Presentation slides base on my paper about Complex System and Information Theory

University of Essex Department of Mathematical Sciences

MA830/831: Capstone Project H A N D B O O K

(Academic Year 2018/19)

Last updated: 7 February 2019

MA830/831 Capstone Project Handbook

Capstone project coordinators: 1. Dr Chris Antonopoulos (module leader)

[Office: STEM 5.8, E-mail: [email protected], Ext. tel.: 3018]

2. Dr Georgi Grahovski [Office: STEM 5.37, E-mail: [email protected], Ext. tel.: 3033]

3. Dr Chris Saker (teaching projects) [Office: STEM 5.14, E-mail: [email protected], Ext. tel.: 2961]

The present handbook should be read in conjunction with the following University doc- uments and regulations:

• MA830/831 module description: https://www1.essex.ac.uk/modules/Default.aspx?coursecode=

MA830&level=6&period=SP&campus=CO&year=18

https://www1.essex.ac.uk/modules/Default.aspx?coursecode=

MA831&level=6&period=FY&campus=CO&year=18

• Maths Undergraduate student handbook: https://www1.essex.ac.uk/students/study-resources/documents/

2017_Department%20of%20Mathematical%20Sciences_UG.pdf

• University’s rules of assessment: https://www1.essex.ac.uk/students/exams-and-coursework/ppg/

ug/default.aspx

• University’s marking policy: https://www1.essex.ac.uk/quality/Documents/Marking_Policy_

2017_18.pdf

• University’s late submission coursework policy: https://www.essex.ac.uk/students/exams-and-coursework/

late-submission.aspx

• University’s academic appeals procedure: https://www1.essex.ac.uk/students/exams-and-coursework/ppg/

ug/appeals.aspx

• University’s academic offences policy: http://www.essex.ac.uk/see/academic-offence

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Contents

1 Introduction 5

2 Project calendar 6

3 Main stages 6 3.1 Allocation of supervisors and assessors . . . . . . . . . . . . . 6 3.2 Choice of topic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.3 Supervision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.4 Submission of project . . . . . . . . . . . . . . . . . . . . . . . . 8

3.4.1 Typesetting using LATEX . . . . . . . . . . . . . . . . . . 9 3.4.2 Duties of supervisor and assessor after submission . . 10

3.5 Oral presentation . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.6 Post-assessment and feedback . . . . . . . . . . . . . . . . . . . 11

4 Assessment guidelines 12 4.1 General guidelines . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2 Final presentation . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Responsibilities 15 5.1 Capstone project coordinator(s) . . . . . . . . . . . . . . . . . . 15 5.2 Supervisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.3 Assessor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.4 Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6 Advice to students on tackling the project 16 6.1 Planning and preparing your project . . . . . . . . . . . . . . . 17

6.1.1 Choosing your topic . . . . . . . . . . . . . . . . . . . . 17 6.1.2 Making the most of your supervisor . . . . . . . . . . . 19

6.2 A rough guide to Mathematical writing . . . . . . . . . . . . . 20 6.2.1 Style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6.2.2 Punctuation and language . . . . . . . . . . . . . . . . . 22

6.3 How to write a good project . . . . . . . . . . . . . . . . . . . . 24

7 List of project topics and supervisors for 2018/19 26 7.1 List of projects with Dr Chris Antonopoulos . . . . . . . . . . . 27 7.2 List of projects with Dr Dan Brawn . . . . . . . . . . . . . . . . 33 7.3 List of projects with Prof. Edd Codling . . . . . . . . . . . . . . 35 7.4 List of projects with Dr Georgi Grahovski . . . . . . . . . . . . 38

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7.5 List of projects with Dr Haslifah Hashim . . . . . . . . . . . . . 43 7.6 List of projects with Dr Junlei Hu . . . . . . . . . . . . . . . . . 46 7.7 List of projects with Dr Vanni Noferini . . . . . . . . . . . . . . 49 7.8 List of projects with Dr Christopher Saker . . . . . . . . . . . . 53 7.9 List of projects with Prof. Abdel Salhi . . . . . . . . . . . . . . 55 7.10 List of projects with Dr Hadi Susanto . . . . . . . . . . . . . . . 58 7.11 List of projects with Dr Alexei Vernitski . . . . . . . . . . . . . 59 7.12 List of projects with Dr Gerald Williams . . . . . . . . . . . . . 61

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MA830/831 Capstone Project Handbook

1 Introduction

This handbook is intended for use by students and staff involved in MA830/ 831 Capstone project. It contains all information needed for running the pro- jects and for their assessment1.

