Fourth Year Pure Mathematics 2007 Handbook8 1. THE STRUCTURE OF PURE MATHEMATICS FOUR 1.4. Pure Mathematics 3(A) Courses for 2007 SEMESTER I SEMESTER II Algebra (3962) Complex Analysis

Fourth Year Pure Mathematics

2007 Handbook

School of Mathematics and StatisticsUniversity of Sydney

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www.maths.usyd.edu.au/u/UG/HM/

CONTENTS

Chapter 1. The structure of pure mathematics four. . . . . . . . . . . . . 51.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 51.2. The lecture courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 61.3. Pure Mathematics 4/PG Courses for 2007 . . . . . . . . . . . . . . .. . . 71.4. Pure Mathematics 3(A) Courses for 2007. . . . . . . . . . . . . . .. . . . 81.5. Tailor-made courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 81.6. The essay project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 81.7. The talk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 91.8. Mathematics in other languages . . . . . . . . . . . . . . . . . . . .. . . . . . . 9

Chapter 2. Entry, administration and assessment. . . . . . . . . . . . . . . 102.1. Entry Requirements for Pure Mathematics 4 . . . . . . . . . . . .. . . . 102.2. Actions to be taken. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 102.3. Administrative arrangements . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 112.4. Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 112.5. Honours grades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 122.6. School Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 132.7. Scholarships, Prizes and Awards. . . . . . . . . . . . . . . . . . .. . . . . . . . 14

Chapter 3. Course descriptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .163.1. Fourth Year Courses — Semester I . . . . . . . . . . . . . . . . . . . . .. . . 163.2. Fourth Year Courses — Semester II . . . . . . . . . . . . . . . . . . . .. . . . 19

Chapter 4. The essay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .214.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 214.2. Choosing a supervisor and topic . . . . . . . . . . . . . . . . . . . . .. . . . . . 214.3. Essay content and format . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 224.4. Submission of essay, assessment, corrections . . . . . . .. . . . . . . . 234.5. Time management and progress reports . . . . . . . . . . . . . . .. . . . . 234.6. Your supervisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 244.7. Use of the library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 24

Chapter 5. Sample essay topics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .265.1. Algebra Research Group . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 265.2. Analysis Research Group . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 34

3

4 CONTENTS

5.3. Computational Algebra Research Group . . . . . . . . . . . . . . . .. . . 345.4. Geometry Research Group . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 355.5. Number Theory Research Group . . . . . . . . . . . . . . . . . . . . . . .. . . 375.6. Non-Linear Analysis Research Group . . . . . . . . . . . . . . . . .. . . . . 42

Chapter 6. The talk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .436.1. General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 436.2. Preparing the talk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 436.3. Overhead projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 44

Chapter 7. Your future and mathematics. . . . . . . . . . . . . . . . . . . . . . .457.1. The colloquium and other seminars . . . . . . . . . . . . . . . . . .. . . . . . 457.2. After fourth year, what then? . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 467.3. Higher degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 467.4. Scholarships and other support . . . . . . . . . . . . . . . . . . . .. . . . . . . . 467.5. Further study in another subject . . . . . . . . . . . . . . . . . . .. . . . . . . . 47

Appendix A. Instructions on preparing the Manuscript . . . . . . . . . 48

Mathematicians are like Frenchmen: whatever you say tothem they translate into their own language and forthwith

it is something completely different.Goethe,Maximen und Reflexionen

CHAPTER 1

The structure of pure mathematics four

1.1. Introduction

In linguistics it is increasingly believed that universal features of languageare reflections of the structure of the human brain and its perception ofthe world around us. In a similar fashion, mathematics is a universal lan-guage that has been developed to understand and describe hownature andlife work. Mathematics, both in structure and development,is inextricablybound to our attempts to understand the world around us and our percep-tions of that world. We see this in the mathematical descriptions and for-mulations of models in the theoretical and applied sciences: from physics,computer science and information theory on the one hand, to engineering,chemistry, operations research and economics on the other.

Just as remarkable is the way in which esoteric and abstract mathematicsfinds applications in the applied sciences. Indeed, one of the most excitingdevelopments in science over the past decade has been the re-emergence ofa dynamic interaction between pure mathematicians and applied scientists,which is bringing together several decades of the relatively abstract andseparate development of pure mathematics and the sciences.Examples in-clude the applications of singularity theory and group theory to symmetry-breaking and bifurcation in engineering; number theory to cryptography;category theory and combinatorics to theoretical and computational com-puter science; and, most spectacular of all, the recent developments of gen-eral field theories in mathematical physics based on the mostprofound workin complex analysis and algebraic geometry. Of course, thisinteraction isnot one way. For example, there is the recent discovery of “exotic” differ-ential structures onR4 utilising ideas from Yang-Mills theory.

There are many valid approaches to the study of Pure Mathematics in thefinal Honours Year. Thus, the course may be regarded as usefulin its ownright, or may lead on to an M. Sc or Ph.D. or to a teaching position inUniversity or High School. In another direction, what want asolid basefrom which to continue with studies in computer science or physics, for

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6 1. THE STRUCTURE OF PURE MATHEMATICS FOUR

example. Finally, you may intend to seek employment with theCSIRO orin the operations research field, or in a financial institution. In the lattercircumstances, one well-known advantage of studying mathematics is thatmathematics gives training in a particular way of thinking and an analyticalapproach to problem solving. Mathematicians are highly adaptable (andemployable).

The Fourth Year Honours program in Pure Mathematics caters for the vari-ous needs described above by offering a highly flexible and adaptable pro-gram, which is both interesting and challenging. We offer a combinationof core courses, which introduce the major areas of mathematics, togetherwith a smorgasbord of deeper courses courses which can be arranged to suityour personal requirements.

In brief, the Fourth Year course comprises the equivalent ofseven lec-ture courses, together with an essay project (the equivalent of three lecturecourses) and a 40-45 minute talk on the essay project.

A description of the various components of the course is given below. Fordetailed descriptions of the courses, the essay project, etc., see the appro-priate chapter in this Handbook.

1.2. The lecture courses

Students are required to be assessed on 7 units of approved lecture courses(or equivalent - see below).

In 2007, the courses may be chosen from:

a) three PM4 core courses, each worth11

2units;

b) five other PM4 courses, each worth 1 unit.(These may presume some knowledge of one or more of the corecourses.)

c) six third yearadvancedcourses, each worth 1 unit.(Students in Pure Mathematics 4 may take any 3(A) course whichthey have not previously taken.)

d) Approved substitutions (up to the value of 2 units) by courses givenby other Departments. (See §1.5 below.)

e) Reading courses arranged with staff members (after consultationwith the PM4 coordinator).

1.3. PURE MATHEMATICS 4/PG COURSES FOR2007 7

Read carefully the guidelines in §1.5 below.

In addition, we are negotiating with UNSW over the possibility of sharingsome PM4 Courses. (The likely UNSW courses include “Graph Theory"and “Banach Algebras"; the potential difficulties are in timetabling.)

Overall, the lecture courses offered at the level of PM4 and above are in-tended to introduce students to the major divisions of modern mathematicsand provide a knowledge of some of the main ideas needed for the under-standing of much of contemporary mathematics, while still reflecting theresearch interests within the pure mathematics research groups.

The “core" of Fourth Year is considered to include Commutative Algebra,Functional Analysis and Algebraic Topology. Each of these has 3 lecturesper week with no tutorial and counts as11

2units. Students are strongly

advised to take all of the core courses.

1.3. Pure Mathematics 4/PG Courses for 2007

SEMESTERI

Algebraic Topology PaunescuCommutative Algebra Lehrer (weeks 1-6)Functional Analysis Dancer

SEMESTERII

Algebraic Geometry LaiCommutative Algebra Lehrer (weeks 1-7)Partial Differential Equations DanersRepresentations of the Symmetric Groups Molev

If you are unsure about the combination of courses you shouldtake, consultwith your supervisor or the course coordinator. In any case,you are verywelcome to attend all the lecture courses.

8 1. THE STRUCTURE OF PURE MATHEMATICS FOUR

1.4. Pure Mathematics 3(A) Courses for 2007

SEMESTERI SEMESTERII

Algebra (3962) Complex Analysis (3964)Metric Spaces (3961) Differential Geometry (3968)

Lebesgue/Fourier Analysis (3969)Representation Theory (3966)

1.5. Tailor-made courses

Students may also take certain courses of an essentially mathematical na-ture other than these (for instance, in Applied Mathematics, Econometrics,Formal Logic, Statistics, etc.)with the approval of the Course Coordi-nator. (Normally we require that at least five of the seven course units bechosen from the Pure Mathematics options). Details of the Applied Math-ematics 4 and Mathematical Statistics 4 options may be obtained from theCoordinators, Dr Peter W Buchen (Applied Maths - tel. 9351-2965, e-mail:[email protected])) and Dr Shelton Peiris (Math. Stats. - tel. 9351-5764, e-mail:[email protected])).

A number of staff are usually willing to supervise a reading course in theirparticular area of interest. Consult the course coordinatorif you have aspecial topic in mind that might be acceptable as a reading course. Readingcourses are generally a matter between the student and a willing member ofthe department, subject to the approval of the course coordinator.

If you wish to do areading coursein Pure Mathematics, or substitute acourse from outside Pure Mathematics, you should ask the lecturer to pre-pare asummary of the course and description of theassessment. Thisshould then be submitted to the course coordinator for approval. Providedthe guidelines sketched above are followed, and a satisfactory balance over-all in the Fourth year courses is maintained, approval will normally begranted.

