# Korner Part Iii Essay Topics

**Part III of the Mathematical Tripos** (officially **Master of Mathematics/Master of Advanced Study**) is a one-year Masters-level taught course in mathematics offered at the Faculty of Mathematics, University of Cambridge. It is regarded as one of the hardest and most intensive mathematics courses in the world and is taken by approximately 200 students each year.^{[1]}^{[2]} Roughly one third of the students take the course as a fourth year of mathematical study at Cambridge (after Parts IA, IB, and II), whilst the remaining two thirds take the course as a one-year course.^{[3]}

## History[edit]

The Smith's Prize Examination was founded by bequest of Robert Smith upon his death in 1768 to encourage the study of more advanced mathematics than that found in the undergraduate course. T. W. Körner notes

Only a small handful of students took the Smith's prize examination in the nineteenth century. When Karl Pearson took the examination in 1879, the examiners were Stokes, Maxwell, Cayley, and Todhunter and the examinees went on each occasion to the examiner's house, did a morning paper, had lunch there, and continued their work on the paper in the afternoon.

— Thomas William Körner

^{[3]}

In 1883 this was replaced by an exam called *Part III* and the Smith's Prize awarded for an essay rather than examination. In 1886 this exam was renamed *Part II*, and later in 1909 *Part II, Schedule B*. In 1934 it was again renamed *Part III*.^{[3]}

In the 1980s the *Certificate of Advanced Study in Mathematics* was introduced; for those students successfully completing Part III of the Mathematical Tripos in Easter Term 2011 CASM was replaced by two new degrees, the *Master of Mathematics* (M.Math.) and *Master of Advanced Study* (M.A.St.). All who have passed the course since 1962 are entitled to these new degrees.^{[3]} The first retrospective M.Math and M.A.St. degrees were conferred as part of a celebration of the University's 800th anniversary.^{[4]} The course is often still referred to as Part III.

## Master of Mathematics vs Master of Advanced Study[edit]

Students who have completed their undergraduate degree at Cambridge will be awarded both a Bachelor of Arts (B.A.) and the Master of Mathematics (M.Math.) degree for four years of study, provided they have not previously graduated with a B.A.^{[5]} This allows Cambridge graduates to remain eligible for government funding for the course. Progression from Part II of the Mathematical Tripos to Part III normally requires either a first in Part II or very good performances in Parts IB and Part II.^{[6]} Students who complete Part III of the Mathematical Tripos, but did not complete undergraduate studies at Cambridge (or have previously graduated with a B.A.) will be awarded the Master of Advanced Study (M.A.St.) in Mathematics degree for the one-year course.^{[7]}^{[8]}

The program previously resulted in a Certificate of Advanced Study in Mathematics instead of a Master's degree.^{[9]}

## Current course structure[edit]

The course lasts one year, divided into three eight-week terms. There are a wide variety of lectures on both pure and applied maths, mostly concentrated in the first two terms. The third term is primarily for examinations (and revision for said examinations) which, together with the option of writing a part III essay (introduced in the 1970s, a miniature thesis of sorts, often in the form of a literature review), determine one's final grade entirely.

## Grading[edit]

The grades available are Fail, Pass (Honours), Merit, and Distinction (the Merit grade was introduced in 2000). Cambridge recognises that in Part III of the mathematical tripos a merit is equivalent to a First Class in the other parts of the Tripos.^{[10]}^{[11]} The level of achievement required for a distinction is yet higher. Traditionally, results are announced in the University's Senate House. Standing on the balcony, the examiner reads out the class results for each student, and printed copies of the results are then thrown to the audience below. The students' exact rankings are no longer announced, but highest-ranked student is still identified, nowadays by the tipping of the examiner's academic hat when the relevant name is read out.

