed in the first class will have demonstrated agood command and secure understanding ofexaminable material.They will have presented standard arguments accurately, showedskill in applyingtheir knowledge, and generally will have producedsubstantially correct solutions to a significant numberof morechallenging questions.
Upper Second Class
Candidates placed in the upper second class will havedemonstrated good knowledge and understandingof examinablematerial. They will have presented standard arguments accuratelyand will have shownsome ability to apply their knowledge to solveproblems. A fair number of their answers to bothstraightforward andmore challenging questions will have been substantiallycorrect.
Lower Second Class
Candidates placed in the lower second class will havedemonstrated knowledge but sometimes imperfectunderstanding ofexaminable material. They will have been aware of relevantmathematical issues, buttheir presentation of standard argumentswill sometimes have been fragmentary or imperfect. Theywill haveproduced substantially correct solutions to some straightforwardquestions, but will have had
limited success at tackling more challenging problems.
Third Class
Candidates placed in the third class will have demonstrated someknowledge but little understandingof the examinable material. Theywill have made reasonable attempts at a small number ofquestions,but will have lacked the skills to complete many ofthem.
Examiners’ reports
For each part of the Tripos, the examiners (internal andexternal) write a joint report. In addition, theexternal examinerseach submit a report addressed to the Vice-Chancellor. The reportsof the externalexaminers are scrutinised by the General Board ofthe University’s Education Committee.
All the reports, the examination statistics (number of attemptsper question, etc), student feedback onthe examinations and lecturecourses (via the end of year questionnaire and paperquestionnaires), andother relevant material are considered by theFaculty Teaching Committee at the start of the Michaelmasterm. TheTeaching Committee includes two student representatives, and mayinclude other students(for example, previous members of theTeaching Committee and student representatives of theFacultyBoard). The Teaching Committee compiles a lengthy reportincluding various recommendations for theFaculty Board to considerat its second meeting in the Michaelmas term. This report alsoforms thebasis of the Faculty Board’s response to the reports ofthe external examiners.
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INTRODUCTION 4
MISCELLANEOUS MATTERS
Numbers of supervisions
Directors of Studies will arrange supervisions for each courseas they think appropriate. Lecturerswill hand out examples sheetswhich supervisors may use if they wish. According to FacultyBoardguidelines, the number of examples sheets for 24-lecture,16-lecture and 12-lecture courses should be 4,
3 and 2, respectively.
Transcripts
In order to conform to government guidelines on examinations,the Faculty is obliged to produce, foruse in transcripts, data thatwill allow you to determine roughly your position within eachclass. Theexaminers officially do no more than place each candidatein a class, but the Faculty authorises apercentage mark to begiven, via CamSIS, to each candidate for each examination. Thepercentagemark is obtained by piecewise linear scaling of meritmarks within each class. The 2/1, 2.1/2.2, 2.2/3and 3/failboundaries are mapped to 69.5%, 59.5%, 49.5% and 39.5% respectivelyand the mark of the5th ranked candidate is mapped to 95%. If, afterlinear mapping of the first class, the percentage markof anycandidate is greater than 100, it is reduced to 100%. Thepercentage of each candidate is thenrounded appropriately tointeger values.
The merit mark mentioned above, denoted byM, isdefined in terms of raw mark m, number of alphas,α, andnumber of betas, β, by
M=
30α + 5β+ m − 120 for candidates in the first class,or in the upper second class withα≥8,15α +5β+ m otherwise
Faculty Committees
The Faculty has two committees which deal with matters relatingto the undergraduate Tripos: theTeaching Committee and theCurriculum Committee. Both have student representatives.
The role of the Teaching Committee is mainly to monitor feedback(questionnaires, examiners’ reports,etc) and make recommendationsto the Faculty Board on the basis of this feedback. It alsoformulatespolicy recommendations at the request of the FacultyBoard.
The Curriculum Committee is responsible for recommending (to theFaculty Board) changes to theundergraduate Tripos and to theschedules for individual lecture courses.
Student representatives
There are three student representatives, two undergraduate andone graduate, on the Faculty Board,and two on each of the theTeaching Committee and the Curriculum Committee. They arenormallyelected (in the case of the Faculty Board representatives)or appointed in November of each year. Their
role is to advise the committees on the student point of view,to collect opinion from and liaise with thestudent body. Theyoperate a website: http://www.maths.cam.ac.uk/studentreps and their emailaddressis [emailprotected].
Feedback
Constructive feedback of all sorts and from all sources iswelcomed by everyone concerned in providingcourses for theMathematical Tripos.
There are many different feedback routes.
• Each lecturer hands out a paper questionnaire towardsthe end of the course.• There are brief web-basedquestionnaires after roughly six lectures of each course.•Students are sent a combined online questionnaire at the endof each year.
• Students (or supervisors) can e-mail[emailprotected]at any time. Such e-mailsareread by the Chairman of the Teaching Committee and forwarded inanonymised form to the ap-propriate person (a lecturer, forexample). Students will receive a rapid response.
• If a student wishes to be entirely anonymous and doesnot need a reply, the web-based commentform atwww.maths.cam.ac.uk/feedback.html can be used (clickable from thefaculty web site).
• Feedback on college-provided teaching (supervisions,classes) can be given to Directors of Studiesor Tutors at anytime.
The questionnaires are particularly important in shaping thefuture of the Tripos and the Faculty Boardurge all students torespond.
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PART IA 5
Part IA
GENERAL ARRANGEMENTS
Structure of Part IA
There are two options:
(a) Pure and Applied Mathematics;(b) Mathematics withPhysics.
Option (a) is intended primarily for students who expect tocontinue to Part IB of the MathematicalTripos, while Option (b) isintended primarily for those who are undecided about whether theywillcontinue to Part IB of the Mathematical Tripos or change toPart IB of the Natural Sciences Tripos(Physics option).
For Option (b), two of the lecture courses (Numbers and Sets,and Dynamics and Relativity) are replacedby the complete Physicscourse from Part IA of the Natural Sciences Tripos; Numbers andSets becauseit is the least relevant to students taking thisoption, and Dynamics and Relativity because muchof this material iscovered in the Natural Sciences Tripos anyway. Students wishing toexamine theschedules for the physics courses should consult thedocumentation supplied by the Physics department,
for example on http://www.phy.cam.ac.uk/teaching/.
Examinations
Arrangements common to all examinations of the undergraduateMathematical Tripos are given onpages 1 and 2 of this booklet.
All candidates for Part IA of the Mathematical Tripos take fourpapers, as follows.Candidates taking Option (a) (Pure and AppliedMathematics) will take Papers 1, 2, 3 and 4 of theMathematicalTripos (Part IA).Candidates taking Option (b) (Mathematics withPhysics) take Papers 1, 2 and 3 of the MathematicalTripos (Part IA)and the Physics paper of the Natural Sciences Tripos (Part IA);they must also submitpractical notebooks.
For Mathematics with Physics candidates, the marks and qualitymarks for the Physics paper are scaledto bring them in line withPaper 4. This is done as follows. The Physics papers of theMathematics withPhysics candidates are marked by the Examiners inPart IA NST Physics, and Mathematics Examiners
are given a percentage mark for each candidate. The classborderlines are at 70%, 60%, 50% and 40%.All candidates for Paper 4(ranked by merit mark on that paper) are assigned nominally toclassesso that the percentages in each class are 30, 40, 20, 10(which is the Faculty Board rough guidelineproportion in each classin the overall classification). Piecewise linear mapping of the thePhysicspercentages in each Physics class to the Mathematics meritmarks in each nominal Mathematics classis used to provide a meritmark for each Mathematics with Physics candidate. The merit mark isthenbroken down into marks, alphas and betas by comparison (foreach candidate) with the break down forPapers 1, 2 and 3.
Examination Papers
Papers 1, 2, 3 and 4 of Part IA of the Mathematical Tripos areeach divided into two Sections. There arefour questions in SectionI and eight questions in Section II. Candidates may attempt all thequestionsin Section I and at most five questions from Section II,of which no more than three may be on thesame lecture course.
Each section of each of Papers 1–4 is divided equally betweentwo courses as follows:
Paper 1: Vectors and Matrices, Analysis IPaper 2: DifferentialEquations, ProbabilityPaper 3: Groups, Vector CalculusPaper 4:Numbers and Sets, Dynamics and Relativity.
Approximate class boundaries
The following tables, based on information supplied by theexaminers, show the approximate borderlines.