The Capstone projects are opportunities for a student to study a topic in Mathematics, Statistics and related areas (Mathematical Physics, Data Sci- ence, Modelling and so on) and to develop skills such as writing reports and giving presentations. The student will be monitored by a supervisor, who will periodically set tasks and discuss the progress of the work. The assign- ments run over one (MA830-Spring) or two terms (MA831). Final examina- tion (100% count) will be based on the final report submitted at the end of the academic year and an oral presentation together with the supervisor’s comments.

The amount of work required for Capstone project modules depends on the number of credits. For all modules the standard University rules imply that a 15-credit module involves 120 hours of student work. The main fea- tures of the two module options are summarised in the following table:

Module Credits Meeting/ Private Approx./Typical Supervision hours study hours report length

MA830 15 10/15 120 25 - 40 pages MA831 30 20/30 240 50 - 80 pages

These notes are intended as guidance, and should be read in conjunc- tion with the relevant module description2 in the University Catalogue and Maths Undergraduate student handbook3.

One aim of project work is that students should enjoy an opportunity to study independently and hence should be allowed a certain amount of lee- way how they do the project. This is best done by agreement between stu- dents and supervisors and so although these notes contain certain rules (e.g., supervisors should meet students in Weeks 1-2 of the Autumn term and sign the Capstone project allocation proforma) it is important that both parties feel able to negotiate between themselves an appropriate course of action in

1If this is not the case, or if you find any typos, inaccuracies or contradictions, then contact Dr Chris Antonopoulos at [email protected].

2Available at: https://www1.essex.ac.uk/modules/Default.aspx?coursecode=MA830& level=6&period=SP&campus=CO&year=18 or https://www1.essex.ac.uk/modules/Default.aspx?coursecode=MA831&level=6&period=

FY&campus=CO&year=18

3Available at: https://www1.essex.ac.uk/students/study-resources/documents/2017_ Department%20of%20Mathematical%20Sciences_UG.pdf

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most instances. A key aim is that students should be given the opportunity to show their strengths.

2 Project calendar

Time Action Pre academic year Module coordinator updates Moodle page Autumn term: Week 2 Informal allocation of supervisors

(based on students’ submitted proforma) Weeks 3-4 Final allocation of supervisors

Students and supervisors meet Week 7 Final allocation of assessors Spring term: Week 23 Module coordinator contacts students

and supervisor to check progress/issues Summer term: Week 30 Final reports for MA830 and MA831 Friday submitted via Faser (26 April 2019, 16:00) Weeks 32, 36 and 37 Oral presentations June Documentation prepared for Examination Board

3 Main stages

3.1 Allocation of supervisors and assessors

The process of allocating supervisors starts informally before the academic year begins as many students like to start their projects early and so that supervisors can plan their teaching.

The allocation to a supervisor will be made as far as possible to accommo- date the wishes of the student as to subject area, though latecomers may have more limited choice. Once each student has been allocated to a supervisor, the precise topic to be studied will be agreed between the two.

It is the responsibility of the module coordinator to ensure that each stu- dent has a supervisor by the end of Week 2 of the Autumn term, although in exceptional circumstances this may not be possible. The ideal situation is one in which all supervisors are allocated before the start of Week 4. An ap- propriate assessor should be agreed by end of the third week of the Autumn

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term. The module coordinator should contact all students and supervisors in

Week 2 of the Autumn term to check whether there are any problems. This is an important opportunity to catch problems before they become serious. Students should be aware that it is very difficult to change supervisors once the module has started and so any issues, such as change of supervisor or project, should be dealt with as early as possible.

3.2 Choice of topic

Lists of topics are available at the beginning of the Summer term of the pre- ceding academic year and will be circulated to all second year students.

The level and amount of material for the 15-credit and 30-credit modules should approximately correspond to one and two standard 15 credit mod- ules, respectively. Much of the material will usually be available in text- books, although the development of a topic should not follow any one text- book completely, and the student will be expected to consult other literature including research articles in journals.

In Statistics an assignment may not neatly fit the above description, but may include other characteristics such as:

(i) a substantive analysis of a specific dataset,

(ii) a comparison of the strengths and weakness of different statistical meth- ods,

(iii) writing computer code in R to implement a statistical analysis or to sim- ulate data.