## Prizes[edit]

In addition to the grades, there are five associated prizes. Four of these may be awarded at the discretion of the examiners: the Mayhew Prize for Applied Mathematics, the Tyson Medal for mathematics and astronomy, the Bartlett Prize for applied probability^{[12]} and the Wishart Prize for statistics.^{[13]} Several notable astronomers and astrophysicists have been awarded the Tyson Medal in the history of Part III maths, including Jayant Narlikar, Ray Lyttleton and Edmund Whittaker. In addition, the Thomas Bond Sprague Prize is awarded by the Rollo Davidson Trust for distinguished performance in actuarial science, finance, insurance, mathematics of operational research, probability, risk and statistics.

## References[edit]

### Fourier Analysis

and### Exercises in Fourier Analysis

These ever popular works are still available from CUP. First read**Fourier Analysis**(ISBN-13: 9780521389914, cost £51) and then work through

**Exercises in Fourier Analysis**(ISBN13: 9780521438490, cost £33.99). If you prefer your books in Japanese, there is a translation of Fourier Analysis by Yorchiro Takanashi.

## Helpful Guides and Sundries

However, the main purpose of this home page is to give access to the most recent version of various collections of well meant advice that I have produced.

The documents are available in dvi, ps, pdf and latex form. (pdf documents can be read using acrobat.)

#### Helpful guides on

- Applying for Cambridge Research Fellowships: DVIPSPDFLATEX
- Part III: DVIPSPDFLATEX
- How to write a Part III Essay: DVIPSPDFLATEX
- Supervising undergraduates: DVIPSPDFLATEX
- How to listen to a maths lecture: DVIPSPDFLATEX (For those who wish to read it in Dutch PDF with splendid illustration.)
- The Naked Lecturer, thoughts on giving lectures: PDF
- How to write an essay in an undergraduate mathematics exam: DVIPSPDFLATEX
- A first draft of some advice for students from overseas wishing to do a first degree in mathematics at Cambridge: DVIPSPDFLATEX
- A second edition (even better than the first) of an advice sheet by Mark Joshi for Maths PhDs thinking of going into finance (he says add `suitable disclaimers blaming him for anything in it' so I do): PDF
- A strongly recommended collection of advice from Terence Tao to young mathematicians. http
- A practice based M-level degree. DVIPSPDFLATEX
- A letter claimed to be by the Duke of Wellington. (An obvious spoof, I think, but if you can prove that it is genuine or know of an `unimproved' letter on which is based, please tell me.): DVIPSPDFLATEX
- Plot synopsis for Jaws: DVIPSPDFLATEX
- An illustration of lion hunting drawn by an Olympic silver medallist. PS
- Demonstrating the `boarding house reach' to a female admirer JPJ
- The Cambridge Festival of Ideas: PDF
- Delaware `The triumph of hope' PS
- Not from the Laurie Taylor column press and from the Department of Canute Studies pdf.
- Have you got the time?press
- Chirality of airport carouselspress
#### Great musical moments

Next notes on courses presently given in the Tripos.

Notes for the Part II course on Codes and Cryptography in DVIPSPDFLATEX. (There is a log of corrections in DVIPSPDFLATEX.)

Notes for the Part II course Linear Analysis in DVIPSPDFLATEX.

Notes for the Part II course Topics in Analysis. (Note that the lecturer has more flexibility in this course than in most, so topics may vary from year to year.) There are two sets of notes the first contains an account of the results DVIPSPDFLATEX, and the second contains proofs of the results DVIPSPDFLATEX. There is log of corrections (starting 1st October 2015, those with older notes should download the present version) at PDF

Notes for the 1B 'Topology and Metric Spaces' course. DVIPSPDFLATEX. There is a log of corrections at PDF.

My Part 1A Analysis notes in DVIPSPDFLATEX.

### Courses close to those presently given

Next notes on courses corresponding to past schedules which are still close enough to course presently given.My Part 1A second half of the old course Algebra and Geometry (fairly close to the new course 1A Groups) in DVIPSPDFPSPDFLATEX.

My Part 1B Complex Methods (P3) notes in DVIPSPDFLATEX. (There are now new complex variable courses with different selections of material.)

My Part 1B Analysis (C9) notes in DVIPSPDFLATEX. These notes have been extended by Richard Johnson to include proofs in LATEXPDF. (Note that the analysis syllabuses have been rewritten since these C9 notes were written).