For convenience, we define M1 andM2 by
M1= 30 α + 5β+ m − 120, M2=15 α + 5β+ m.
M1 is related to the primary classificationcriterion for the first class and M2 is relatedto the primaryclassification criterion for the upper and lowersecond classes.
The second column of each table shows a sufficient criterion foreach class. The third and fourthcolumnsshowsM1(for the first class) orM2 (for the other classes), raw mark, number ofalphas and number ofbetas of two representative candidatesplaced just above the borderline. The sufficient condition foreachclass is not prescriptive: it is just intended to be helpfulfor interpreting the data. Each candidate neara borderline isscrutinised individually. The data given below are relevant to oneyear only; borderlinesmay go up or down in future years.
Part IA 2013
Class Sufficient condition Borderline candidates1 M1> 7 27 73 1/3 91, 14 ,8 7 32/ 377 ,1 5,5
2.1 M2> 480 481/311,7,13483/303,9,9
2.2 M2> 384 385/245,6,10385/265,5,9
3 2α + β > 1 2 25 0/1 80, 1, 11 2 62/ 182 ,3 ,7
Part IA 2014
Class Sufficient condition Borderline candidates
1 M1> 625 605/335,12,6 608/358,11,8
2.1 M2> 438 420/260,7,11 422/287,6,9
2.2 M2> 328 328/213,5,8334/209,5,103 2α + β > 11 247/167,3,7 273/183,5,3
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PART IA 6
GROUPS 24 lectures, Michaelmas term
Examples of groups
Axioms for groups. Examples from geometry: symmetry groups ofregular polygons, cube, tetrahedron.Permutations on a set; thesymmetric group. Subgroups and hom*omorphisms. Symmetry groupsassubgroups of general permutation groups. The Möbius group;cross-ratios, preservation of circles, thepoint at infinity.Conjugation. Fixed points of Möbius maps and iteration. [4]
Lagrange’s theoremCosets. Lagrange’s theorem. Groups of smallorder (up to order 8). Quaternions. Fermat-Euler theoremfrom thegroup-theoretic point of view. [5]
Group actionsGroup actions; orbits and stabilizers.Orbit-stabilizer theorem. Cayley’s theorem (every groupisisomorphic to a subgroup of a permutation group). Conjugacyclasses. Cauchy’s theorem. [4]
Quotient groups
N or ma l s ub gr ou ps, qu ot ie nt gr ou ps a nd t he i so morp hi sm t heo re m. [4 ]
Matrix groupsThe general and special linear groups; relationwith the M öbius group. The orthogonal and specialorthogonalgroups. Proof (in R3) that every element of the orthogonal group isthe product of reflectionsand every rotation in R3 has an axis. B as is change as an e xample of c onjugati on. [ 3]
Permutations
Permutations, cycles and transpositions. The sign of apermutation. Conjugacy in Sn and inAn.Simple groups; simplicity ofA5. [4]
Appropriate books
M.A. ArmstrongGroups and Symmetry. Springer–Verlag1988 (£33.00 hardback)† Alan F Beardon Algebra and Geometry.CUP 2005 (£21.99 paperback, £48 hardback).
R.P. Burn Groups, a Path to Geometry. CambridgeUniversity Press 1987 (£20.95 paperback)J.A. Green Sets andGroups: a first course in Algebra. Chapman and Hall/CRC 1988(£38.99 paper-
See AlsoCondensate Retention Effects on the Air-Side Heat Transfer - [PDF Document]Encyclopedia of Women's Folklore and Folklife 2 volumes - PDF Free Downloadback)W. LedermanIntroduction to Group Theory.Longman 1976 (out of print)Nathan Carter Visual GroupTheory. Mathematical Association of America Textbooks(£45)
VECTORS AND MATRICES 24 lectures, Michaelmas term
Complex numbers
Review of complex numbers, including complex conjugate, inverse,modulus, argument and Arganddiagram. Informal treatment of complexlogarithm, n-th roots and complex powers. deMoivre’stheorem. [2]
VectorsReview of elementary algebra of vectors in R3,including scalar product. Brief discussion of vectorsin Rnand Cn; scalar product and the Cauchy–Schwarz inequality.Concepts of linear span, linearindependence, subspaces, basis anddimension.
Suffix notation: including summation convention, δij and ϵijk . Vector product and tripleproduct:definition and geometrical interpretation. Solution oflinear vector equations. Applications of vectorsto geometry,including equations of lines, planes and spheres. [5]
MatricesElementary algebra of 3 × 3 matrices, includingdeterminants. Extension to n × n complexmatrices.Trace, determinant, non-singular matrices and inverses.Matrices as linear transformations; examplesof geometrical actionsincluding rotations, reflections, dilations, shears; kernel andimage. [4]
Simultaneous linear equations: matrix formulation; existence anduniqueness of solutions, geometricinterpretation; Gaussianelimination. [3]
Symmetric, anti-symmetric, orthogonal, hermitian and unitarymatrices. Decomposition of a general
matri x i nto i sotropic , s ymme tric trace -f re e and antisymme tric parts . [ 1]
Eigenvalues and EigenvectorsEigenvalues and eigenvectors;geometric significance. [2]
Proof that eigenvalues of hermitian matrix are real, and thatdistinct eigenvalues give an orthogonal basisof eigenvectors. Theeffect of a general change of basis (similarity transformations).Diagonalizationof general matrices: sufficient conditions; examplesof matrices that cannot be diagonalized. Canonicalforms for 2 × 2matrices. [5]Discussion of quadratic forms, including change ofbasis. Classification of conics, cartesian and polarforms. [1]
Rotation matri ce s and Lorentz trans formations as transformation groups. [ 1]
Appropriate books
Alan F Beardon Algebra and Geometry. CUP 2005(£21.99 paperback, £48 hardback)Gilbert StrangLinearAlgebra and Its Applications. Thomson Brooks/Cole, 2006(£42.81 paperback)
Richard Kaye and Robert Wilson Linear Algebra.Oxford science publications, 1998 (£23 )D.E. Bourne and P.C.KendallVector Analysis and Cartesian Tensors. NelsonThornes 1992 (£30.75
paperback)E. SernesiLinear Algebra: A Geometric Approach.CRC Press 1993 (£38.99 paperback)James J. Callahan TheGeometry of Spacetime: An Introduction to Special and GeneralRelativity.
Springer 2000 (£51)
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PART IA 7
NUMBERS AND SETS 24 lectures, Michaelmas term
[Note that this course is omitted from Option (b) of PartIA.]
Introduction to number systems and logicOverview of the naturalnumbers, integers, real numbers, rational and irrational numbers,algebraic andtranscendental numbers. Brief discussion of complexnumbers; statement of the Fundamental Theoremof Algebra.
Ideas of axiomatic systems and proof within mathematics; theneed for proof; the role of counter-examples in mathematics.Elementary logic; implication and negation; examples of negation ofcom-pound statements. Proof by contradiction. [2]
Sets, relations and functions
Union, intersection and equality of sets. Indicator(characteristic) functions; their use in establishingsetidentities. Functions; injections, surjections and bijections.Relations, and equivalence relations.Counting the combinations orpermutations of a set. The Inclusion-Exclusion Principle. [4]
The integersThe natural numbers: mathematical induction and thewell-ordering principle. Examples, includingthe Binomial Theorem.[2]
Elementary number theoryPrime numbers: existence and uniquenessof prime factorisation into primes; highest common factorsand leastcommon 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. ChineseRemainder Theorem. Wilson’s Theorem;the Fermat-Euler Theorem.Public key cryptography and the RSA algorithm. [8]
The real numbersLeast upper bounds; simple examples. Least upperbound axiom. Sequences and series; convergenceof bounded monotonicsequences. Irrationality of
√2 and e. Decimal expansions. Construction of a
transcendental number. [4]
Countability and uncountabilityDefinitions of finite, infinite,countable and uncountable sets. A countable union of countable setsiscountable. Uncountability ofR . Non-existence of abijection from a set to its power set. Indirect proofofexistence of transcendental numbers. [4]
Appropriate books
R.B.J.T. AllenbyNumbers and Proofs.Butterworth-Heinemann 1997 (£
19.50 paperback)R.P. Burn Numbers and Functions: stepsinto analysis. Cambridge University Press 2000(£21.95paperback)
H. Davenport The Higher Arithmetic. CambridgeUniversity Press 1999 (£19.95 paperback)A.G. Hamilton Numbers, sets and axioms: the apparatus of mathematics.Cambridge University Press
1983 (£20.95 paperback)C. Schumacher Chapter Zero:Fundamental Notions of Abstract Mathematics. Addison-Wesley2001
(£42.95 hardback)I. Stewart and D. Tall The Foundations ofMathematics. Oxford University Press 1977 (£22.50 paper-
back)
DIFFERENTIAL EQUATIONS 24 lectures, Michaelmas term
Basic calculus
Informal treatment of differentiation as a limit, the chainrule, Leibnitz’s rule, Taylor series, informaltreatmentofO and o notation and l’Hôpital’srule; integration as an area, fundamental theorem ofcalculus,integration by substitution and parts. [3]
Informal treatment of partial derivatives, geometricalinterpretation, statement (only) of symmetryof mixed partialderivatives, chain rule, implicit differentiation. Informaltreatment of differentials,including exact differentials.Differentiation of an integral with respect to a parameter. [2]
First-order linear differential equationsEquations with constantcoefficients: exponential growth, comparison with discreteequations, seriessolution; modelling examples including radioactivedecay.