1.6. The essay project

The essay project counts as the equivalent of three PM4 units. Work onthe essay project proceeds throughout the year and the finished essay issubmitted near the end of the second semester. Note that it isalso possible

1.8. MATHEMATICS IN OTHER LANGUAGES 9

for the project to be supervised by a member of another department (orjointly).

1.7. The talk

As part of the essay project, students are required to give a talk about theirproject. Talks are normally scheduled to take place in September.

1.8. Mathematics in other languages

Ability to read mathematics in at least one approved foreignlanguage isno longer a requirement for Pure Mathematics 4. An increasing proportionof mathematical papers are now written in English, there arestill a signif-icant number of important mathematical works not written inEnglish. Inaddition, many older mathematical works written in other languages havenot yet been translated into English. For these reasons, students who areseriously thinking of pursuing further studies in mathematics are stronglyencouraged to acquire a reading knowledge of mathematics inat least oneforeign language. Indeed, such knowledge is often a requirement of M. Sc.and Ph.D. courses in mathematics (especially in the U. S. A.).

In particular, a working knowledge of mathematical French is extremelyuseful (and relatively easy to acquire). At present, Russianis less useful, asthe collapse of the Soviet system has obliged many of the bestmathemati-cians from the former Soviet Union to work in the West, and in any casethere are English translations of most of the Russian language mathematicsjournals. (However, the Carslaw library still subscribes tothe Russian orig-inals in most cases, as they are much cheaper than the translations). On theother hand German may again become the preferred language ofpublica-tion for mathematicians in an united Germany. Although the mathematicalschools of China and Japan are large and increasingly important acquiring areading knowledge of the languages of these countries is difficult and time-consuming for most adults.

The Departments of French and German offer reading courses that enableone to acquire a working reading knowledge of French and German. Thesecourses begin in the first semester and participants must register with therelevant Department before the start of the course.

I fear that I bore you with these details, but I have to letyou see my little difficulties, if you are to understand the

situation.Sherlock Holmes,A Scandal in Bohemia

CHAPTER 2

Entry, administration and assessment

2.1. Entry Requirements for Pure Mathematics 4

Students who have fulfilled the requirements of the faculty in which theyare enrolled and satisfied conditions (a), and (b) or (c) below are eligible toenrol in Pure Mathematics 4:

a) taken 24 credit points of third year mathematics units (see the seniorpure and applied mathematics handbook) with at least 16 of thesecredit points in pure mathematics;

b) obtained a distinction average or better in 24 credit points of thirdyear mathematics units

or,c) obtained a credit average in 24 credit points of third yearmathe-

matics units,includinga credit in at least one Pure Mathematics 3Advancedunit.

Entry to PM4 is also subject to the approval of the Head of School

Note Since we advise all PM4 students to take the core courses (commuta-tive algebra, algebraic topology and functional analysis), the natural prereq-uisites for PM4 are: Metric Spaces; Algebra 1; and Lebesgue and FourierAnalysis. Students without this background should expect to do some pre-liminary reading over the summer (see the course coordinator for advice ifnecessary).

2.2. Actions to be taken

All students intending to take Pure Mathematics 4 in 2007 should see thePM4 Course Coordinator, Dr Laurentiu Paunescu (Carslaw 816, tel. 9351-2969, e-mail:[email protected]) at their earliest opportunity, andin any case well before the beginning of the new teaching year. The Course

10

2.4. ASSESSMENT 11

Coordinator will advise you about choosing a supervisor and atopic for theessay project (see also section 4.2).

2.3. Administrative arrangements

The Course Coordinator is in charge of Pure Mathematics 4 and should beconsulted about any organisational problems that may arise.

In particular, students should note that the Course Coordinator’s permissionshould be obtained if you wish to substitute courses from outside, or take areading course or a postgraduate course. In the first instance, however, youshould discuss such matters with your supervisor. Providedyou can agree,the Course Coordinator’s permission wouldnormallybe a formality.

Please take particular note of the procedure to be followed if you are sickor other circumstances arise that may lead to late submission of your essay(see §4.4). Also note that at the end of first semester a progress report mustbe given to the Course Coordinator (see Chapter 5).

When we know that you are enrolled for PM4 you will be given a com-puter account.The usual way in which messages for PM4 students will bedistributed will be via e-mail. Please remember to check your e-mail regu-larly. This will become second nature once you start to type up your essays.

2.4. Assessment

The possible results for Fourth year are First Class Honours,Second ClassHonours division 1, Second Class Honours Division 2, Third Class Honoursand No Award (Fail), usually abbreviated I, II-1, II-2, III and F. The last twoare rarely awarded.

Each Fourth Year course is assessed at a time and in a manner arrangedbetween the lecturer and the class. Usually, a written examination is heldduring the exam period immediately following the course; however, somecourses are assessed entirely by assignment. It is undesirable to have ex-aminations during term or to have many papers deferred to theend of theyear. Each PM3 advanced course is assessed in the usual way. Students areinformed about their performance as information becomes available.

Marks provided by the lecturers are scaled according to the lecturer’s judge-ment as to what constitutes a First Class, II-1, etc., performance on eachcourse. The marks from the best7/71

2units are counted towards the final

12 2. ENTRY, ADMINISTRATION AND ASSESSMENT

mark. The essay is equivalent to 3 PM4 units; it accounts for 30% of theyears assessment.

As well as assessing the Fourth Year performance, the Department is re-quired to make a recommendation for a grade of Honours based on theperformance in all subjects over the four years. In exceptional cases, thegrade of Honours awarded could differ from the level of performance in theFourth Year.

2.5. Honours grades

The Faculty of Science has given the following guidelines for assessmentof student performance in fourth year.

95–100 Outstanding First Class quality of clear Medal standard, demonstrat-ing independent thought throughout, a flair for the subject,compre-hensive knowledge of the subject area and a level of achievementsimilar to that expected by first rate academic journals. This markreflects an exceptional achievement with a high degree of initiativeand self-reliance, considerable student input into the direction of thestudy, and critical evaluation of the established work in the area.

90-94 Very high standard of work similar to above but overallperformanceis borderline for award of a Medal. Lower level of performance incertain categories or areas of study above.

Note: An honours mark of 90+ and a third year WAM of 80+ arenecessary but not sufficient conditions, for the award of theMedal.Examiners are referred to the Academic Board Guidelines on theaward of Medals found in the general policy pages at the frontofthe Examiners’ Manual.

80-89 Clear First Class quality, showing a command of the field both broadand deep, with the presentation of some novel insights. Student willhave shown a solid foundation of conceptual thought and a breadthof factual knowledge of the discipline, clear familiarity with andability to use central methodology and experimental practices of thediscipline, and clear evidence of some independence of thought inthe subject area. Some student input into the direction of the study ordevelopment of techniques, and critical discussion of the outcomes.

2.6. SCHOOL FACILITIES 13

75-79 Second class honours, first division - student will have shown acommand of the theory and practice of the discipline. They willhave demonstrated their ability to conduct work at an independentlevel and complete tasks in a timely manner, and have an adequateunderstanding of the background factual basis of the subject. Stu-dent shows some initiative but is more reliant on other people forideas and techniques and project is dependent on supervisor’s sug-gestions. Student is dedicated to work and capable of undertaking ahigher degree.

70-74 Second class honours, second division - student is proficient in thetheory and practice of their discipline but has not developed com-plete independence of thought, practical mastery or clarity of pre-sentation. Student shows adequate but limited understanding of thetopic and has largely followed the direction of the supervisor.

65-69 Third class honours - performance indicates that the student has suc-cessfully completed the work, but at a standard barely meeting hon-ours criteria. The student’s understanding of the topic is extremelylimited and they have shown little or no independence of thought orperformance.

The award of a medal isnot made just on the basis of a numerical mark orformula. The merits of each eligible candidate are debated by the Board ofExaminers of the relevant Faculty.

2.6. School Facilities

Pure Mathematics 4 students traditionally enjoy a number ofprivileges.These include:

• Office space and a desk in the Carslaw Building.• A computer account with access to e-mail and the World-Wide Web,

as well as TEX and laser printing facilities for the preparation ofessays and projects.

• A photocopying account paid by the School for essay/projectsourcematerial.

• After-hours access to the Carslaw Building. (A deposit is payable.)• A pigeon-hole in room 728 - please inspect it regularly as lecturers

often use it to hand out relevant material.• Participation in the School’s social events.

14 2. ENTRY, ADMINISTRATION AND ASSESSMENT

• Class representative at School meetings.

2.7. Scholarships, Prizes and Awards

The following scholarships and prizes may be awarded to PureMathemat-ics 4 students of sufficient merit. (Note that unless the conditions of theprize state otherwise, as in the David G.A.Jackson Prize andthe A.F.U.W.Prize, these prizes are also open to all Honours students in the School ofMathematics and Statistics.)

2.7.1. Joye Prize in Mathematics. Value: $5650

To the most outstanding student completing fourth year honours in theSchool of Mathematics and Statistics, $5650 plus medal and shield.

2.7.2. George Allen Scholarship in Pure Mathematics.Value:$400

To a student proceeding to Honours in Pure Mathematics who has showngreatest proficiency in at least 24 credit points of Senior units of study inthe School of Mathematics and Statistics.

2.7.3. Barker Prize. Value: $375

Awarded at the fourth (Honours) year examiner’s meetings for proficiencyin Pure Mathematics, Applied Mathematics or Mathematical Statistics.