3.3 Supervision

At their first meeting in weeks 1-2 of the Autumn term, the supervisor will provide the student with an outline of the agreed topic, together with appro- priate reading such as textbooks and/or articles. This should be considered as a guide, since students may have their own suggestions.

The student will be able to consult the supervisor in case of difficulty. The time for consultation is officially limited to ten or twenty hours for the 15- credit and 30-credit assignments, respectively. However, most supervisors are flexible about this and a weekly meeting is not uncommon. The student and supervisor should agree the timing of meetings.

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3.4 Submission of project

The final report should be submitted via FASER4. The Capstone project coor- dinators (or the relevant administrator from the Department of Mathemati- cal Sciences office) will promptly pass the report to the relevant assessor. The electronic versions should be in Word or PDF format (electronically scanned copies of a wholly handwritten report are not permitted).

The submission must be accompanied by an academic integrity form/dec- laration (the will be done electronically via FASER) and will be checked for plagiarism using the standard University software. If any plagiarism is sus- pected, the standard University procedures on plagiarism will be followed. Information on what constitutes plagiarism can be found in the University Taught Student Guide.

The deadline for submission is the end of Week 30 of the Spring term. For the academic year 2018/19 this is

Module Week Date MA830 (Spring) Week 30 Friday, 26 April 2019

16:00 MA831 Week 30 Friday, 26 April 2019

16:00

In accordance with University rules, late submission results in a mark of zero5. If a student is unable to submit the work, for whatever reason, then they should contact their supervisor or module coordinator as soon as pos- sible. Students are reminded that information on extenuating circumstances can be found on the webpage of the Department Mathematical Sciences6.

The final project should include a summary (typically, in the introductory section) describing the scope of the work and the main results, indicating the main sources used.

Pages and sections should be numbered for easy reference and the title page should contain student name, student number, module code and title, title of project, supervisor and the date of submission.

The report should include a contents page and a full bibliography, indi- cating all the books, articles or websites used.

4According to the new University rules, starting from 2017/18 academic year, printed copies are not re- quired anymore.

5Details of the University’s late submission of coursework policy can be found here: https://www. essex.ac.uk/students/exams-and-coursework/late-submission.aspx.

6This information is available also at: https://www1.essex.ac.uk/students/ exams-and-coursework/ext-circ.aspx.

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Recall that, in the text, if a lemma or theorem is taken from some source, there must be an indication of the source, and whether it has been signifi- cantly expanded or adapted. Apart from short statements, students should not copy directly from sources, and everything must be expressed in the stu- dent’s own words, except where explicitly stated. Students whose work is too closely based on their sources will have marks deducted.

Key points of the final report

1. Title page: Name, number, module, title, supervisor, date.

2. Academic integrity form (electronic).

3. Contents page.

4. Numbered pages (and sections).

5. Summary or work and sources (usually within the introduction).

6. Bibliography with the sources used.

7. Upload on FASER, before the deadline, the Word or PDF file of the report.

3.4.1 Typesetting using LATEX

It is recommended that you use the LATEX typesetting package to produce your report, which is available on the University network (the best version is called TeXworks), or can be downloaded freely from the web. As you may know, LATEX is a typesetting package designed for writing mathematical reports. There are many LATEX tutorials along with other documentation on the web. For example, you may find it helpful to consult the relevant section of Capstone Projects Moodle page.

There are also many LATEX editors that you may find useful and facilitate the writing of your report. A good LATEX editor is TeXstudio that you can freely download from http://texstudio.sourceforge.net/.

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MA830/831 Capstone Project Handbook

3.4.2 Duties of supervisor and assessor after submission

After submission of the report, the Capstone project coordinator or the rele- vant Departmental administrator will promptly pass the report to the asses- sors. The supervisor is presumed to be one of the assessors, in addition to the main assessor.

The project will be read and marked independently by the assessors. It is recommended that the main assessor discusses the report with the supervi- sor before the oral presentation: assessor’s comments on the initiative and independence should be based on the input from the supervisor.

The mark components, which are informed by the presentation and ques- tioning, will be finalised after the oral presentation.

3.5 Oral presentation

A time and venue for the presentations will be arranged for each student dur- ing Weeks 32, 36 and 37 (immediately after the examination period). There will be opportunity to practise these presentations beforehand. Capstone Project coordinator will inform students in good time of the date and place of the presentations.