E quations with non-c onstant c oe ffic ie nts: s ol ution by integrati ng f ac tor. [ 2]
Nonlinear first-order equations
Separable equations. Exact equations. Sketching solutiontrajectories. Equilibrium solutions, stabilityby perturbation;examples, including logistic equation and chemical kinetics.Discrete equations: equi-l ib ri um so lu ti on s, st ab il ity ; exam ples in cl ud in g t he lo gi st ic ma p. [4 ]
Higher-order linear differential equations
Complementary function and particular integral, linearindependence, Wronskian (for second-orderequations), Abel’stheorem. Equations with constant coefficients and examplesincluding radioactivesequences, comparison in simple cases withdifference equations, reduction of order, resonance, tran-sients,damping. hom*ogeneous equations. Response to step and impulsefunction inputs; introductionto the notions of the Heavisidestep-function and the Dirac delta-function. Series solutionsincludingstatement only of the need for the logarithmic solution.[8]
Multivariate functions: applicationsDirectional derivatives andthe gradient vector. Statement of Taylor series for functions onRn. Localextrema of real functions, classification using theHessian matrix. Coupled first order systems: equiv-alence to singlehigher order equations; solution by matrix methods. Non-degeneratephase portraitslocal to equilibrium points; stability.
Simple examples of first- and second-order partial differentialequations, solution of the wave equationin the formf(x+ ct) + g(x − ct). [5]
Appropriate books
J. Robinson An introduction to Differential Equations.Cambridge University Press, 2004 (£33)
W.E. Boyce and R.C. DiPrima Elementary DifferentialEquations and Boundary-Value Problems (andassociated website: google Boyce DiPrima). Wiley, 2004 (£34.95hardback)
G.F.SimmonsDifferential Equations (with applications andhistorical notes). McGraw-Hill 1991 (£43)D.G. Zill and M.R.Cullen Differential Equations with Boundary Value Problems. Brooks/Cole 2001
(£37.00 hardback)
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PART IA 8
ANALYSIS I 24 lectures, Lent term
Limits and convergence
Sequences and series in R and C. Sums, products and quotients.Absolute convergence; absolute conver-gence implies convergence.The Bolzano-Weierstrass theorem and applications (the GeneralPrincipleo f C onve rge nce) . C omp ar iso n a nd r at io t est s,al ter na ti ng se rie s t es t. [6 ]
ContinuityContinuity of real- and complex-valued functionsdefined on subsets of R and C. Theintermediatevalue theorem. A continuous function on a closedbounded interval is bounded and attains its bounds.
[3]
DifferentiabilityDifferentiability of functions from Rto R. Derivative of sums and products. The chain rule.Derivativeof the inverse function. Rolle’s theorem; the mean valuetheorem. One-dimensional version of theinverse function theorem.Taylor’s theorem from R to R ; Lagrange’s form ofthe remainder. Complexdifferentiation. [5]
Power seriesComplex power series and radius of convergence.Exponential, trigonometric and hyperbolic functions,and relationsbetween them. *Direct proof of the differentiability of a powerseries within its circle ofconvergence*. [4]
IntegrationDefinition and basic properties of the Riemannintegral. A non-integrable function. Integrabilityofmonotonic functions. Integrability of piecewise-continuousfunctions. The fundamental theorem ofcalculus.Differentiation of indefinite integrals. Integration by parts. Theintegral form of the remainderin Taylor’s theorem. Improperintegrals. [6]
Appropriate books
T.M. Apostol Calculus, vol 1. Wiley 1967-69(£181.00 hardback)† J.C. Burkill A First Course inMathematical Analysis. Cambridge University Press 1978(£27.99 pa-
perback).D.J.H.Garling A Course in Mathematical Analysis(Vol 1). Cambridge University Press 2013 (£30
paperback)J.B. ReadeIntroduction to Mathematical Analysis.Oxford University Press (out of print)M.SpivakCalculus. Addison–Wesley/Benjamin–Cummings 2006(£35)David M. Bressoud A Radical Approach to Real Analysis . Mathematical Association of America
Textbooks (£
40)
PROBABILITY 24 lectures, Lent term
Basic concepts
Classical probability, equally likely outcomes. Combinatorialanalysis, permutations and combinations.Stirling’s formula(asymptotics for log n! proved). [3]
Axiomatic approach
Axioms (countable case). Probability spaces. Inclusion-exclusionformula. Continuity and subadditiv-ity of probability measures.Independence. Binomial, Poisson and geometric distributions.Relationbetween Poisson and binomial distributions. Conditionalprobability, Bayes’s formula. Examples, in-cluding Simpson’sparadox. [5]
Discrete random variablesExpectation. Functions of a randomvariable, indicator function, variance, standard deviation.Covari-ance, independence of random variables. Generatingfunctions: sums of independent random variables,random sum formula,moments.
Conditional expectation. Random walks: gambler’s ruin,recurrence relations. Difference equationsand their solution. Meantime to absorption. Branching processes: generating functions andextinctionp ro ba bil ity. C omb in at or ia l a pp li ca tio ns of g en er at in g f un ct ion s. [7 ]
Continuous random variablesDistributions and density functions.Expectations; expectation of a function of a randomvariable.Uniform, normal and exponential random variables.Memoryless property of exponential distribution.
Joint distributions: transformation of random variables(including Jacobians), examples. Simulation:generating continuousrandom variables, independent normal random variables. Geometricalprobabil-ity: Bertrand’s paradox, Buffon’s needle. Correlationcoefficient, bivariate normal random variables.
[6]
Inequalities and limitsMarkov’s inequality, Chebyshev’sinequality. Weak law of large numbers. Convexity: Jensen’sinequalityfor general random variables, AM/GM inequality.
Moment generating functions and statement (no proof) ofcontinuity theorem. Statement of centrall im it t heo rem an d sketch o f p ro of . E xa mp les , in clu din g sa mp lin g. [3 ]
Appropriate books
W. Feller An Introduction to Probability Theory and itsApplications, Vol. I. Wiley 1968 (£73.95hardback)
† G. Grimmett and D. Welsh Probability: An Introduction. Oxford University Press 2nd Edition 2014
(£33.99 paperback).† S. RossA First Course in Probability.Prentice Hall 2009 (£39.99 paperback).
D.R. Stirzaker Elementary Probability. CambridgeUniversity Press 1994/2003 (£19.95 paperback)
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PART IA 9
VECTOR CALCULUS 24 lectures, Lent term
Curves in R3
Parameterised curves and arc length, tangents and normals tocurves in R3, the radius of curvature.[1]
Integration in R2 and R3
Line integrals. Surface and volume integrals: definitions,examples using Cartesian, cylindrical andspherical coordinates;change of variables. [4]
Vector operatorsDirectional derivatives. The gradient of areal-valued function: definition; interpretation as normal tolevelsurfaces; examples including the use of cylindrical, spherical∗and general orthogonal curvilinear∗
coordinates.
Divergence, curl and∇2 in Cartesian coordinates, examples;formulae for these operators (statementonly) 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’ssecond theorem: statements; infor-mal proofs; examples; applicationto fluid dynamics, and to electromagnetism including statementofMaxwell’s equations. [5]
Laplace’s equationLaplace’s equation in R2 and R3:uniqueness theorem and maximum principle. Solution ofPoisson’sequation by Gauss’s method (for spherical and cylindricalsymmetry) and as an integral. [4]
Cartesian tensors in R3
Tensor transformation laws, addition, multiplication,contraction, with emphasis on tensors of secondrank. Isotropicsecond and third rank tensors. Symmetric and antisymmetric tensors.Revision ofprincipal axes and diagonalization. Quotienttheorem. Examples including inertia and conductivity.