2.7.4. Ashby Prize. Value: $250

Offered annually for the best essay, submitted by a student in the Faculty ofScience, that forms part of the requirements of Pure Mathematics 4, AppliedMathematics 4 or Mathematical Statistics 4.

2.7.5. Norbert Quirk Prize No IV. Value: $225

Awarded annually for the best essay on a given mathematical subject by astudent enrolled in a fourth year course in mathematics (Pure Mathematics,Applied Mathematics or Mathematical Statistics) providedthat the essay isof sufficient merit.

2.7. SCHOLARSHIPS, PRIZES AND AWARDS 15

2.7.6. David G.A.Jackson Prize. Value: $200

Awarded for creativity and originality in any undergraduate Pure Mathe-matics unit of study.

2.7.7. Australian Federation of University Women (NSW) Prize inMathematics. Value: $100

Awarded annually, on the recommendation of the Head of the School ofMathematics and Statistics, to the most distinguished woman candidate forthe degree of BA or BSc who graduates with first class Honours inPureMathematics, Applied Mathematics or Mathematical Statistics.

2.7.8. University Medal.

Awarded to Honours students who perform outstandingly. Theaward issubject to Faculty rules, which require a Faculty mark of 90 or more inPure Mathematics 4 and a Third year WAM of 80 or higher. More than onemedal may be awarded in any year.

I dislike arguments of any kind. They are always vulgar,and often convincing.

Oscar Wilde,The importance of being earnest

CHAPTER 3

Course descriptions

The three “core” Fourth Year courses (Commutative Algebra, AlgebraicTopology and Functional Analysis) are 3 lecture per week andcount 11

2

units. The other five courses offered in 2007 are 2 lecture perweek coursesand count as 1 unit. Some of the Semester II courses have one ofthe corecourses as a prerequisite. In addition, all 3(A) courses, not previously ex-amined, are be available for credit. Each 3(A) course is run at 3 lectures perweek (plus tutorial(s)) and counts as 1 unit. For substitutions by courses notgiven by Pure Mathematics see Section 1.3.

3.1. Fourth Year Courses — Semester I

Algebraic Topology — L. Paunescu

Algebraic topology has advanced more rapidly than any otherbranch ofmathematics during the twentieth century. Its influence on other branches,such as algebra, number theory, algebraic geometry, differential geometry,and analysis has been enormous.

The typical problems of topology such as whetherRm is homeomorphic to

Rn or whether the projective plane can be embedded inR

3 or whether wecan choose a continuous branch of the complex logarithm on the whole ofC\{0} may all be interpreted as asking whether there is a suitable contin-uous map. The goal of Algebraic Topology is to construct invariants bymeans of which such problems may be translated into algebraic terms. Thehomotopy groupsπn(X) and homology groupsHn(X) of a spaceX aretwo important families of such invariants. The homotopy groups are easyto define but in general are hard to compute; the converse holds for thehomology groups.

We begin with simplicial homology theory. Then we define singular ho-mology theory, and over several weeks develop the properties which aresummarized in the Eilenberg-Steenrod axioms. (These give an axiomatic

16

3.1. FOURTH YEAR COURSES— SEMESTERI 17

characterization of homology for reasonable spaces). We then apply ho-mology to various examples, and conclude this part of the course with twoor three lectures on cohomology and differential forms on open subsets ofR

n. The final third of the course shall be devoted to the fundamental group,its relationship with covering space theory and elementaryideas of combi-natorial group theory.

This course will be offered both as a 1-unit course (26 lectures on homol-ogy) and as a full11

2-unit core course.

Prerequisite: Metric spaces.

Assessment:3 assignments, containing 10 questions in all (total 20%);final exam (80%). (The assessment for the 1-unit version shall involve thefirst two assignments and part of the final exam).

References:Algebraic Topology: A First Courseby M. J. Greenberg andJ. Harper, Benjamin/Cummings (1981)Algebraic Topologyby Allan Hatcher, Cambridge UP (2002).

There are also about 70 pages of notes (by Dr Hillman) available throughthe School web site.

Commutative Algebra — G. Lehrer

General: One of the most significant mathematical innovations of the 20thcentury was the development of “context-free geometry”. The key idea ofthis is that the study of the simultaneous solutions of polynomial equationssuch asx4 + y4 + z4 − 5x2y2 = 0 may be carried out in a way whichis independent of the domain in which the variablesx, y, z, . . . lie. Forexample they may lie inR, C, Z or Fq, and prior to these developments thestudy of solutions in those domains would have been regardedas separatedisciplines. Thus new common ground now exists between geometry overvarious domains. There is also now a much better understanding of muchstudied concepts such as the “multiplicity” of a higher order intersection ofcurves or their higher dimensional analogues.

The foundations for these spectacular advances were laid largely by theParis school of Grothendieck in the 1950’s and 60’s buildingon the work ofmany mathematicians over many centuries, but most importantly on that ofHilbert in the 1920’s. Its basis is the abstraction of geometry by algebra, andthis course, on commutative algebra, is intended to be an introduction to the

18 3. COURSE DESCRIPTIONS

basic ideas of the subject. The course will cover the basic concepts of com-mutative algebra, and illustrate them by giving geometric interpretations asfar as possible.

Lectures: Note that in 2007 only, there will be 6 weeks of lectures in firstsemester (weeks 1-6 of first semester), and 8 weeks of lectures in secondsemester (weeks 1–7 of second semester).

Prerequisites: A thorough knowledge of linear algebra. The third yearadvanced algebra courses would be a distinct advantage.

Assessment:Assignments, and a written examination, to be held at the endof semester 2.

References: David Eisenbud,Commutative algebra with a view towardalgebraic geometry, Springer-Verlag 1994.

M.F. Atiyah and I.G. Macdonald,Introduction to commutative algebra,Addison-Wesley 1969.

N. Bourbaki,Algèbre commutative, Ch. 8–9, Masson, NY 1983.

S. Lang,Algebra (3rd edn), Addison-Wesley 1993. (For background inalgebra).

Course content:The course will include topics from the following.

1. Basic ideas. Commutative algebras over a field; affine varieties andcommutative algebras; examples. Noetherian rings, Hilbert basis theorem;ideals, prime and primary ideals, decomposition theory. Localisation.

2. Integral extensions, the Nulstellensatz, geometric consequences. Defini-tion of Spec(R). Noether normalisation. Filtered and graded rings; com-pletions; flatness. Homological functors: Ext and Tor.

3. Dimension theory; Poincaré or Hilbert series; morphismsof varieties andtheir fibres.

4. Tangent and cotangent spaces; local properties of morphisms. Examples:deteminantal varieties, group schemes,

3.2. FOURTH YEAR COURSES— SEMESTERII 19

Functional Analysis —N. Dancer

Functional analysis is one of the indispensable tools in themodern theory ofpartial differential equations, mathematical physics, probability theory etc.It grew out of the idea that functions with certain properties (for instancecontinuous functions on an interval) can be considered as a “point” in a“vector space,” giving the field its name.

Modern functional analysis is the study of infinite dimensional vector spaces,and linear transformations acting between them. It can be thought of linearalgebra in infinite dimensional spaces. Unlike inR

n a topology compatiblewith the vector space structure is no longer unique, and linear transforma-tions need not be continuous. This leads to a very rich theory.

In this course we introduce Banach and Hilbert spaces, the most imme-diate generalisations ofRn. Moreover we consider linear transformationsbetween such spaces, and investigate their properties. Thetheory of eigen-values and eigenvectors extends to a “spectral theory.” We illustrate thetheory with many examples.

Prerequisites: Metric spaces, Lebesgue/Fourier Analysis, a knowledge ofLp-spaces is of advantage.

Assessment:Assignments and exam.

Reference:K. Yosida,Functional Analysis, Springer Verlag (1980).

3.2. Fourth Year Courses — Semester II

Algebraic Geometry—K. Lai

This is an elementary introduction to algebraic geometry.We shall cover the following topics:Affine spaces. Projective spaces. Variety.Sheaves. Spectrum of a ring.Schemes. Integral schemes.Gluing. Fibred product.Subscheme. Singularities.

20 3. COURSE DESCRIPTIONS

Reference:The general reference is: Hartshorne, Algebraic geometry SpringerVerlag.Assessment:Final exam of 2 hours.

Partial Differential Equations—D. Daners

The course is an introduction to the modern theory of partialdifferentialequations. The major types of equations will be studied including elliptic,parabolic and if time permits also hyperbolic equations. Wewill look atweak solutions, maximum principles, existence and uniqueness of solutionsto linear and non-linear equations. Assumed knowledge is the functionalanalysis from first semester and some knowledge of the Lebesgue integraland Fourier Transforms.

Assessment:Assignments and final exam.

Reference: L.C. Evans, Partial Differential Equations, American Mathe-matical Society, 1998 (Library 515.353/25)

Representations of the symmetric groups—A. Molev

This would include:An explicit construction of the irreducible representations of the symmetricgroupCombinatorial interpretations: Robinson-Schensted algorithmA recent combinatorial proof of the hook formulaCharacter formulas: Murnaghan-Nakayama rule

Assessment:One assignment and exam.

Reference:B.E. Sagan "The symmetric group: representations, combina-torial algorithms, and symmetric functions".