My rather old Linear Mathematics notes (WARNING the new syllabus contains substantially more.) DVIPSPDFLATEX

### Part III courses

Next various Part III courses.

My 2010/11 Part III course Topics in Analysis in DVIPSPDFLATEX.

Notes for my Part III Topics in Fourier Analysis and Complex Variable course (2009/10) in DVIPSPDFLATEX,

Notes for my Part III Functional Analysis course. in DVIPSPDFLATEX,

Notes for my Part III Topological Groups course (note that the last part follows Rudin's book very closely) in DVIPSPDFLATEX,

Notes for my 2012/13 Part III Fourier Analysis course in DVIPSPDFLATEX.

Notes for my Part III Complex Variable course in DVIPSPDFLATEX,

Please note that my Part III notes are particularly error prone.

### Notes of little useful relation to current courses

My IIB course on Partial Differential Equations (based very closely on notes by Joshi and Wassermann) in DVIPSPDFLATEX. (The course has now been completely rewritten.)

Notes on a course on a no longer existent Rings and Modules in DVIPSPDFLATEX.

Notes on a no longer existent Calculus and Methods course in DVIPSPDFLATEX.

Skeleton notes for an introductory course in Fourier Analysis given at the 2003 PCMI conference in DVIPSPDFLATEX. Solutions for some of the exercises for this course kindly contributed by Mihai Stoiciu DVIPSPDFLATEX.

### Notes describing my own work during the past few years

These notes give the background to and proofs of most of the results I have obtained in the last few years. DVIPSPDFLATEX.Caveat emptor, these notes come without guarantees of any kind. All contain misprints and some (particularly the Part III notes) much worse things. Corrections are gratefully received.

If you have any problems obtaining these documents write to me at

DPMMS, Wilberforce Road, Cambridge CB3 0WB

and I will send whichever one you want by classical means. (Because of the interaction between my computer illiteracy and the constant modernisation of our computer facilities weird things occasionally happen. Please e-mail me if they do.)

I would be happy to receive comments and suggestions for improvements. I am particularly anxious for suggestions of possible sources of funding for Part III students.

## And Finally

You can e-mail me by pressing SCHAZAM! or if that does not work by using the address **twk@dpmms.cam.ac.uk**. My favourite e-mail ends with the words `this message needs no reply'.

You may wish to go to the Faculty of Mathematics home page.

The guide to the University issued by the Administration begins by saying that the heart of the University `is a central administration team'. Later the guide informs us that the title of emeritus is a title applied `to a .... Professor ... who has retired after the age of sixty. ... the term is not conferred as a ... mark of distinction.' I am an emeritus professor. I have also retired from being junior Director of Mathematical Studies at Trinity Hall. The senior Director of Mathematical Studies is Tadashi Tokieda. His web page includes several cheerful hedgehogs (or, perhaps, several pictures of the same cheerful hedgehog) and much useful information in two of the many languages that he speaks. Those of a romantic disposition are directed to A Valentine from Möbius. If you would like a statement of the college's admission policy for mathematics undergraduates, press here. (Of course colleges are always fiddling with their web pages so, if this does not work, google Trinity Hall and start from there.)

You have read the book and studied the pamphlets BUT NOW in glowing LMScolour you can watch the film (video actually). In response to overwhelming lack of demand the video of my lecture `Marrying, Voting, Choosing' is now available for the price of £10 including postage and packing (a reader writes `Worth it for the postage and packing alone') from the LMS, De Morgan House, 57-58 Russell Square, London WC1B 4HP. Be the first person on your street to have your own copy!

Or, if you are one of those who wants everything to be free you can go to Gresham College and watch me pontificating on `Mathematics and Smallpox' (a reviewer writes `Mathematics and Smallpox, what's not to like?').

`There is no limit to what people will believe if they see it on the internet' Alan Turing 1936.

If you believe that the OED contains every word in the English language, the following is a counterexample.

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