[5]
Appropriate books
H. Anton Calculus. Wiley Student Edition 2000(£33.95 hardback)T.M. Apostol Calculus. Wiley StudentEdition 1975 (Vol. II £37.95 hardback)M.L.BoasMathematical Methods in the Physical Sciences.Wiley 1983 (£32.50 paperback)
† D.E. Bourne and P.C. Kendall Vector Analysis andCartesian Tensors. 3rd edition, Nelson Thornes1999 (£29.99paperback).
E. KreyszigAdvanced Engineering Mathematics. WileyInternational Edition 1999 (£30.95 paperback,£97.50 hardback)
J.E. Marsden and A.J.Tromba Vector Calculus. Freeman1996 (£35.99 hardback)P.C. Matthews Vector Calculus. SUMS (Springer Undergraduate Mathematics Series) 1998 (£18.00
paperback)† K. F. Riley, M.P. Hobson, and S.J. Bence Mathematical Methods for Physics and Engineering. Cam-
bridge University Press 2002 (£27.95 paperback, £75.00hardback).H.M. Schey Div, grad, curl and all that: aninformal text on vector calculus. Norton 1996 (£16.99
paperback)M.R. SpiegelSchaum’s outline of Vector Analysis.McGraw Hill 1974 (£16.99 paperback)
DYNAMICS AND RELATIVITY 24 lectures, Lent term
[Note that this course is omitted from Option (b) of PartIA.]
Familarity with the topics covered in the non-examinableMechanics course is assumed.
Basic conceptsSpace and time, frames of reference, Galileantransformations. Newton’s laws. Dimensional analysis.Examples offorces, including gravity, friction and Lorentz. [4]
Newtonian dynamics of a single particleEquation of motion inCartesian and plane polar coordinates. Work, conservative forcesand potentialenergy, motion and the shape of the potential energyfunction; stable equilibria and small oscillations;effect ofdamping.
Angular velocity, angular momentum, torque.
Orbits: the u(θ) equation; escape velocity; Kepler’s laws;stability of orbits; motion in a repulsivepotential (Rutherfordscattering).
Rotating frames: centrifugal and coriolis forces. *Briefdiscussion of Foucault pendulum.* [8]
Newtonian dynamics of systems of particles
Momentum, angular momentum, energy. Motion relative to thecentre of mass; the two body problem.Variable mass problems; therocket equation. [2]
Rigid bodies
Moments of inertia, angular momentum and energy of a rigid body.Parallel axes theorem. Simplee xample s of motion i nvolving bothrotation and trans lati on (e. g. rol li ng). [ 3]
Special relativityThe principle of relativity. Relativity andsimultaneity. The invariant interval. Lorentz transformationsin (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 andLorentz invariants. Proper time. 4–velocity and 4–momentum.Conservation of 4–momentum in particle decay. Collisions. TheNewtonianlimit. [7]
Appropriate books
† D.GregoryClassical Mechanics. Cambridge UniversityPress 2006 (£26 paperback).A.P. French and M.G. EbisonIntroduction to Classical Mechanics. Kluwer 1986(£33.25 paperback)T.W.B Kibble and F.H. BerkshireIntroductionto Classical Mechanics. Kluwer 1986 (£33 paperback)M. A. LunnA First Course in Mechanics. Oxford University Press1991 (£17.50 paperback)G.F.R. Ellis and R.M. Williams Flatand Curved Space-times. Oxford University Press 2000(£24.95
paperback)† W. RindlerIntroduction to Special Relativity.Oxford University Press 1991 (£19.99 paperback).
E.F. Taylor and J.A. Wheeler Spacetime Physics:introduction to special relativity. Freeman 1992(£29.99paperback)
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PART IA 10
C OMP UTAT IONA L PROJ ECT S 8 l ec tu re s, Ea ste r te rm o fPa rt I A
The Computational Projects course is examined in Part IB.However introductory practical sessionsare offered at the end ofLent Full Term and the beginning of Easter Full Term of the Part IAyear(students are advised by email how to register for a session),and lectures are given in the Easter FullTerm of the Part IA year.The lectures cover an introduction to algorithms and aspects of theMATLABprogramming language. The projects that need to be completedfor credit are published by the Faculty
in a manual usually by the end of July at the end of the Part IAyear. The manual contains details of theprojects and information about course administration. The manual is available on the Facultywebsiteat http://www.maths.cam.ac.uk/undergrad/catam/. Full creditmay obtained from the submission ofthe two core projects anda further two additional projects. Once the manual is available,these projectsmay be undertaken at any time up to the submissiondeadlines, which are near the start of the FullLent Term in the IByear for the two core projects, and near the start of the FullEaster Term in theIB 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 fewer thanthree A-level Mechanics modules (or the equivalent).Thematerial is prerequisite for Dynamics and Relativity inthe Lent term.
Lecture 1
Brief introduction
Lecture 2: Kinematics of a single particlePosition, velocity,speed, acceleration. Constant acceleration in one-dimension.Projectile motion intwo-dimensions.
Lecture 3: Equilibrium of a single particleThe vector nature offorces, addition of forces, examples including gravity, tension ina string, normalreaction (Newton’s third law), friction. Conditionsfor equilibrium.
Lecture 4: Equilibrium of a rigid bodyResultant of severalforces, couple, moment of a force. Conditions for equilibrium.
Lecture 5: Dynamics of particles
Newton’s second law. Examples of pulleys, motion on an inclinedplane.
Lecture 6: Dynamics of particlesFurther examples, includingmotion of a projectile with air-resistance.
Lecture 7: EnergyDefinition of energy and work. Kinetic energy,potential energy of a particle in a uniform gravitationalfield.Conservation of energy.
Lecture 8: MomentumDefinition of momentum (as a vector),conservation of momentum, collisions, coefficient ofrestitution,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 particlehanging on a string. Simple harmonic motion of a particle for smalldisplacement fromequilibrium.
Lecture 10: Motion in a circleDerivation of the centralacceleration of a particle constrained to move on a circle. Simplependulum;motion of a particle sliding on a cylinder.
Appropriate books
J. Hebborn and J. Littlewood Mechanics 1, Mechanics 2 andMechanics 3 (Edexel). Heinemann, 2000(£12.99 eachpaperback)
Peter J O’Donnell Essential Dynamics and Relativity.CRC Press, 2014 (£31.99 Paperback)Anything similar to theabove, for the other A-level examination boards
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PART IB 11
CONCEPTS IN THEORETICAL PHYSICS (non-examinable) 8 lectures,Easter term
This course is intended to give a flavour of the some of themajor topics in Theoretical Physics. It will be of interest toallstudents.
The list of topics below is intended only to give an idea ofwhat might be lectured; the actual content will be announcedinthe first lecture.
Principle of Least Action
A better way to do Newtonian dynamics. Feynman’s approach toquantum mechanics.
Quantum Mechanics
Principles of quantum mechanics. Probabilities and uncertainty.Entanglement.
Statistical MechanicsMore is different: 1̸= 1024. Entropyand the Second Law. Information theory. Black holeentropy.Electrodynamics and RelativityMaxwell’s equations. Thespeed of light and relativity. Spacetime. A hidden symmetry.
Particle Physics
A new periodic table. From fields to particles. From symmetriesto forces. The origin of mass and theHiggs boson.
SymmetrySymmetry of physical laws. Noether’s theorem. Fromsymmetries to forces.
General RelativityEquivalence principle. Gravitational timedilation. Curved spacetime. Black holes. Gravity waves.
Cosmology
From quantum mechanics to galaxies.
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PART IB 12
Part IB
GENERAL ARRANGEMENTS
Structure of Part IB
Seventeen courses, including Computational Projects, areexamined in Part IB. The schedules for Com-plex Analysis andComplex Methods cover much of the same material, but from differentpoints ofview: students may attend either (or both) sets oflectures. One course, Optimisation, can be taken inthe Easter termof either the first year or the second year. Two other courses,Metric and TopologicalSpaces and Variational Principles, can alsobe taken in either Easter term, but it should be noted thatsome ofthe material in Metric and Topological Spaces will prove useful forComplex Analysis, and thematerial in Variational Principles forms agood background for many of the theoretical physics coursesin PartIB.