A man ought to read just as the inclination leads him; forwhat he reads as a task will do him little good

Samuel Johnson

CHAPTER 4

The essay

4.1. Introduction

The essay project has several objectives. First and foremost, it is intended toprovide an essentially open-ended framework whereby you may pursue, de-velop and discover your interests in mathematics unencumbered by syllabusand the prospect of eventual written examination. Basic to this process isthe use of the library and communication with others, most especially yoursupervisor. The writing of the essay is a most valuable part of the project.The very act of writing is an invaluable aid to comprehension. A good essayshould be carefully organised, clear, readable by others, laid out well, prop-erly referenced and convey the essential ideas. Attainmentof such writingskills is of great benefit whether or not you elect to stay in mathematics.

One point should, perhaps, be emphasized: the essay projectis not intendedto be a contribution to original research; however, the essay must clearlydemonstrate that you understand and have mastered the material. Original-ity in presentation or view in the essay is required.

4.2. Choosing a supervisor and topic

Choosing a supervisor and topic are the first two things that you should do,and are really not two choices, but one. It is recommended that you beginin the long vacation (preceding your fourth year) by seekingout membersof staff and asking them about their interests and topics they would be keenon supervising. (See also Chapter 5 below). It is a good idea toask themabout their particular method of supervising and other questions importantto you. Do not feel you must settle for the first person you talkto!

All staff members, lecturer and above, are potential supervisors.

There is not necessarily any correlation between supervising style and lec-turing style. Also, the subject a lecturer taught you may notbe their realarea of interest. You should try to decide on a supervisor andtopic before

21

22 4. THE ESSAY

the start of first semester. Most Department members will be available dur-ing the last two weeks of the long vacation and so, if you have not arrangeda topic and supervisor at the beginning of the long vacation,you will proba-bly have to organise your supervisor and topic during these last two weeks.

Changes in supervisor and/or topic are possible during the year (the earlierthe better!). If you do change supervisors then you must notify the fourthyear coordinator.

It is a good idea to have a provisional topic and supervisor inmind at thebeginning of the long vacation. Your potential supervisor can then suggestsome reading over the vacation and, if you have second thoughts about thetopic or supervisor, it is then easy to change before the firstsemester starts.

4.3. Essay content and format

The essay must start with an introduction describing the objective and con-tents of the essay. The essay may end with a summary or conclusion; how-ever, this is optional. Should you wish to make any acknowledgements,they should appear on a separate page, following the introduction.

You should aim at the best scholarly standards in providing bibliographicreferences. In particular, clear references to cited worksshould be made,where appropriate, throughout the text. Furthermore, it isnot acceptableto base large portions of your essay on the existing literature and wheneverpart of your essay closely follows one of your sources this must beexplicitlyacknowledgedin the text. References should not appear in the bibliographyunless they are referred to in the text. For the format of the references seethe appendix.

The essay should be clear, coherent, self contained and something that oth-ers (your fellow students and other non-specialists in the topic) can readwith profit. The essay should not exceed (the equivalent of) 60 pages oneand a half spaced type of normal TEX font size (i.e., as on this sheet). About40–50 pages would normally be acceptable. Students are asked to try tokeep their essays within these limits; overly long essay maybe penalized.Supervisors should advise their students accordingly.

Take pains over style: especially clarity, precision and grammar. Aim atreadability for the non-specialist. Avoid starting sentences with symbols.Aim for succinct statements of theorems and lemmas. Break up long proofs

4.5. TIME MANAGEMENT AND PROGRESS REPORTS 23

into lemmas. Cross reference previous results and notation as this markedlyimproves readability.

Finally, the essay must be typed or printed and prepared in accordance withthe instructions listed in the appendix. These days most Fourth Year stu-dents prepare their essays using a word-processing programsuch as LATEX.

4.4. Submission of essay, assessment, corrections

Three copies of the essay should be given to the Course Coordinator formarkingnot later than the second Friday following the mid-semester breakin second semester.

Students should be aware that at least two weeks is set aside for marking ofthe essay and if the essay is not marked before the examiners meeting at theend of November then it willnot count towards the final mark obtained inFourth Year.

If, during the year, illness or other personal circumstances seem likely toincrease the probability of late submission of the essay, such matters shouldbe reported to the Course Coordinator through your supervisor. Do notpresent such evidence at the last minute!

Each essay will be read independently by at least two membersof the De-partment. (The number of readers will depend on the staff available). Oneof the readers may or may not be the candidate’s supervisor. The mark-ers may suggest corrections should be made to the manuscript. If correc-tions are required, a final corrected copy of the essay shouldbe given tothe Course Coordinator as a Departmental record of the essay. If no correc-tions are required, one of the markers’ copies will normallybe kept by thedepartment and the remaining two copies returned to the candidate.

4.5. Time management and progress reports

At the end of the first semester you should write a summary (approximatelyone page in length) of your essay project and progress and give this to theCourse Coordinator. Here are some rough guidelines and deadlines:-

• Select supervisor and topic - Before beginning of first semester• Reading, discussion and understanding - first semester• Start work on first draft - beginning of second semester

24 4. THE ESSAY

• Final proofreading - mid-semester break

The essay should be submitted by the second Friday followingthe secondsemester break.

Do not underestimate the time it takes you to do the actual writing. Oftenit is not until you start writing that you will settle on a finalview, or realisethat you have misunderstood a particular part of the theory.Allow yourselfsufficient time both for the typing and proof reading of the manuscript.

4.6. Your supervisor

To get the most benefit from the course, you should work closely with yoursupervisor. To this end, you may set up a regular hour each week to meetand discuss progress and problems with your essay project. Alternatively,you might come to some more informal arrangement.

You can expect your supervisor to:

• Help you select or modify your topic;• Direct you to useful sources on your topic;• Explain difficult points;• Provide feedback on whether you are going in the right direction;• Advise you on other course matters.

4.7. Use of the library

The Mathematics Library is located on the eighth floor of the CarslawBuilding (Hours (subject to change): 9:00 - 5:00, Monday to Friday). Itis among the most complete mathematical libraries in Australia. A photo-copier is available to copy journal articles, although not all projects requireextensive use of these.

The most useful source isMathematical Reviews, which is available on–lineat http://ams.rice.edu/mathscinet. Mathematical Reviewsisupdated monthly and contains reviews of a paragraph or so, describing thecontents of recent books and research articles, grouped together by researcharea. It also has extensive subject and author search facility. Every searchfor information on what has been done in a given area should begin withMathematical Reviews. Your supervisor can help you learn how to effec-tively use reference sources likeMathematical Reviewsand, together with

4.7. USE OF THE LIBRARY 25

the Librarian, will gladly assist you with any other problemconcerning li-brary use.

Every Friday the new mathematical periodicals and books aredisplayed(near the photocopier). It is a worthwhile and enjoyable habit to glanceat each new journal and book to see if it contains a relevant orinterestingarticle.

A magazine of general interest that frequently has excellent articles on de-velopments in contemporary mathematics is the monthlyMathematical In-telligencer. Two other journals that usually carry expository articlesof highquality are theAmerican Mathematical MonthlyandL’Enseignement Math-ématique. (The majority of the articles in the latter journal are in English).

The paradox is now fully established that the utmostabstractions are the true weapons with which to control

our thought of concrete fact.A. Whitehead,Science and the Modern World

CHAPTER 5

Sample essay topics

Here are some topics or areas of interest suggested by members of the De-partment for 2007. Please note that this list isnot intended to be complete,the topics suggested are perhaps best regarded as a guide to the likely in-terests of the proposer, and other staff members are willingto act as essaysupervisors. The topics are grouped according to the Research Group towhich each staff member belongs.

5.1. Algebra Research Group

Dr David Easdown — Carslaw 619.

a) Computational geometry and perceptronsPerceptrons are a primi-tive parallel computing device, described in an interesting book byMinsky and Papert. Some topics in the book could make a goodessay, by filling out details (and correcting mistakes), forexample,theorems about computational power of perceptrons, and conver-gence when training a perceptron to recognize patterns. Themathe-matics involves a mixture of algebra, logic and analysis. There areinteresting open questions about polynomials, the answersto whichmay make some proofs constructive.

b) Transformation representations of semigroupsThere is a well de-veloped theory of permutation groups. However very little is knownabout transformation semigroups, and, in particular, how agivensemigroup might be represented “efficiently” by transformations.There are a few published and unpublished results that couldbe thecore of an essay, and possibly lead to new results.

c) Minimal representations of inverse semigroups by partial one-onemappingsInverse semigroups are an abstraction of collections ofpartial one-one mappings of a set closed under composition and in-version. When the mappings are total then these become permuta-tion groups. In concretely representing a given inverse semigroup

26

5.1. ALGEBRA RESEARCHGROUP 27

one would like to minimize the size of the set on which the par-tial mappings act. A discussion of the general problem, its refor-mulation in special cases, and even detailed solutions in particularexamples would make a rounded essay.

d) Idempotents of semigroupsThe set of idempotents may be thoughtof as forming the “skeleton” of a semigroup, and such skeletonshave been characterized axiomatically (Easdown 1985) as biorderedsets, and may be visualized (analogous to Hasse diagrams forlat-tices). An essay could describe the axiomatics and known tech-niques for constructing biordered sets. No-one has exhaustivelyconstructed (using a machine) biordered sets of small order, so thereare opportunities to break new ground.

e) Fundamental semigroupsThese are "basic building blocks" in semi-group theory (as simple groups are in group theory), and there is atechnique for constructing them from biordered sets (putting "fleshon the skeleton"), which relies on being able to calculate automor-phism groups of biordered sets. The basic theory (much of whichis still unpublished) could make a good essay, which might alsoinclude calculations of automorphism groups of some interestingbiordered sets.