The format of the presentation will vary depending on the subject. In Pure and Applied Mathematics students generally give a 15-20 minute talk and there is 15-20 minutes for questions from the panel. In Statistics and Data Science the talk may be shorter and the question section longer.

The key to a presentation is that the student should demonstrate under- standing. Given the time constraints it will be impossible for students to explain every detail from their report. Thus students should choose some aspect, e.g., a theorem or set of examples, and use that to demonstrate that they understand the material in the report.

1. Students should demonstrate that they understand and have mastered the material.

2. Students should demonstrate their presentation skills.

Students can make the presentation in any format they like, e.g. LATEX (Beamer) slides, OHP slides, PowerPoint, black/white board, video, ip-chart, etc. Students should inform the module coordinator at least one week ahead of time of their requirements for the presentation. It cannot be assumed that the room for the presentation will contain a computer with every possible

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computer program. Students are advised to bring electronic files on a mem- ory stick, as occasionally wi-fi connections etc might not work.

The assessment panel of the project consists of

1. module coordinator,

2. supervisor,

3. assessor,

4. other members of staff (for example, for moderation purposes).

All members of the assessment panel can ask questions, make comments and give their opinions. Other students are not allowed to attend.

Whilst being sensitive to a student struggling during a presentation, su- pervisors should avoid answering questions for them (panel members often ask questions not because they want to know the answer but because they want to know if the student knows the answer!)

After the presentation the student will leave and the panel will discuss the presentation and report. An initial mark will be agreed. After all projects have been assessed, the module coordinators can moderate marks taking into consideration fairness and consistency. The module coordinators are responsible for the final module marks presented to the exam board in June.

The final presentation/interview is inseparable part of the assessment process – Capstone projects cannot be assessed based on the final report / dissertation. Failure to attend the final interview is formally considered as a failure to attend an exam.

3.6 Post-assessment and feedback

The assessors are responsible for filling out the assessment and feedback form, with the input of the supervisor to the ’Initiative and Independence’ section.

The Capstone project coordinators are responsible for ensuring that all marks are collected, moderated, and submitted to the Department of Mathe- matical Sciences Undergraduate Office by the relevant University deadlines. They are responsible for collecting the reports from the assessors and ensur- ing that they are ready for the external examiners.

A copy of the final assessment reports will be available to the student after the Exam board in June, therefore should contain comments and constructive criticism (assessors should avoid unhelpful phrases like ‘Could have been better’, for example).

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Assessors and supervisors should be aware that all assessment documen- tation written by them may be made available to the student and the external examiner.

4 Assessment guidelines

4.1 General guidelines

Broadly speaking, projects can be divided into several different types:

• Reading several sources and presenting some mathematical ideas. The mathematics is expected to be correct and substantial, and the presenta- tion coherent.

• Investigating a mathematical or statistical model using numerical/sta- tistical methods.

• Developing some new mathematical ideas or details, where statements and/or proofs are not in the literature. Compared to the previous types, the amount of mathematics can be less, but should be no less accurate.

• Essay style projects - for example a historical project. The amount of ac- tual mathematics would be lower, but there should be a correspondingly greater amount of analysis and criticism.

• Industry-oriented projects (internships). The variety and nature of in- dustry based projects is very broad in general. However, it should be tailored to the business of the company and demonstrably applicable to their routine operations. A project oriented in this way could potentially provide valuable experience and has an obvious input into employabil- ity beyond any particular business.

• Teaching projects. These projects can focus on one or more aspects of mathematics education, they could, for example include a school place- ment with the write up taking the form of an essay style project or they could involve the creation of mathematics education resources for use in schools or undergraduate study such as e-assessment materials, interac- tive resources in packages like GeoGebra or videos explaining mathe- matical problems/concepts.

Other types of project are also possible, and many projects will be a com- bination of more than one of these aspects. The purpose of this section is

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provide unique criteria for assessment ensuring equal treatment of project of all types.

The general structure of the project assessment is as follows:

Type Count Deadline for submission Assessment 2. Final Presentation 100 % Week 30 Weeks 36, 37

4.2 Final presentation

The following is meant as a guidance to what constitutes a particular mark. Comments on the assessment pro-forma should refer to some of these crite- ria.

Class I, 70 - 100 %:

• Extremely well organised and presented.