The Faculty Board guidance regarding choice of courses in PartIB is as follows:
Part IB of the Mathematical Tripos provides a wide range ofcourses from which students
should, in consultation with their Directors of Studies, make aselection based on their in-
dividual interests and preferred workload, bearing in mind thatit is better to do a smaller
number of courses thoroughly than to do many courses scrappily. The table of dependencies
on the next page is intended to help you choose your Part IBcourses.
Computational Projects
The lectures for Computational Projects will normally beattended in the Easter term of the first year,the ComputationalProjects themselves being done in the Michaelmas and Lent terms ofthe secondyear (or in the summer, Christmas and Eastervacations).
No questions on the Computational Projects are set on thewritten examination papers, credit forexamination purposes beinggained by the submission of notebooks. The maximum creditobtainableis 160 marks and there are no alpha or beta qualitymarks. Credit obtained is added directly to thecredit gained in thewritten examination. The maximum contribution to the final meritmark is thus160, which is roughly the same (averaging over thealpha weightings) as for a 16-lecture course.
Examination
Arrangements common to all examinations of the undergraduateMathematical Tripos are given onpages 1 and 2 of this booklet.
Each of the four papers is divided into two sections. Candidatesmay attempt at most four questionsfrom Section I and at most sixquestions from Section II.
The number of questions set on each course varies according tothe number of lectures given, as shown:
Number of lectures Section I Section II
24 3 416 2 3
12 2 2
Examination Papers
Questions on the different courses are distributed among thepapers as specified in the following table.The letters S and Lappearing the table denote a question in Section I and a questionin Section II,respectively.
Paper 1 Paper 2 Paper 3 Paper 4
Linear Algebra L+S L+S L L+SGroups, Rings and Modules L L+S L+SL+S
Analysis II L L+S L+S L+SMetric and Topological Spaces L S SL
Complex AnalysisComplex Methods
L+S* L* L
SS
LGeometry S L L+S L
Variational Principles S L S LMethods L L+S L+S L+S
Quantum Mechanics L L L+S SElectromagnetism L L+S L S
Fluid Dynamics L+S S L LNumerical Analysis L+S L L S
Statistics L+S S L LOptimization S S L L
Markov Chains L L S S
*On Paper 1 and Paper 2, Complex Analysis and Complex Methodsare examined by means of commonquestions (each of which may containtwo sub-questions, one on each course, of which candidatesmayattempt only one (‘either/or’)).
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PART IB 13
Approximate class boundaries
The following tables, based on information supplied by theexaminers, show the approximate borderlines.
For convenience, we define M1 andM2 by
M1= 30 α + 5β+ m − 120, M2=15 α + 5β+ m.
M1 is related to the primary classificationcriterion for the first class and M2 is relatedto the primaryclassification criterion for the upper and lowersecond and third classes.
The second column of each table shows a sufficient criterion foreach class (in terms of M1 for thefirstclass and M2 for the other classes). The thirdand fourth columns showM1 (for the first class)orM2 (for the other classes), raw mark, number ofalphas and number of betas of two representativecandidates placedjust above the borderline.
The sufficient condition for each class is not prescriptive: itis just intended to be helpful for interpretingthe data. Eachcandidate near a borderline is scrutinised individually. The datagiven below are relevantto one year only; borderlines may go up ordown in future years.
PartIB 2013
Class Sufficient condition Borderline candidates
1 M1> 8 06 8 07 /42 2,1 5, 11 8 11/426 ,1 5,1 1
2.1 M2> 451 453/308,7 ,8 458/288,8,10
2.2 M2> 311 312/202,5 ,7 317/212,5,6
3 2α + β≥10 190/130,2,6 209/139,3,5PartIB2014
Class Sufficient condition Borderline candidates
1 M1> 78 3 7 84/ 494 ,1 2,1 0 7 89/4 74, 13, 9
2.1 M2> 480 481/366,4 ,11 484/364,5,9
2.2 M2> 336 337/232,4 ,9 345/225,6,6
3 M2> 228 229/199,0 ,6 255/185,1,11
Part II dependencies
The relationships between Part IB courses and Part II coursesare shown in the following tables. Ablank in the table means thatthe material in the Part IB course is not directly relevant to thePart IIcourse.The terminology is as follows:
Essential: (E) a good understanding of the methods andresults of the Part IB course is essential;
Desirable: (D) knowledge of some of the results of thePart IB course is required;
Background: (B) some knowledge of the Part IB course wouldprovide a useful background.
Linear
Algebr
a
Groups
,Rin
gsand
Modul
es
Analy
sisII
Metric
and
Topolog
icalS
paces
Compl
exAnaly
sis
Compl
exMetho
ds
Geom
etry
Varia
tionalP
rinciples
Metho
ds
Quantum
Mecha
nics
Electro
magnetism
Fluid
Dynamics
Num
ericalA
naly
sis
Statistics
Optim
isatio
n
Marko
vChain
s
Number TheoryTopics in Analysis B BCoding and Cryptography DEStatistical Modelling EMathematical BiologyFurther Complex MethodsEClassical Dynamics ECosmology E ELogic and Set TheoryGraphTheoryGalois Theory D ERepresentation Theory E ENumber Fields EDAlgebraic Topology E DLinear Analysis E E ERiemann Surfaces DE
Algebr aic Geometry EDifferential Geometry E DProb. and MeasureEApplied Prob. EPrinc. of Stats EStochastic FM’s D D DOpt. andControl D D DPartial DEs E D D D E EAsymptotic Methods E DDynamicalSystemsIntegrable Systems D EPrinciples of QM D EApplications of QMB EStatistical Physics EElectrodynamics D E
General Relativity D DFluid Dynamics II E EWaves E DNumericalAnalysis D D D E
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PART IB 14
LINEAR ALGEBRA 24 lectures, Michaelmas term
Definition of a vector space (over R or C),subspaces, the space spanned by a subset. Linear indepen-d enc e, base s, d imen sio n. Di rec t s ums a nd co mp le me nt ar y su bspace s. [3 ]
Linear maps, isomorphisms. Relation between rank and nullity.The space of linear maps from Uto V,
repre se ntation by matri ce s. Change of bas is . Row rank andc ol umn rank. [ 4]Determinant and trace of a square matrix.Determinant of a product of two matrices and of the inversematr ix.Determinant of an endomorphism. The adjugate matrix. [3]
Eigenvalues and eigenvectors. Diagonal and triangular forms.Characteristic and minimal polynomials.Cayley-Hamilton Theorem overC. Algebraic and geometric multiplicity of eigenvalues.Statement andillustration of Jordan normal form. [4]
Dual of a finite-dimensional vector space, dual bases and maps.Matrix representation, rank anddeterminant of dual map [2]
Bilinear forms. Matrix representation, change of basis.Symmetric forms and their link with quadraticforms. Diagonalisationof quadratic forms. Law of inertia, classification by rank andsignature. ComplexHermitian forms. [4]
Inner product spaces, orthonormal sets, orthogonal projection, V = W⊕ W⊥.Gram-Schmidt or-thogonalisation. Adjoints. Diagonalisation ofHermitian matrices. Orthogonality of eigenvectors andproperties ofeigenvalues. [4]
Appropriate books
C.W. CurtisLinear Algebra: an introductory approach.Springer 1984 (£38.50 hardback)P.R.HalmosFinite-dimensional vector spaces. Springer 1974(£31.50 hardback)K. Hoffman and R. Kunze Linear Algebra.Prentice-Hall 1971 (£72.99 hardback)
GROUPS, RINGS AND MODULES 24 lectures, Lent term
Groups
Basic concepts of group theory recalled from Part IA Groups.Normal subgroups, quotient groupsand isomorphism theorems.Permutation groups. Groups acting on sets, permutationrepresentations.Conjugacy classes, centralizers and normalizers.The centre of a group. Elementary properties offinitep-groups. Examples of finite linear groups and groupsarising from geometry. Simplicity ofAn.
Syl ow s ubgroups and Syl ow the orems. Appli cati ons, groupsof s mall order. [ 8]
Rings
Definition and examples of rings (commutative, with 1). Ideals,hom*omorphisms, quotient rings, iso-morphism theorems. Prime andmaximal ideals. Fields. The characteristic of a field. Field offractionsof an integral domain.