f) Idempotents of ringsIt is well known that no finite set of axioms candescribe the multiplicative semigroups of rings. However it is notknown whether their sets of idempotents (biordered sets) can be ax-iomatically described. An essay might include the above result formultiplicative semigroups of rings, and describe known necessaryconditions for their idempotents, including calculationsin interest-ing special cases.

g) Braid semigroupsIn 1925 Artin defined a braid group and found apresentation for it, which uses just the Coxeter relations describingthe symmetric group minus the relations that the generatorsare invo-lutions. If one considers braids with strings “missing” oneobtainsan inverse semigroup, which has a nice presentation, again whichcan be obtained by deleting certain relations from a presentation forthe symmetric inverse semigroup of all partial one-one mappings.This was discovered recently by Easdown and Lavers. A discussionof this theorem and related questions would make an interesting andtopical essay.

h) Group and semigroup algebrasThe matrix representation theory ofa finite group or semigroup is encoded in its algebra formed bytak-ing linear combinations of group or semigroup elements withscalarsfrom some field. Maschke’s Theorem says that if the characteristic

28 5. SAMPLE ESSAY TOPICS

of the field doesn’t divide the order of the group then the represen-tation theory is well-behaved, reflected in the semisimplicity of thegroup algebra. Modular representation theory for groups deals withthe situation where this condition on the field fails. The semisim-plicity of semigroup algebras appears to be even more delicate, andthe discussion of any of a number of recent results would makeafine essay.

i) Equational axioms for regular languagesBloom and Esik (1993)find a list of equations that axiomatize the set of recognizable lan-guages over a given alphabet as a universal algebra with 2 binary,1 unary and 2 nullary operations. Their beautiful proof exploits thestandard theory of minimal automata recognizing regular languages,and the fact that the behaviour of an automaton can be expressedusing matrix multiplication. Their list of equations is necessarilyinfinite. Redko (1964) and Conway (1971) proved that no finite listof equations can suffice. An essay could give an account of theseresults with as much detail as space allows.

Dr Anthony Henderson — Carslaw 805.

I would be happy to supervise an Honours essay in some area of represen-tation theory or related parts of combinatorics and geometry. The followingtopics could be tailored to the student’s prior knowledge.

a) Symmetric functions.This is a crucial area of combinatorics which hasconnections to representations of the symmetric group and polynomial rep-resentations of the general linear group. An especially interesting feature isthe operation of plethysm and its generalizations.

b) Representations of quivers.This theory systematizes linear algebra prob-lems such as classifying pairs of linear maps between vectorspaces of afixed dimension. A highlight is Gabriel’s Theorem, which reveals the as-tonishing connection with reflection groups and Lie algebras.

c) Partition combinatorics.A partition is merely a finite nonincreasing se-quence of positive integers. Many sets of interest in representation theoryare in bijection with the set of partitions (or collections of partitions) satis-fying certain properties. Equalities derived from representation theory thenimply results in combinatorics and vice versa.

d) Representations of Lie algebras.Beyond the classical foundations, therehas been exciting recent research on such problems as that offinding canon-ical bases for various kinds of irreducible representations.


e) Finite groups of Lie type.The leading example of this class is the gen-eral linear group over a finite field, whose character table was computedby Green in one of the seminal papers of the second half of the twentiethcentury.

f) Schubert varieties.The study of these varieties, their intersections andsingularities, is of great importance in algebraic geometry. It is now deeplyembedded in representation theory as well, via the ubiquitous Kazhdan-Lusztig polynomials, a master key to all the above topics.

A/Prof. Bob Howlett — Carslaw 523.

I would be happy to supervise a fourth year essay on any topic related tomy areas of research expertise. These are all various branches of algebra,involving group theory and/or representation theory. The four main areasare as follows.

a) Representation theory of finite groups of Lie typeLie theory encom-passes a large variety of topics that are in one way or anotherrelatedto the work of the 19th century mathematician Sophus Lie on con-tinuous transformation groups. The central objects of study in thisarea are these days known asLie algebras, although in the 19th cen-tury they were calledinfinitesimal groups. It turns out that there areseven types of simple Lie algebras, imaginatively called typesA, B,C, D, E, F andG. As a finite group theorist, I am interested infinite groups of these same types; they are basically obtained by re-placing the field of complex numbers by various finite fields. Thesefinite groups of Lie type include matrix groups (such as the gen-eral linear group, orthogonal groups and symplectic groups) as wellas the slightly more mysterious groups obtained from the so-calledexceptional Lie algebras (typesE, F andG).

b) Finite Coxeter groups and Iwahori-Hecke algebrasAssociated witheach simple Lie algebra over the complex field is a finite groupknown as theWeyl groupof the Lie algebra. It turns out that theseWeyl groups can be realized as finite groups generated by reflec-tions acting on Euclidean space. The Canadian mathematicianH. S.M. Coxeter studied and classified finite Euclidean reflection groups,and these are now known as Coxeter groups in his honour. Theyinclude a few groups that are not Weyl groups. There is a beautifuland extensive theory of Coxeter groups, and it seems that almost allquestions in Lie theory lead back to questions about Coxeter groups.


It has also been discovered that each Coxeter group has a “deforma-tion”, or “q-analogue”, known as an Iwahori-Hecke algebra, and itis profitable to study these at the same time as one studies Coxetergroups.

c) Infinite Coxeter groupsAlthough finite Euclidean reflection groupsprovided the original motivation for studying Coxeter groups, thereare also infinite Coxeter groups, and they are becoming ever moreimportant in generalizations and extensions of Lie theory.Most ofmy work in recent years has been concerned with advancing thetheory of infinite Coxeter groups, attempting to find the correct waysto extend the well-established theory of finite Coxeter groups.

d) Representation theory of finite soluble groupsUnlike topics (a), (b)and (c) above, this topic is not derived from Lie theory. The basicproblem in representation theory of finite groups is to find ways ofconstructing and describing the irreducible representations of anygiven finite groupG. Here a “representation” ofG means a ho-momorphism fromG to a group of linear operators on a vectorspace, and the representation in “irreducible” if there areno non-trivial proper invariant subspaces. It is hard to find general methodsthat apply to all finite groups, but for soluble groups a theory can bedeveloped. The crucial fact is that ifG is soluble thenG has a seriesof normal subgroups{1} = G0 < G1 < · · · < Gn = G such thatfor all i the quotient groupGi/Gi−1 is Abelian. There are theoremsthat describe the relationships between the irreducible representa-tions of a groupG and those ofN andG/N , whereN is a normalsubgroup ofG. This topic is known asClifford theory. Repeateduse of Clifford theory provides a method for proving many differenttheorems about representations of soluble groups.

For more information, see Dr Howlett’s web page:http:\\www.maths.usyd.edu.au\u\bobh\.

Professor Gus Lehrer — Carslaw 813.

There are several possible topics, which come from two basicthemes.

a) The first theme is that ofsymmetries of algebraic varieties; the solu-tion sets of polynomial equations often have symmetries (for exam-ple when the defining polynomials are permuted by a group). Thisleads to actions on the homology of interesting topologicalspaces,


such as discriminant varieties and toric varieties, the latter being de-fined by monomials associated with polyhedral cones. Topicsavail-able here include: computing actions via rational points; geometryof certain classical algebraic varieties; topological andgeometricproblems arising from reflection groups.

b) The second theme is therepresentation theory of associative alge-bras. There are many semisimple algebras (with easily describedrepresentations) which may be deformed into non-semisimple onesby variation of parameters. These occur in algebra (Brauer, Heckealgebras), in mathematical physics (Temperley-Lieb algebras) andtoplology (BMW algebras, etc). “Cellular theory” allows one toreduce deep questions about these deformations to (usuallyhard)problems in linear algebra. There are several possibilities for essaytopics in this area.

Dr Andrew Mathas — Carslaw 635.

I would be happy to supervise a fourth year essay on any topic in represen-tations theory, or combinatorics. My main research interests are the rep-resentation theory of the symmetric groups and related algebras (such asHecke algebra, Ariki Koike algebras, Schur algebras, general linear groups,Brauer algebras, Solomon’s descent algebras...), with an emphasis of thenon-semisimple case—which is where things start to get interesting, andmore difficult!

Possible topics include:

a) The modular representation theory of finite groupsIn characteristiczero every representation of a finite group can be decomposed, ina unique way, as a direct sum of irreducible representations. Forfields of positive characteristic this is no longer the case,but never-theless the number of times that a given irreducible module can ariseas a composition factor of a representation is uniquely determined.Possible projects in this area range from classifying the number ofirreducible representations of a finite group, to studying the Brauerand Green correspondences.

b) Representations of symmetric groupsThe representation theory ofthe symmetric group is a rich and beautiful subject which involvesa lot of algebra and combinatorics. Possible projects here includecharacter formulae, classifying homomorphisms, computing decom-position matrices, Murphy operators, the Jantzen sum formula...


c) Brauer algebrasThe Brauer algebras arise naturally from th= e rep-resentation theory of the symplectic and orthogonal groups, but theycan also be understood from a purely combinatorial viewpoint interms of = a “diagram calculus”. Possible topics in this areain-clude character formulae, classifying semisimplicity, branching the-orems,...

d) Seminormal formsFor many algebras it is possible to give “nice"generating matrices for the irreducible representations in the semi-simple case. These explicit matrix representations are called semi-normal forms. The study of the seminormal forms, and the resultingcharacter formulae, for one or more algebras would make an inter-esting essay topic.

