Application of Index theorem in Dirac Operators

Guidance:

A dissertation is usually expository, collecting together results from several research papers into a coherent whole. Sometimes dissertations contain original research, and this is encouraged where appropriate. The dissertation should normally be produced in TEX or LaTEX.

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Length: Dissertations should normally be no longer than 15,000 words or 40 pages in length (including titlepage, abstract, and references). The principal reason is that it is considered too difficult to write a dissertation to a high standard that is any longer in the time available. Dissertations substantially longer than this length are likely not to be of the highest standard and may additionally be marked down on the basis of length.

Level of detail: The dissertation should contain enough detail for a reader familiar with the material taught in the MSc to understand what you have done. If necessary, material which you feel is too technical for the main text can be included in the form of appendices.

Structure of the dissertation

Dissertations need to contain title page, abstract, main text, list of references. A table of contents before the main text, which e.g. LaTEX can produce automatically, is also helpful.

The abstract is a short summary, one or two paragraphs long, of the context of the research and the main results. It is not a description of the layout of the dissertation.

A possible skeleton structure for the main text is:

1. Introduction: This should explain the context of the dissertation and situate it in the broader field of mathematics and should give an overview of the structure of the remainder of the report. It should also state clearly which area the dissertation will cover, whether there is any original work orwhether it is purely an overview or comparison of existing research results.

2. Main Body: This will comprise the bulk of the dissertation.

3. Discussion and conclusions: This should be a summary and critical evaluation of the dissertation. For a review of existing knowledge you should include a critical review of the material covered, its relevance, importance and realm of application. If you have produced original results you should explain how these relate to results of other people. You should also say how the dissertation could be improved or developed further in future work.

4. Referencing: The purpose of referencing is not to show that you have done some extended reading. It is to explain the current state of knowledge, to evaluate and debate (if necessary) existing knowledge, and to make clear which work is new and which is existing (to avoid charges of plagiarism). The list of references should be formatted consistently and contain all the information which a reader would need to retrieve the items referred to. For example, for an article from a journal you would normally list, in order: Authors, title of article, title of journal, volume number, page numbers, year of publication (e.g. “A. N. Other and B. Someone, Gnus versus gnats, Journal of Gnuology, 23, 110-113, 2002.”). For books also list the publisher; for edited volumes such as conference proceedings also the editors. References to websites should be used sparingly, since URLs tend to have a rather short lifespan, and should mention at least the author(s), title, and give the full URL. All references need to be referred to in the main text.

Attribution of othersʼ work: whenever you discuss, refer to or quote from other work, the appropriate reference must be given. This also applies to, for example, explanations of background material that are taken from lecture notes or books. Dissertations often tend to cite too little; for every statement that is not your own a citation must clearly identify the source. This applies even when the same work is referred to several times; just citing once on the first occasion is not sufficient. There are only very few exceptions to this general rule; e.g. if a whole section is devoted to a review of a particular paper or set of papers, it is sufficient to state this at the beginning of the section.

5. Style: The dissertation is expected to be written in the fairly formal language of a good scientific paper or text book. A chatty style such as “I found this wouldnʼt work, got fed up and tried a new idea” is frowned upon but the language used should not be so convoluted as to obscure the meaning. If in doubt of how to express something, write it as you would say it.

If you are in doubt about anything regarding the structure of your dissertation it is good to discuss and clarify it with your supervisor.

Finally some good piece of advice: It is important that you do not leave the writing up of your report too late; it frequently happens that good work cannot be given full credit because it is poorly described in the dissertation. It is therefore a good idea to write up continuously as you work on the project; e.g. chapters on background material can normally be written quite early on.

Marking criteria and policies

The project will be assessed by a dissertation. Two internal examiners, one of them your supervisor, will assess the dissertation using the criteria

Originality and independence of the work

Mastery and understanding of the topic

The extent to which the topic is dealt with

Presentation

Each of the four criteria has equal weighting. Both internal examiners must agree on a final mark for the project and an external examiner will give an opinion on the proposed mark of the two internal examiners.

Here are some questions an examiner might ask:

Does the introduction say clearly what the dissertation is about?

How well have you explained the area of work and summarised the relevant literature

Does the dissertation show evidence of learning beyond the material of the taught courses and options?

If the dissertation is mainly a survey, is it complete and up to date, and have you shown clearly, by summarising and comparing the literature in your own words, that you have mastered the subject?

If the dissertation involves numerical work, is it described clearly enough for someone else to reproduce the calculations, if required, and do the conclusions demonstrate that you understand what has been done?

Does the dissertation have a logical structure?

Are books and journal articles adequately referenced? Note that a complete bibliography is important for a good dissertation and should not be ignored

Where the results of numerical and other work are being discussed is this just a statement of what the results are or does the text contain real interpretation of the results. For example, does the text explain why the results are as they are?

Below are copies of the marking criteria for MSc dissertations from King’s College London and the Faculty of Natural and Mathematical Sciences.

Once you have received your project mark, you are able to request individual project feedback from your project supervisor if you wish.

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