Factorization in rings; units, primes and irreducibles. Uniquefactorization in principal ideal domains,and in polynomial rings.Gauss’ Lemma and Eisenstein’s irreducibility criterion.
Rings Z[α] of algebraic integers as subsetsofC and quotients ofZ[x]. Examples ofEuclidean domainsand uniqueness and non-uniqueness offactorization. Factorization in the ring of Gaussianintegers;representation of integers as sums of two squares.
Ideals in polynomial rings. Hilbert basis theorem. [10]
Modules
Definitions, examples of vector spaces, abelian groups andvector spaces with an endomorphism. Sub-
modules, hom*omorphisms, quotient modules and direct sums.Equivalence of matrices, canonical form.Structure of finitelygenerated modules over Euclidean domains, applications to abeliangroups andJordan normal form. [6]
Appropriate books
P.M.Cohn Classic Algebra. Wiley, 2000 (£29.95paperback)P.J. Cameron Introduction to Algebra. OUP(£27 paperback)J.B. FraleighA First Course in AbstractAlgebra. Addison Wesley, 2003 (£47.99 paperback)B. Hartleyand T.O. HawkesRings, Modules and Linear Algebra: a furthercourse in algebra. Chapman
and Hall, 1970 (out of print)I. Herstein Topics inAlgebra. John Wiley and Sons, 1975 (£45.99 hardback)P.M.Neumann, G.A. Stoy and E.C. Thomson Groups and Geometry.OUP 1994 (£35.99 paperback)M. Artin Algebra.Prentice Hall, 1991 (£53.99 hardback)
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PART IB 15
ANALYSIS II 24 lectures, Michaelmas term
Uniform convergence
The general principle of uniform convergence. A uniform limit ofcontinuous functions is continuous.Uniform convergence and termwiseintegration and differentiation of series of real-valuedfunctions.Local uniform convergence of power series. [3]
Uniform continuity and integrationContinuous functions on closedbounded intervals are uniformly continuous. Review of basic factsonRiemann integration (from Analysis I). Informal discussion ofintegration of complex-valued and Rn-
valued functions of one variable; proof that∥∫ba
f(x) dx∥ ≤ ∫ba∥f(x)∥ dx. [2]
Rn as a normed spaceDefinition of a normed space. Examples,including the Euclidean norm on Rn and the uniformnormonC[a, b]. Lipschitz mappings and Lipschitz equivalenceof norms. The Bolzano–Weierstrass theoremin Rn.Completeness. Open and closed sets. Continuity for functionsbetween normed spaces. Acontinuous function on a closed bounded setin Rn is uniformly continuous and has closed boundedi mage .All norms on a fini te -dimensi onal s pace are Lipsc hi tz equivale nt. [ 5]
Differentiation from Rm to Rn
Definition of derivative as a linear map; elementary properties,the chain rule. Partial derivatives; con-tinuous partialderivatives imply differentiability. Higher-order derivatives;symmetry of mixed partialderivatives (assumed continuous). Taylor’stheorem. The mean value inequality. Path-connectedness
for subsets ofRn; a function having zero derivative on apath-connected open subset is constant. [6]
Metric spacesDefinition and examples. *Metrics used inGeometry*. Limits, continuity, balls, neighbourhoods, openandclosed sets. [4]
The Contraction Mapping TheoremThe contraction mapping theorem.Applications including the inverse function theorem (proof ofcon-tinuity of inverse function, statement of differentiability).Picard’s solution of differential equations.
[4]
Appropriate books
† J.C. Burkill and H. Burkill A Second Course inMathematical Analysis. Cambridge University Press2002 (£29.95paperback).
A.F. Beardon Limits: A New Approach to Real Analysis.Springer 1997 (£22.50 hardback)D.J.H.Garling A Coursein Mathematical Analysis (Vol 3). Cambridge University Press2014 (£30
paperback)† W. RudinPrinciples of Mathematical Analysis.McGraw–Hill 1976 (£35.99 paperback).
W.A. Sutherland Introduction to Metric and TopologicalSpaces. Clarendon 1975 (£21.00 paperback)A.J. WhiteRealAnalysis: An Introduction.Addison–Wesley 1968 (out ofprint)T.W. KörnerA companion to analysis. AMS, 2004()
ME TRI C A ND T OP OLO GI CA L SPACES 12 l ec tu re s, Ea st erte rm
Metrics
Definition and examples. Limits and continuity. Open sets andneighbourhoods. Characterizing limitsand continuity usingneighbourhoods and open sets. [3]
Topology
Definition of a topology. Metric topologies. Further examples.Neighbourhoods, closed sets, conver-gence and continuity. Hausdorffspaces. Homeomorphisms. Topological and non-topologicalproperties.Completeness. Subspace, quotient and product topologies.[3]
Connectedness
Definition using open sets and integer-valued functions.Examples, including intervals. Components.The continuous image of aconnected space is connected. Path-connectedness. Path-connectedspacesare connected but not conversely. Connected open sets inEuclidean space are path-connected. [3]
Compactness
Definition using open covers. Examples: finite sets and [0, 1].Closed subsets of compact spaces arecompact. Compact subsets of aHausdorff space must be closed. The compact subsets of the realline.Continuous images of compact sets are compact. Quotientspaces. Continuous real-valued functions ona compact space arebounded and attain their bounds. The product of two compact spacesis compact.T he co mpa ct s ub set s o f E uc li dea n sp ac e. Seq uent ial co mp act ne ss. [3 ]
Appropriate books
† W.A. Sutherland Introduction to metric and topologicalspaces. Clarendon 1975 (£21.00 paperback).D.J.H.GarlingA Course in Mathematical Analysis (Vol 2). CambridgeUniversity Press 2013 (Septem-
ber) (£30 paperback)A.J. WhiteReal analysis: anintroduction. Addison-Wesley 1968 (out of print)B.MendelsonIntroduction to Topology. Dover, 1990 (£5.27paperback)
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PART IB 16
COMPLEX ANALYSIS 16 lectures, Lent term
Analytic functions
Complex differentiation and the Cauchy-Riemann equations.Examples. Conformal mappings. Informaldiscussion of branch points,examples of log z and z c. [3]
Contour integration and Cauchy’s theorem
Contour integration (for piecewise continuously differentiablecurves). Statement and proof of Cauchy’stheorem for star domains.Cauchy’s integral formula, maximum modulus theorem, Liouville’stheorem,fundamental theorem of algebra. Morera’s theorem. [5]
Expansions and singularities
Uniform convergence of analytic functions; local uniformconvergence. Differentiability of a powerseries. Taylor and Laurentexpansions. Principle of isolated zeros. Residue at an isolatedsingularity.Classification of isolated singularities. [4]
The residue theorem
Winding numbers. Residue theorem. Jordan’s lemma. Evaluation ofdefinite integrals by contourintegration. Rouch́e’s theorem,principle of the argument. Open mapping theorem. [4]
Appropriate books
L.V. AhlforsComplex Analysis. McGraw–Hill 1978(£30.00 hardback)† A.F. Beardon Complex Analysis. Wiley(out of print).
D.J.H.Garling A Course in Mathematical Analysis (Vol 3). Cambridge University Press 2014 (£30paperback)
† H.A. Priestley Introduction to Complex Analysis.Oxford University Press 2003 (£19.95 paperback).I. Stewartand D. Tall Complex Analysis. Cambridge UniversityPress 1983 (£27.00 paperback)
COMPLEX METHODS 16 lectures, Lent term
Analytic functions
Definition of an analytic function. Cauchy-Riemann equations.Analytic functions as conformal map-pings; examples. Application tothe solutions of Laplace’s equation in various domains. Discussionoflog z and za. [5]
Contour integration and Cauchy’s Theorem[Proofs of theorems inthis section will not be examined in this course. ]Contours,contour integrals. Cauchy’s theorem and Cauchy’s integral formula.Taylor and Laurentseries. Zeros, poles and essential singularities.[3]
Residue calculusResidue theorem, calculus of residues. Jordan’slemma. Evaluation of definite integrals by contourintegration.[4]
Fourier and Laplace transforms
Laplace transform: definition and basic prop erties; inversiontheorem (proof not r equired); convolutiontheorem. Examples ofinversion of Fourier and Laplace transforms by contour integration.Applicationsto differential equations. [4]
Appropriate books
M.J. Ablowitz and A.S. Fokas Complex variables:introduction and applications. CUP 2003 (£65.00)
G. Arfken and H. Weber Mathematical Methods forPhysicists. Harcourt Academic 2001 (£38.95 pa-perback)
G. J. O. Jameson A First Course in Complex Functions.Chapman and Hall 1970 (out of print)T. NeedhamVisualcomplex analysis. Clarendon 1998 (£28.50 paperback)
† H.A. Priestley Introduction to Complex Analysis.Clarendon 1990 (out of print).† I. Stewart and D. Tall Complex Analysis (the hitchhiker’s guide to the plane).Cambridge University
Press 1983 (£27.00 paperback).