Dr Alex Molev — Carslaw 531.

a) Quasideterminants and their applications. In this project we wouldconsider matrices whose entries are elements of an arbitrary (notnecessarily commutative) ring. The classical problem is todefinean analog of the determinant of such a matrix. About a decade agoGelfand and Retakh came up with natural noncommutative analogsof the determinant called quasideterminants. We would examinenoncommutative analogs of the classical theorems on minorsanddeterminants and their applications.

b) Lie algebras and quantum groups. This project would deal with ba-sic properties of the matrix Lie algebras and their "quantizations"known as quantum groups. The algebraic structure and representa-tions of these objects will be investigated in the project.

Dr Bill Palmer — Carslaw 521.

Possible topics include:

a) Bhaskar Rao designsA Bhaskar Rao design is a matrix with groupelement entries, which has certain very interesting properties. Thereare many open questions concerning the existence of Bhaskar Raodesigns over finite groups. I am currently working on the existenceproblem for Bhaskar Rao designs over dihedral groups.

b) Latin squaresJ. Dénes and A. D. Keedwell’s recent bookLatinSquares(North-Holland, 1991), reveals some interesting topics con-cerning the existence of Latin squares and the connections betweenthese squares and codes and non–associative binary systems.

c) Complete mappings, transversals and Latin squares


d) A transversalof the Cayley table of a group of ordern is a setof n cells, one in each row, one in each column, such that no twoof the cells contain the same symbol. Acomplete mappingof agroup is equivalent to a transversal in a Cayley table of the group.For some large families of finite groups, complete mappings areknown to exist. A long-standing conjecture of Hall and Paige: Afinite groupG whose Sylow2−subgroup is non-cyclic possesses acomplete mappingis the subject of current interest. An interestedstudent would find the book:Orthomorphism Graphs of Groups,Anthony B. Evans, Springer-Verlag, 1992, and its review inMathe-matics Reviewswell worth reading.

e) Designs, in generalThe recent 1996 survey:CRC Handbook ofCombinatorial Designsby Charles J. Colbourn and Jeffrey H. Dinitz,contains excellent up-to-date summaries of exciting advances in re-search in design theory. An interested student would be sureto finda worthwhile essay topic in these748 pages.

A/Prof Ruibin Zhang — Carslaw 722.

I work on the representation theory of Lie algebras and quantum groups,and applications of these algebraic structures in quantum physics. I amhappy to supervise any projects in this general area. Some possible essaytopics are:

a) Affine Kac-Moody Algebras and Vertex OperatorsLevel 1 repre-sentations of affine Kac-Moody algebras can be realized on Fockspaces of quantum fields by using vertex operators. These repre-sentations played important roles in quantum field theory, and thevertex operator construction gave birth to the subject of vertex op-erator algebras. Level 1 representations of quantum affine algebrascan also be constructed similarly.

b) Quantum Groups and Deformations of Universal Enveloping Alge-brasQuantum groups are a special type of deformations of universalenveloping algebras of Lie algebras. A distinctive property of quan-tum groups is their braided structure, namely, the existence of uni-versal R-matrices satisfying the Yang-Baxter equation. For the uni-versal enveloping algebras of the finite dimensional semi-simple Liealgebras, it is possible to classify all the deformations with braiding.


5.2. Analysis Research Group

Dr Donald Cartwright — Carslaw 620.

a) Analysis on treesA homogeneous treeT is an infinite graph, inwhich there are no “loops", and in which each vertex has the samenumber of neighbours. BecauseT is highly symmetric, it has avery large group of automorphisms, and many nice representationsof this group can be defined in a natural way. Research in this areahas been very active over the last decade, but the subject is quiteaccessible. Preliminary reading:Harmonic Analysis and Represen-tation Theory for Groups acting on Homogeneous Trees, by A. Figá-Talamanca and C. Nebbia (Cambridge University Press, 1991).

b) Analysis on the hyperbolic planeThe “hyperbolic plane" is the unitdisc in the ordinary plane, endowed with a different idea of distance:hyperbolic distance. It is an example of a “symmetric space", whichmeans roughly that there is a large group of (hyperbolic-)distancepreserving transformations acting on it. These give rise tobeautifultesselations of the disc which inspired many designs by the artistEscher. An interplay of geometry, complex analysis, numbertheoryand a little bit of group theory. Preliminary reading: Chapter IIIof Harmonic Analysis on Symmetric Spaces and Applications. IbyA. Terras, Springer-Verlag (1985).

c) Random walks on groupsThe classical example is of a point mov-ing on the groupZ of integers, at each time hopping one to the rightor one to the left with probability 1/2. One studies what happens"in the long run". Considering the natural extension to more generalgroups leads to an interesting interplay between analysis and grouptheory, which has received much attention in recent years. Somebackground in probability and measure theory would be useful, butnot essential. Preliminary reading:Random Walks and Electric Net-worksby P. Doyle and L. Snell, Mathematical Association of Amer-ica (1984).

5.3. Computational Algebra Research Group

Professor John Cannon — Carslaw 618.

a) Computational Number TheoryFor example:- Primality testing and factorization- Constructive algebraic number theory

5.4. GEOMETRY RESEARCHGROUP 35

- Computation of Galois groupsb) Computational Group Theory

- Algorithmic methods for finitely presented groups- Algorithmic methods for permutation groups- Computational representation theory- Constructive invariant theory

c) Computational Differential AlgebraFor example:- The Risch algorithm for indefinite integration

5.4. Geometry Research Group

Dr Emma Carberry—Carslaw 609.

Dr Carberry will be on leave in Semester 1 of 2007.

Dr Jonathan Hillman — Carslaw 617.

My research interests are in Algebraic Topology; more specifically, in theapplication of commutative algebra and group theory to the study of mani-folds. In low dimensions (i.e.,≤ 4) it is often possible to minimize the needfor algebra by direct “geometric" arguments. In particular, I am interestedin the classification of knots, links and manifolds of dimensions 3 and 4.Prerequisite for any such topic: “Metric Spaces 3961". (Someknowledgeof Group Theory or Differential Geometry might also be useful, and any-one working with me would be expected to take the Algebraic Topologycourse).

Essays that I have supervised in recent years include “Symmetries and KnotPolynomials" (in 2001), "K-Theory and the Frobenius Conjecture" (2002),"Presentations of knot groups and representations onto metacyclic groups"(2004), "Surgery recovers the knot module" (2005) and "Intersection Formson closed 4-manifolds" (2006). The following topics are offered as a start-ing point for discussion.

a) Knots and LinksKnots and links are the basic building blocks for 3- and 4-manifolds,

as well as being of interest in their own right. Their study has manydifferent aspects: algebraic, combinatorial, geometrical, topologi-cal, and an Honours essay could focus on one of these or combineseveral.


General references: there are many books on knots in our li-brary. For instance, seeOn Knotsby L.H.Kaufman, Annals of Math.Studies 115, Princeton University Press (1987). (Kauffmanhas avery original approach to knot theory that does not rely on elaboratetechniques). I shall provide specific references after consultationwith interested students.

Possible topics include:Representation varieties for knot groupsReferences: Le Thuy Quoc Thang, Heusener, ...Knots and singularities of real polynomialsReferences: Looijenga, Perron, Bennedeti/ShimadaReidemeister torsion and branched cyclic coversReferences: Milnor, Porti, TuraevConstructing knots with given polynomials;References: Levine, Davis and Livingston, Sakai,...Milnor invariants of linksReferences: Cochran, Milnor, Hillman ...

b) Normal surfaces in 3-manifolds.Surfaces also play a central role in the study of 3-manifolds. The

notion of normal surfaces has been developed in order to find algo-rithms for deciding when two manifolds are homeomorphic, etc.This notion is well suited to computational exploration.

References:“PL equivariant surgery and invariant decompositions of 3-manifolds"

by W.Jaco and J.H.Rubinstein, Advances in Math. 73 (1989), 149-191 (especially Sections 1 and 2).

“An equivariant sphere theorem" by M.J.Dunwoody, Bull. Lon-don Math. Soc. 17 (1985), 437-448.

c) Triangulation of surfaces.

References:"All 2-manifolds have finitely many minimal triangulations",by

D.W.Barnette and A.L.Edelson, Israel J. Math. 67 (1989), 123-128;"A short proof that compact 2-manifolds can be triangulated",

by P.H.Doyle and D.A.Moran, Inventiones Math. 5 (1968), 160-162.

d) Handlebody theoryWhile I usually work with low-dimensional manifolds I am quite

willing to advise students on high-dimensional differential topologyor geometric topology. Here the main problem is that the exist-ing references are almost too good; it is difficult to escape their

5.5. NUMBER THEORY RESEARCHGROUP 37

influence. Almost any book by John Milnor is worth reading (al-though I would advise against starting with his book on Character-istic Classes). There are two approaches: through differential topol-ogy (Morse theory) and through piecewise linear topology (polyhe-dra). It is fair to say that the study of high-dimensional manifolds nolonger has the relative status it had in the 1950s and 1960s, math-ematical tastes having returned to lower dimensions (at once bothmore difficult and more concrete). Nevertheless the work of that erawas stunningly successful and the ideas developed then influencecurrent research.

References:Morse Theoryby J.W.MilnorLectures on theh-Cobordism Theoremby J.W.MilnorIntroduction to Piecewise-Linear Topologyby C.P.Rourke and

B.J.Sanderson

Dr Laurentiu Paunescu — Carslaw 816.