• Project could serve as a basis for a course at the appropriate level.

• Excellent choice of examples and logical flow.

• Required little help from supervisor (relative to difficulty of topic).

• Good evidence of originality and independent thinking.

• Mastery of material.

To achieve a Class I students do not need to have achieved mastery or excellence in all the above. Greater marks will be given for originality and evidence of independent thinking.

The full range of marks from 70 to 100 should be considered with the following criteria in mind:

Class I, 95-100 Excellent in all criteria. Of publishable quality.

Class I, 85-94 Excellent in most criteria and highly competent in others, shows mastery of the material. Could be used as a basis for a course in the material without many changes.

Class I, 75-84 Excellent in many criteria and competent in others, demon- strating a high degree of mastery of material.

Class I, 70-74 Excellent in many criteria and competent in others, though demonstrating a high degree of mastery with some minor gaps.

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Class II (upper), 60-69%:

• Well organised and presented.

• Good choice of examples and logical flow.

• Required a reasonable amount of help from supervisor (relative to diffi- culty of topic).

• Some evidence of independent thinking. Follows standard texts some- times.

• Sound understanding of material.

• Project could with some significant corrections be used as a basis for a course on the material.

Class II (lower), 50-59%:

• Adequately organised and presented.

• Reasonable choice of examples and logical flow.

• Required a substantial amount of help from supervisor (relative to diffi- culty of topic).

• Little evidence of independent thinking. Tends to follow sources.

• Some good understanding of material.

Class III, 40-49%:

• Poorly organised and presented.

• Poor choice of examples and logical flow.

• Required a significant amount of help from supervisor (relative to diffi- culty of topic).

• No evidence of independent thinking. Slavishly follows sources.

• Some understanding of material.

Fail, 0-39%:

• Almost non-existent organisation and presentation, for example sections missing.

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• No new examples. Illogical arguments.

• Required a substantial amount of help from supervisor (relative to diffi- culty of topic) or did not see supervisor.

• No significant understanding of material demonstrated.

5 Responsibilities

5.1 Capstone project coordinator(s)

1. Ensure that the present handbook is available on the Department of Mathematical Sciences webpage and that the Project’s Moodle page have all materials relevant to the new academic year before it begins.

2. Matching of supervisors and students by end of Week 2.

3. Assign assessors by end of Week 3.

4. Check for problems in Week 6 by emailing students and supervisors.

5. To send email reminders to students and supervisors regarding dead- lines, if necessary. This can be partially automated using Moodle.

6. Organise presentations. Ask students for their presentation require- ments.

7. Ensure fairness and consistency in the assessment process. Moderate marks if necessary.

8. Prepare marks and projects for external examiner.

9. Send completed assessment reports to the student (after the Exam board).

5.2 Supervisor

1. To meet with student in Weeks 1-2 of Autumn term to discuss project.

2. Provide project outline to student and (later) to assessor.

3. Provide advice and support to students during meetings/supervisions.

4. To report any concerns regarding student work (such as no supervision) to Capstone project coordinator(s) in a good time.

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5.3 Assessor

1. To read the final report in good time.

2. Complete the relevant sections of the assessment form before the final presentation/interview.

5.4 Student

1. Contact potential supervisor(s) in good time to allocate a project.

2. Arrange and attend supervision meetings.

3. To be aware of deadlines.

4. Submit work by deadlines.

5. Not plagiarise.

6. Ensure to be registered for the relevant module (MA830/MA831) in good time. Read and reply to e-mails.

7. Communicate their needs for presentation (such as Powerpoint, black- board) to Capstone project coordinator(s) at least one week before the presentation.

6 Advice to students on tackling the project

For most of your life so far, the only kind of writing you have done in Maths classes has been on homework and tests, and for most of your life you have explained your work to people that know more mathematics than you do (that is, to your teachers).

Writing is a significant and essential part of being a mathematician, and anyone who enters the profession will find their time occupied with the writ- ing of mathematics papers, grant proposals, letters of recommendation, ref- eree reports, and a variety of other items.

But most of all, one of the simplest reasons for writing a project in Maths and Science is that writing helps you to learn and understand better. By explaining a difficult concept to other people, you end up explaining it to yourself.

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6.1 Planning and preparing your project

Your Capstone research project may involve months of work, so planning and preparing can be one of the most important stages in the process.