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PART IB 17
GEOMETRY 16 lectures, Lent term
Parts of Analysis II will be found useful for this course.
Groups of rigid motions of Euclidean space. Rotation andreflection groups in two and three dimensions.Lengths of curves.[2]
Spherical geometry: spherical lines, spherical triangles and theGauss-Bonnet theorem. Stereographicprojection and Möbiustransformations. [3]
Triangulations of the sphere and the torus, Euler number.[1]
Riemannian metrics on open subsets of the plane. The hyperbolicplane. Poincaré models and theirmetrics. The isometry group. Hyperbolic triangles and the Gauss-Bonnet theorem. Thehyperboloidmodel. [4]
Embedded surfaces in R3. The first fundamental form.Length and area. Examples. [1]
Length and energy. Geodesics for general Riemannian metrics asstationary p oints of the energy.First variation of the energy andgeodesics as solutions of the corresponding Euler-Lagrangeequations.Geodesic polar coordinates (informal proof of existence).Surfaces of revolution. [2]
The second fundamental form and Gaussian curvature. For metricsof the form du2 +G(u, v)dv2,expression of thecurvature as
√Guu/
√G. Abstract smooth surfaces and isometries. Eulernumbers
a nd st at eme nt o f G au ss- Bo nn et t heo rem , e xa mp lesa nd ap pl ic at ion s. [3 ]
Appropriate books
† P.M.H. Wilson Curved Spaces. CUP, January 2008(£60 hardback, £24.99 paperback).
M. Do Carmo Differential Geometry of Curves and Surfaces. Prentice-Hall, Inc., Englewood Cliffs,N.J., 1976 (£42.99hardback)
A. PressleyElementary Differential Geometry.Springer Undergraduate Mathematics Series, Springer-VerlagLondon Ltd., 2001 (£19.00 paperback)
E. ReesNotes on Geometry. Springer, 1983 (£18.50paperback)M. Reid and B. Szendroi Geometry and Topology.CUP, 2005 (£24.99 paperback)
VARIATIONAL PRINCIPLES 12 lectures, Easter Term
Stationary points for functions on Rn. Necessary andsufficient conditions for minima and maxima.Importance ofconvexity. Variational problems with constraints; method ofLagrange multipliers. TheLegendre Transform; need for convexity toensure invertibility; illustrations from thermodynamics.
[4]
The idea of a functional and a functional derivative. Firstvariation for functionals, Euler-Lagrangeequations, for bothordinary and partial differential equations. Use of Lagrangemultipliers and multi-plier functions. [3]
Fermat’s principle; geodesics; least action principles,Lagrange’s and Hamilton’s equations for particlesand fields.Noether theorems and first integrals, including two forms ofNoether’s theorem for ordinarydifferential equations (energy andmomentum, for example). Interpretation in terms ofconservationlaws. [3]
S eco nd var ia tio n fo r fu nc tio na ls; as so cia te d e igenval ue p rob lem . [2 ]
Appropriate books
D.S. LemonsPerfect Form. Princeton Unversity Press1997 (£13.16 paperback)C. LanczosThe Variational Principlesof Mechanics. Dover 1986 (£9.47 paperback)R. Weinstock Calculus of Variations with applications to physics andengineering. Dover 1974 (£7.57
paperback)I.M. Gelfand and S.V. Fomin Calculus ofVariations. Dover 2000 (£5.20 paperback)W. Yourgrau and S.Mandelstam Variational Principles in Dynamics and QuantumTheory. Dover
2007 (£6.23 paperback)S. Hildebrandt and A. Tromba Mathematics and Optimal Form. Scientific American Library1985
(£16.66 paperback)
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PART IB 18
METHODS 24 lectures, Michaelmas term
Self-adjoint ODEs
Periodic functions. Fourier series: definition and simpleproperties; Parseval’s theorem. Equationsof second order.Self-adjoint differential operators. The Sturm-Liouville equation;eigenfunctions andeigenvalues; reality of eigenvalues andorthogonality of eigenfunctions; eigenfunction expansions(Fourierseries as prototype), approximation in mean square,statement of completeness. [5]
PDEs on bounded domains: separation of variablesPhysical basisof Laplace’s equation, the wave equation and the diffusionequation. General methodof separation of variables in Cartesian,cylindrical and spherical coordinates. Legendre’sequation:derivation, solutions including explicit formsofP0, P1 and P2,orthogonality. Bessel’s equation ofinteger order as anexample of a self-adjoint eigenvalue problem with non-trivialweight.
Examples including potentials on rectangular and circulardomains and on a spherical domain (axisym-metric case only), waveson a finite string and heat flow down a semi-infinite rod. [5]
Inhom*ogeneous ODEs: Green’s functions
Properties of the Dirac delta function. Initial value problemsand forced problems with two fixed endpoints; solution usingGreen’s functions. Eigenfunction expansions of the delta functionand Green’sfunctions. [4]
Fourier transforms
Fourier transforms: definition and simple properties; inversionand convolution theorems. The discrete
Fourier transform. Examples of application to linear systems.Relationship of transfer function toGreen’s function for initialvalue problems. [4]
PDEs on unbounded domainsClassification of PDEs in twoindependent variables. Well posedness. Solution by the methodofcharacteristics. Green’s functions for PDEs in 1, 2 and 3independent variables; fundamental solutionsof the wave equation,Laplace’s equation and the diffusion equation. The method ofimages. Applicationto the forced wave equation, Poisson’s equationand forced diffusion equation. Transient solutionsofdiffusion problems: the error function. [6]
Appropriate books
G. Arfken and H.J. WeberMathematical Methods forPhysicists. Academic 2005 (£39.99 paperback)M.L.BoasMathematical Methods in the Physical Sciences.Wiley 2005 (£36.95 hardback)J. Mathews and R.L. Walker Mathematical Methods of Physics. Benjamin/Cummings 1970(£68.99
hardback)K. F. Riley, M. P. Hobson, and S.J.BenceMathematical Methods for Physics and Engineering:a
comprehensive guide. Cambridge University Press 2002(£35.00 paperback)Erwin Kreyszig Advanced EngineeringMathematics. Wiley ()
QUANTUM MECHANICS 16 lectures, Michaelmas term
Physical background
Photoelectric effect. Electrons in atoms and line spectra.Particle diffraction. [1]
Schrödinger equation and solutionsDe Broglie waves.Schrödinger equation. Superposition principle. Probabilityinterpretation, density
and current. [2]Stationary states. Free particle, Gaussian wavepacket. Motion in 1-dimensional potentials, parity.Potential step,square well and barr ier . Harmonic oscillator. [4]
Observables and expectation valuesPosition and momentumoperators and expectation values. Canonical commutation relations.Uncer-tainty principle. [2]
Observables and Hermitian operators. Eigenvalues andeigenfunctions. Formula for expectation value.[2]
Hydrogen atom
Spherically symmetric wave functions for spherical well andhydrogen atom.
O rbital angul ar momentum ope rators . General s ol ution ofhydroge n atom. [ 5]
Appropriate books
Feynman, Leighton and Sandsvol. 3 Ch 1-3 of the Feynmanlectures on Physics. Addison-Wesley 1970(£87.99paperback)
† S. GasiorowiczQuantum Physics. Wiley 2003 (£34.95hardback).P.V. Landshoff, A.J.F. Metherell and W.G Rees Essential Quantum Physics. Cambridge University
Press 1997 (£21.95 paperback)† A.I.M. RaeQuantumMechanics. Institute of Physics Publishing 2002 (£16.99paperback).