I am interested in the applications of singularity theory todifferential equa-tions, and in using the combinatorics of Toric Modificationsin investigatingthe equisingularity problem. My main research interests are:

a) Singularities of complex and real analytic functionsb) Stratified Morse theoryc) Toric resolution of singularities

5.5. Number Theory Research Group

Dr Martine Girard — Carslaw 638.

I will be happy to supervise a 4th year essay on the arithmeticof curves.

The arithmetic of curves both provides an introduction to modern numbertheory and algebraic geometry, and is ideally suited for development as anhonours project. The study of rational points (i.e. points with coordinatesin Q) can be the starting point of a project. Finding the set can be a difficulttask. Subsequently proving that a given set of points is complete presentseven more theoretical challenges.

On an elliptic curve, the set of points forms a group which is finitely gener-ated (Mordell-Weil theorem), while on curves of higher genus, the numberof such points is finite. A project could start with the case ofelliptic curves


and the Mordell-Weil theorem. The problem of finding a complete set ofgenerators represents both computational and theoreticalchallenges, whichwould be the subject of investigation.

Taking another direction, for curves of higher genus, thereare two mainmethods, of Chabauty and Dem’janenko-Manin, for providing aproof thata given set of rational points is complete. An understandingof the theoryof divisors, of height functions and of the construction of the Jacobian of acurve will be outcomes of research on curves of higher genus.

References: Rational Points on Elliptic Curves, by Silverman and Tate,Springer (1992).The Arithmetic of Elliptic Curves, by J.Silverman, Springer (1986).

Dr David Kohel — Carslaw 625.

My research concerns algorithmic and computational aspects of numbertheory. The former component of this work focuses on areas ofmodernmathematics where the proofs can be made effective through atheoreticaldescription of steps which produce a solution to a class of problems.

The computational aspect of this work concerns turning sucha theoreticalsequences of steps into practice with the view of better understanding thealgorithm or of gaining insight into further theory throughempirical inves-tigations.

Consistent with these research interests, I am willing to supervise eitherstudents who are oriented towards purely theoretical investigations into in-teresting aspects of modern number theory — but who do not shyawayfrom explicit computation — or those who have experience with moderncomputer languages and take an interest in combining mathematical andcomputational investigations.

a) Arithmetic Geometry1) Picard group algorithms and computationsThis project is an in-vestigation of algorithms for computing the Picard group ofellipticcurves, hyperelliptic curves, and cyclic covers of the projective line.Emphasis will be on the representation of elements and the grouplaw and application development of efficient algorithms forcomput-ing in torsion subgroups. A related project forp-adic point countingis described below under the auspices of cryptography.2) Modular curves and modular formsModular curves serve to pa-rametrize elliptic curves with some prescribed structure,of which


the class of curvesX0(N) play a significant role in number the-ory and arithmetic geometry, including the recent proof of Fermat’sLast Theorem. This project will delve into the theory of modularcurves and the modular forms associated to them, and be directedtoward understanding the explicit parametrization of elliptic curveisogenies, or to a focus on modular forms and modp Galois repre-sentations.3) Finite abelian group schemesThe goal of this project would be tounderstand the concept of a scheme and of group schemes, thentoaddress the question of how finite groups schemes generalizefiniteabelian groups, and why they are nontrivial objects of study. Thetheory of torsion subgroup schemes on abelian varieties andgroupsschemes over number rings will motivate this research.

b) Number Theory1) Arithmetic of integral quadratic formsThe theory of quadraticforms serves as an introduction to modern number theory. Thegoalfor this project is to understand the background for the local-globalequivalence which holds for quadratic forms over a number field,and in particular for conics, and how this equivalence failsfor curvesof higher genus. The project will investigate more subtle arithmeticquestions of integral equivalence, with a view to understand the clas-sification of the genus and spinor genus of integral quadratic forms,and relations with their automorphism groups.2) Arithmetic of quaternion algebrasThe central questions of num-ber theory concern the structure of ideal class groups and units oforders of number fields. The analogous study of non–commutativealgebras requires differentiation between the left, right, and 2-sidedideals. Quaternion algebras provide the first nontrivial examples ofnon–commutative algebras for study, and their arithmetic relates toquestions in diverse areas of mathematics, from elliptic curves tograph theory and hyperbolic geometry.

c) Cryptography and Coding Theory1) p-adic point counting algorithmsA novel series of algorithmsfor point counting on elliptic curves was introduced by Satoh in1999, using canonicalp-adic lifts of elliptic curves. Efficient vari-ants, such as the AGM of Mestre in characteristic 2, have followed,as well as generalizations to hyperelliptic curves by Kedlaya andothers. These have had significant impact on the practicality of us-ing elliptic curves for public key cryptography. The student who isinterested in a challenging topic could look into generalizations ofthese algorithms either for elliptic or hyperelliptic curves, or appli-cations to CM constructions.


2) Attacks on popular public key cryptosystemsPublic key cryptog-raphy relies on a class of maps which are easy to apply yet whoseinverse is difficult to compute. The companion problems of inte-ger multiplication and factorization, and group exponentiation anddiscrete logarithms are the principle examples of current use in cryp-tography. This project would first review known algorithms for dis-crete logarithms, then progress to algorithms for ellipticcurves suchas the MOV attack, attacks on anomalous curves, and use of Weilrestriction. The direction of the research could then focuson ad-vancing one or more of these algorithms or towards formulating anunderstanding of and guidelines for the use of these attacks, andrelated questions such as the relative security of ellipticcurve andXTR cryptosystems and the use of Weil pairing on elliptic curvesfor digital signatures.3)Algebraic–geometric coding theoryThe introduction of algebraic–geometric coding theory by Goppa in the 1980’s launched a fer-tile new area of research in coding theory. The goal of this projectwill be to develop an understanding of divisors, linear systems, andthe Riemann-Roch theory of curves underlying the mathematicsofGoppa’s construction. This can be supplemented with explicit com-putations and research into explicit constructions which might im-prove on the best known parameters of codes.

Dr King-Fai Lai — Carslaw 633.

My research interest is the theory ofautomorphic forms.

Let us writen · x = for the translationx + n of the real numberx by theintegern. We say that this defines an action of the groupZ of integers onthe real numbers. We say a functionf on the real numbers isperiodic if itsatisfies the following equation :

f(n · x) = f(x)

for all integersn and for all real numbersx. An example of such a functionis f(x) = sin2πx. We call the integersZ the group of periods.

We can ask for functions which are periodic with respect to a bigger groupof periods; for example we can replace the commutative groupZ of integersby the non-commutative groupΓ consisting of matrices

γ =

(

a bc d

)


with integersa, b, c, d satisfying the conditionad − bc = 1. Next we mustintroduce the action of the groupΓ of periods. Let us denote the set ofall complex numbersz with positive imaginary parts byh. The action isdefined by the following equation

γ · z = (az + b)(cz + d)−1.

For a functionf defined onh the periodicity condition now reads as

f(γ · z) = f(z)

for all γ in Γ andz in h. Such a functionf is called a modular function(we have ignored the analyticity and growth conditions for simplicity). Amodular function is an example of an automorphic form which is definedwith respect to discrete subgroupΓ of a Lie group.

If you are interested to find out more about automorphic formsyou canspend the summer months reading : Miyake T, Modular Forms [available inthe library]. Keep a notebook with you while you are reading,write downthe things you don’t understand. When the new term starts and if you arestill interested we can talk.

Two other topics that interest me areTannakian categoriesandformal groups. The theory of formal groups is useful in a number of topicsincluding number theory and group theory.

References:Deligne, P. Catégories tannakiennes, in

The Grothendieck Festschrift,Vol. II, 111–195, Progr. Math., 87, BirkhäuserBoston, Boston, MA, 1990.

Cartiertheorie kommutativer formaler Gruppen, by T.Zink, Teubner (1984).[512.2 2] .

Dr Adrian Nelson — Carslaw 526.

Current research interests include various topics in numbertheory, for ex-ample those concerning zeta functions and Galois module structure. A pos-sible essay topic would be on zeta functions and regularizeddeterminants.See C. Deninger,Local L-factors of motives and regularized determinants,Invent. Math, 107, 135-150 (1992).


5.6. Non-Linear Analysis Research Group

Professor Norm Dancer — Carslaw 717.

Professor Dancer will not be available for supervision in 2007.

Dr Daniel Daners — Carslaw 715.

Areas of interest:

a) partial differential equations (linear or nonlinear)b) ordinary differential equations (linear or nonlinear)c) bifurcation theoryd) analytic semigroup theory and abstract evolution equations. (This is

a theory of "ordinary differential equations" in infinite dimensionalspaces with applications to partial differential equations).

Please see me to negotiate a topic of your interest or for suggestions forspecific projects related to the above areas.

Dr Nigel O’Brian — Carslaw 714.

a) Information Theory and log-optimal portfoliosI am currently interested in financial mathematics, and espe-

cially portfolio theory. This has interesting connectionswith In-formation Theory, as described inReference: T. M. Cover and J. Thomas, Elements of InformationTheory, Wiley 1991).

for example.b) Quantum Information Theory and Quantum Computation

Another area related to Information Theory which might be suit-able for an essay is the newly emerging field of Quantum Informa-tion Theory, with applications to Quantum Computation. Recenttexts include:Reference:M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information, Cambridge 2000).