L.I. SchiffQuantum Mechanics. McGraw Hill 1968(£38.99 hardback)
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PART IB 19
ELECTROMAGNETISM 16 lectures, Lent term
Electromagnetism and Relativity
Review of Special Relativity; tensors and index notation.Lorentz force law. Electromagnetic ten-sor. Lorentz transformationsof electric and magnetic fields. Currents and the conservation ofcharge.Maxwell equations in relativistic and non-relativist ic forms. [5]
ElectrostaticsGauss’s law. Application to spherically symmetricand cylindrically symmetric charge distributions.Point, line andsurface charges. Electrostatic potentials; general chargedistributions, dipoles. Electro-static energy. Conductors. [3]
Magnetostatics
Magnetic fields due to steady currents. Ampre’s law. Simpleexamples. Vector potentials and the Biot-Savart law for generalcurrent distributions. Magnetic dipoles. Lorentz force on currentdistributionsand force between current-carrying wires. Ohm’s law.[3]
Electrodynamics
Faraday’s law of induction for fixed and moving circuits.Electromagnetic energy and Poynting vector.4-vector potential,gauge transformations. Plane electromagnetic waves in vacuum,polarization.
[5]
Appropriate books
W.N. Cottingham and D.A. Greenwood Electricity andMagnetism. Cambridge University Press 1991(£17.95paperback)
R. Feynman, R. Leighton and M. Sands The Feynman Lectureson Physics, Vol 2. Addison–Wesley1970 (£87.99 paperback)
† P. Lorrain and D. Corson Electromagnetism, Principlesand Applications. Freeman 1990 (£47.99 pa-perback).
J.R. Reitz, F.J. Milford and R.W. Christy Foundations ofElectromagnetic Theory. Addison-Wesley1993 (£46.99hardback)
D.J. GriffithsIntroduction to Electrodynamics.Prentice–Hall 1999 (£42.99 paperback)
FLUID DYNAMICS 16 lectures, Lent term
Parallel viscous flow
Plane Couette flow, dynamic viscosity. Momentum equation andboundary conditions. Steady flowsincluding Poiseuille flow in achannel. Unsteady flows, kinematic viscosity, brief description ofviscousboundary layers (skin depth). [3]
KinematicsMaterial time derivative. Conservation of mass and thekinematic boundary condition. Incompressibil-i ty; s treamf unctionf or two-dimensi onal flow. Streaml ines and path l ines . [ 2]
Dynamics
Statement of Navier-Stokes momentum equation. Reynolds number.Stagnation-point flow; discussionof viscous boundary layer andpressure field. Conservation of momentum; Euler momentumequation.Bernoulli’s equation.
Vorticity, vorticity equation, vortex line stretching,irrotational flow remains irrotational. [4]
Potential flowsVelocity potential; Laplace’s equation, examplesof solutions in spherical and cylindrical geometry byseparation ofvariables. Translating sphere. Lift on a cylinder withcirculation.
Expression for pressure in time-dependent potential flows withpotential forces. Oscillations in amanometer and of a bubble.[3]
Geophysical flowsLinear water waves: dispersion relation, deepand shallow water, standing waves in a container,Rayleigh-Taylorinstability.
Euler equations in a rotating frame. Steady geostrophic flow,pressure as streamfunction. Motion in ashallow layer, hydrostaticassumption, modified continuity equation. Conservation of potentialvorticity,Rossby radius of deformation. [4]
Appropriate books
† D.J. Acheson Elementary Fluid Dynamics. OxfordUniversity Press 1990 (£40.00 paperback).G.K. Batchelor AnIntroduction to Fluid Dynamics. Cambridge University Press2000 (£36.00 paper-
back)G.M. Homsey et al. Multi-Media Fluid Mechanics. Cambridge University Press 2008 (CD-ROM for
Windows or Macintosh, £16.99)M. van Dyke An Album ofFluid Motion. Parabolic Press (out of print, but availablevia US Amazon)M.G. Worster Understanding Fluid Flow.Cambridge University Press 2009 (£15.59 paperback)
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PART IB 20
NUMERICAL ANALYSIS 16 lectures, Lent term
Polynomial approximation
Interpolation by polynomials. Divided differences of functionsand relations to derivatives. Orthogonalpolynomials and theirrecurrence relations. Least squares approximation by polynomials.Gaussianquadratur e formulae. Peano kernel theorem andapplications. [6]
Computation of ordinary differential equationsEuler’s method andproof of convergence. Multistep methods, including order, the rootcondition andthe concept of convergence. Runge-Kutta schemes. Stiffequations and A-stability. [5]
Systems of equations and least squares calculations
LU triangular factorization of matrices. Relation to Gaussianelimination. Column pivoting. Fac-torizations of symmetric and bandmatrices. The Newton-Raphson method for systems ofnon-linearalgebraic equations. QR factorization of rectangularmatrices by Gram–Schmidt, Givens and House-h old er t ech ni qu es.A pp li ca tio n t o l in ea r le ast sq uar es ca lc ula tio ns.[5 ]
Appropriate books
† S.D. Conte and C. de Boor Elementary NumericalAnalysis: an algorithmic approach. McGraw–Hill1980 (out ofprint).
G.H. Golub and C. Van Loan Matrix Computations. Johns Hopkins University Press 1996 (out ofprint)
A IserlesA first course in the Numerical Analysis ofDifferential Equations. CUP 2009 (£33, paperback)Endri Suliand David F Meyers An introduction to numerical analysis. Cambridge University Press
2003 (£34, paperback)Anthony Ralston and PhilipRabinowitzA first course in numerical analysis. Dover2001 (£21, paper-
back)M.J.D. Powell Approximation Theory and Methods.Cambridge University Press 1981 (£25.95 paper-
back)
STATISTICS 16 lectures, Lent term
Estimation
Review of distribution and density functions, parametricfamilies. Examples: binomial, Poisson, gamma.Sufficiency, minimalsufficiency, the Rao–Blackwell theorem. Maximum likelihoodestimation. Confi-d enc e int er va ls. U se o f p ri or d ist ribu tio ns a nd B ayesi an i nfe ren ce . [5 ]
Hypothesis testingSimple examples of hypothesis testing, nulland alternative hypothesis, critical region, size, power, typeI andtype II errors, Neyman–Pearson lemma. Significance level ofoutcome. Uniformly most powerfultests. Likelihood ratio, and use ofgeneralised likelihood ratio to construct test statistics forcompositehypotheses. Examples, includingt-testsandF-tests. Relationship with confidence intervals.Goodness-of-fit tests and contingency tables. [4]
Linear models
Derivation and joint distribution of maximum likelihoodestimators, least squares, Gauss-Markov the-orem. Testinghypotheses, geometric interpretation. Examples, including simplelinear regression andone-way analysis of variance.∗Use ofsoftware∗. [7]
Appropriate books
D.A. Berry and B.W. Lindgren Statistics, Theory andMethods. Wadsworth 1995 ()G. Casella and J.O. BergerStatistical Inference. Duxbury 2001 ()M.H. DeGroot andM.J. Schervish Probability and Statistics. PearsonEducation 2001 ()
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PART IB 21
MARKOV CHAINS 12 lectures, Michaelmas term
Discrete-time chains
Definition and basic properties, the transition matrix.Calculation ofn-step transitionprobabilities.Communicating classes, closed classes, absorption,irreducibility. Calculation of hitting probabilitiesand meanhitting times; survival probability for birth and death chains.Stopping times and statementof the strong Markov property. [5]
Recurrence and transience; equivalence of transience andsummability ofn-step transitionprobabilities;equivalence of recurrence and certainty of return.Recurrence as a class property, relation with closedclasses. Simplerandom walks in dimensions one, two and three. [3]
Invariant distributions, statement of existence and uniquenessup to constant multiples. Mean returntime, positive recurrence;equivalence of positive recurrence and the existence of aninvariant distri-bution. Convergence to equilibrium forirreducible, positive recurrent, aperiodic chains *and proofbycoupling*. Long-run proportion of time spent in given state.[3]
Time reve rs al , detai le d bal ance , reve rs ibil ity; randomwalk on a graph. [ 1]
Appropriate books
G.R. Grimmett and D.R. Stirzaker Probability and RandomProcesses. OUP 2001 (£29.95 paperback)J.R. NorrisMarkovChains. Cambridge University Press 1997 (£20.95paperback)
Grimmett, G. and Welsh, D. Probability, An Introduction. Oxford University Press, 2nd edition, 2014(£33.99paperback)
OPTIMISATION 12 lectures, Easter term
Lagrangian methods
General formulation of constrained problems; the Lagrangiansufficiency theorem. Interpretation ofLagrange multipliers asshadow prices. Examples. [2]
Linear programming in the nondegenerate case
Convexity of feasible region; sufficiency of extreme points.Standardization of problems, slack variables,equivalence of extremepoints and basic solutions. The p
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