I am also interested in computer graphics and visualisationof mathematicalstructures. I would be happy to discuss possible topics withanyone havingan interest and computer skills in this area.

Extraordinaire comme les mathématiques vous aident àvous connaître.

Samuel Beckett,Molloy

CHAPTER 6

The talk

6.1. General remarks

Before the essay is submitted at the end of Second Semester, each studentgives a talk on their essay project. In past years these talkshave taken placein the weeks of September leading up to the mid-semester break.

The aim of the talk is to provide training in the explanation to others ofthe purpose and nature of a project, within definite time limits. In recentyears we have allowed forty minutes for each talk, plus fives minutes forquestions.

All members of the Department, Fourth Year and postgraduatestudents areinvited to the Fourth Year talks.

No explicit grade is given for the talk

6.2. Preparing the talk

The purpose of your talk is to convey to yourfellow students(and the aca-demic staff) what you are working on. They probably know verylittle aboutyour essay topic; this comment may also apply to the academicstaff. Donot make the talk too long or ambitious. Aim to convey the essence of yourproject to the audience rather than trying to impress the audience; after all,it is unlikely that you can cover the whole of your project in 40 minutes!.

The key to giving a successful mathematical talk is: “Keep itsimple!” Oneidea, illustrated by one or two examples, is enough for a goodtalk. A specialcase often conveys more than a general, all-encompassing theorem. Forexample, to give the flavour of general fields, a detailed study of a simple,but unfamiliar field, such asGF (9), might be appropriate.

Keep in mind that the audience is swept along with you and thatthey cannotgo back to earlier stages of your talk like when they are reading an article.

43

44 6. THE TALK

You are not giving a lecture, so although some definitions maybe appropri-ate, lengthy technical proofs should be avoided. It is also not a good ideato over-develop the theory at the expense of examples: a well-chosen ex-ample is worth ten thousand theorems. Finally, try and relate your contentto other areas of mathematics or applications; this can makethe talk muchmore interesting for the general audience.

You should aim your talk at a general mathematical audience and avoiddirecting it at the odd specialist in your topic in the audience. Thus a goodtalk is judged by one criterion: you have given the audience,especially yourfellow Fourth Year students, a good idea of your project and its significance.

Discuss the talk with your supervisor.

Having chosen the topic for your talk, prepare a written outline. Somepeople write their talk out in full, while others prefer to use only a writtenoutline and allow improvisations. As it is probably your first talk of thiskind, it is advisable to do a full dress rehearsal the previous evening; sofind a blackboard and an overhead project and go through the completetalk. This will help you in judging the timing of your talk properly: it takesmuch longer to say things than you probably realize. If you can, find asympathetic listener to give you feedback. Your listener does not have tobe mathematically literate: a good talk is almost as much about theatre andpresentation as it is about mathematics.

6.3. Overhead projectors

Decide if an overhead projector is appropriate. This allowspreparation ofcomplicated figures or tables ahead of time, or the inclusionof photocopiesof published material in your exposition. Beware, however, that althoughthe speaker can by this means pass a vast amount of information beforethe eyes of the audience very quickly, the audience will probably not takeit all in. It is important either to write clearly and in largeletters and torefer explicitly to each line (say by gradually revealing line-by-line usingcovering paper) or, in the case of a diagram or complicated formula, toallow your audience time to absorb its detail.

If you are going to use TEX to create slides thenmake sure that you uselarge enough fonts; the easiest way to do this is to use a LATEX package suchasfoiltex, or a program such as powerpoint.

Life is good for only two things, discovering mathematicsand teaching mathematics.

Siméon Poisson

CHAPTER 7

Your future and mathematics

As a fourth year student you are a member of the mathematics departmentand you should take advantage of the facilities it offers. The University ofSydney has one of the top mathematics research departments in the country,and it ranks very highly internationally in several areas. There are alsoa number of prominent international (short and long term) visitors to thedepartment who give seminar talks within the department. Itpays to keepan eye onscnews (the School’s web based bulletin board), for upcomingseminar announcements.

The academic staff, the many postdocs and the visitors to thedepartmentare all usually very happy to talk mathematics talk with interested students:all you have to do is find the courage to ask!

Fourth year students are also very welcome to join the staff and postgradu-ates in the use of the tea room; this can be a good place to meet other peoplein the department.

7.1. The colloquium and other seminars

Most Fridays during the year, a Colloquium is held at 2:00pm ineitherCarslaw Lecture Theatre 375 or in the ‘Red Centre’ UNSW. Topics vary,but the intention is to provide a one-hour talk on a subject ofcontempo-rary mathematical interest to a general audience. Fourth Year students areencouraged to attend the Colloquium and indeed are welcome toany sem-inar run in the Department. For a schedule of upcoming seminars, see theDepartment notice board on Level 7, Carslaw, orread your e–mail.

There are also a number of other active seminars in the department; notably,in algebra, computational algebra algebraic geometry seminar and categorytheory.

45

46 7. YOUR FUTURE AND MATHEMATICS

7.2. After fourth year, what then?

Recent graduates have found employment in a wide variety of occupations:computer related jobs, teaching (University or School), positions in insur-ance and finance. To find out more about where maths can take you:

http://www.careers.usyd.edu.auhttp://www.amsi.org.au

http://www.austms.org.au/Jobshttp://www.ice-em.org.au/careers.html .

Here we shall just outline briefly the postgraduate degree options. For moreinformation consult the departments web pages.

7.3. Higher degrees

A result of II-2 or better is the minimum requirement for entry into a higherdegree at Sydney. However it should be noted that one should not normallycontemplate continuing without a result of at least II-1. Anyone intendingto undertake a higher degree should consult with the Mathematics Postgrad-uate Coordinator (Dr. Andrew Mathas, Carslaw 635) as soon as possible.The usual practice is to enrol for an M. Sc in the first instanceand later toconvert to a Ph.D if it is desired to continue.

7.4. Scholarships and other support

Scholarships, prizes and travel grants are available both for study at Sydneyand for study elsewhere. Full details can be found in the University Calen-dar and from the Scholarships Office (Administration Building). Intendingapplicants should obtain application forms from the Scholarships Office assoon as possible.The closing dates for some scholarships can be as earlyas September

If you are considering further study at an Australian University, you shouldapply for an Australian Postgraduate Research Award. (even for an M. Scby coursework). For study at a university in Britain or Canada,apply fora University of Sydney travelling Scholarship and also apply to the chosenuniversity for employment as a Graduate Assistant.

7.5. FURTHER STUDY IN ANOTHER SUBJECT 47

7.5. Further study in another subject

As mentioned in the introduction to this booklet, it is quitepossible to doFourth Year Pure Mathematics and then continue with a higherdegree inanother subject. Within Australia, prerequisites vary from Department toDepartment and for those intending to follow this path it is advisable toconsult with the Department concerned to determine an appropriate choiceof Fourth Year topics. If you are intending to continue with Postgraduatestudies in another fieldoutsideAustralia, do check prerequisites. Providedyou have done third year courses in the subject at Sydney, youwill probablynot encounter significant problems over prerequisites.

It has long been an axiom of mine that the little things areinfinitely the most important.

Sherlock Holmes,A Case of Identity

APPENDIX A

Instructions on preparing the Manuscript

Essays must be typed using LATEX (or TEX), or a commercial word process-ing program such as word. Amongst professional mathematicians LATEX hasbecome the standard; it produces better quality output thanany word pro-cessing programs program — at least when it comes to mathematics. Thedownside to LATEX is that it takes some time to learn.

The fourth year coordinator will give an introduction to using TEX and LATEXbefore the beginning of second semester. For those wishing to use LATEX DrMathas has written a LATEX class filewhich takes care of the basic layout ofthe essay; for information see

http://www.maths.usyd.edu.au:/u/mathas/pm4/

If you decide not to use this LATEX class file, then your document mustsatisfy the following requirements.

a) A margin of at least 2.5cm must be left at the top, bottom, left- andright-hand side of each page. The margin is determined by thelastletter or character in the longest line on the page.

b) All pages must be numbered (in a consistent way), except for thetitle page.

c) Avoid excessive use of footnotes. They are rarely necessary in math-ematics.

d) Diagrams should be created using appropriate software; hand drawndiagrams are not acceptable.

e) Theorem Propositions, etc. should be labelled consistently through-out the document.

f) ReferencesA consistent scheme should be adopted. Thus, refer-ences may be numbered: [1], [2],... and referred to as such inthetext. Alternatively, by author’s initial: [A], [F1], [F2],...

Sample references for the bibliography are given below:[10] C. Chevalley,Theory of Lie Groups, Princeton Univ. Press,

Princeton (1946).

48

A. I NSTRUCTIONS ON PREPARING THEMANUSCRIPT 49

[15] H. Grauert, On Levi’s problem and the embedding of real an-alytic manifolds, Ann. Math. 68 (1958), 460-472.

[28] J. N. Mather, Stratifications and mappings, inDynamical Sys-tems, Academic Press, New York (1973), 195-232.

Note: instead of using italics, the titles of books may be underlined,or placed in quotation marks. Similarly the titles of journals may beitalicised or underlined.

References should be listed alphabetically.

Notes on the use of TEX and LATEX are available from Dr Howlett and theweb page above.

Documents

Fourth Year Pure Mathematics 2007 Handbook8 1. THE STRUCTURE OF PURE MATHEMATICS FOUR 1.4. Pure Mathematics 3(A) Courses for 2007 SEMESTER I SEMESTER II Algebra (3962) Complex Analysis