6.1.1 Choosing your topic

Sometimes having to choose your own topic can be a difficult task – answer- ing the following questions may help you to focus your thoughts:

Personal interest

• Do you have a genuine interest in the topic? You will have to spend a lot of time over the next few months working on it, so you need to make sure that it will sustain your interest and motivation!

• Think back to previous modules – was there anything that particularly stoked your interest?

• Are there any obvious potential topic areas which fit well with your wider studies and (if applicable) future study plans?

• Are there any topics that fit with your future employment plans? Will it add anything to your CV? Will you be able to talk about it in job appli- cations?

Scope and manageability

• Is the topic overlay ambitious or too broad? Remember you only have a short period of time to do the research.

• Is there sufficient background literature to work with? What are the key sources of information?

• Does your topic have a clear aim? Can you identify the aims and objec- tives?

• How will you go about researching the topic? Will you be carrying out primary research? If so, how? Will you have enough time to gather the information and to process the results?

Skills and methods

• Do you have the right skills to complete the research? Is it realistic or too ambitious?

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• If you answer yes to the above question, how much training will you need to acquire new skills/methods, and where and when you will gain them?

• Do you have access to all of the facilities and equipments necessary to carry out your research? Will it be accessible to you when you need it?

Originality

(These are just all points to consider; your project does not have to do all of these things)

• Can you identify an original angle on the research, and would you be able to make an original contribution?

• What is already known in the topic? Can you strongly challenge existing interpretations and ideas?

• What are the current issues or questions surrounding the topic? Is there a gap in the research that you can potentially fill?

• Can you exploit existing sources or data in new ways?

• Can you look at an unstudied topic which impacts on existing theories?

• Can you exploit new sources or data for old problems?

• Can you employ new methodologies?

• Can you apply approaches from other fields?

Inspiration

• Can you look at past research projects to get inspiration?

• Can you talk to former students who have already carried out similar types of research?

• Have you talked to your supervisor about your ideas?

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6.1.2 Making the most of your supervisor

What should you expect from your supervisor?

In general, the supervisor’s role is advisory: they are there to help you car- rying out your research project autonomously. There are number of ways in which your supervisor may support you:

• Monitor the development of your project and give advice how to de- velop your initial ideas.

• Recommend literature that would be relevant to your project.

• Provide guidance and advice on how to plan, prepare and carry out your research.

• Comment on your research.

Your supervisor is not there to:

• Do the research for you.

• Supply you with books to read.

• Proofread your final piece of work.

• Do check spelling and correct grammar.

• Scrutinise your final piece of work for plagiarism.

What your supervisor will expect from you?

Your supervisor is a great source of support and information, so it is impor- tant that you understand what they expect from you:

• It is your responsibility to make initial contact and to maintain contact with your supervisor – they will not chase you!

• Keep appointment with your supervisor. If you are unable to attend an appointment – let them know!

• Be prepared! Make constructive use of meetings by having a clear idea of what you want to discuss at each session.

• Take ultimate responsibility for your research project.

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6.2 A rough guide to Mathematical writing

Writing mathematics well is a skill that takes time and practice to learn, and the following list is meant to provide the beginner with aspects of style to consider, conventions to be aware of, and common pitfalls to avoid. While much of the following may seem like common sense, keep in mind that it is surprisingly easy, even for experienced writers, to forget the following guide- lines when immersed in writing mathematics.

Below we present a quick guide about the style and punctuation of a generic mathematical text.

6.2.1 Style

1. Use brevity in your writing. Write as simply and directly as possible. Avoid the use of ponderous or pretentious prose, and remove any unnec- essary words or phrases. It is surprising how often one can take a piece of writing and improve it merely by removing portions. This does not mean that your final product must be short, or that you must leave out details. In- stead, you should write so that every word, phrase, and sentence contributes to what you are trying to communicate.

For example, there is no need to say: “and now we prove a lemma”. Simply prove it. Likewise there is no reason to remark: “we have proven the claim” at the end of a proof, since the symbol ⇤ says precisely that already. Do not repeat yourself in your writing, and do not use superfluous phrases. When you proofread a paper for the first time, ask yourself after every sentence whether the reader would be any less informed you removed it. If the answer is “no”, you should take it out.

Furthermore, keep in mind that writing concisely takes far more time and effort than writing at length – it is for this reason that Pascal, once at the end of a long letter, apologised for not having had the time to write a short one. However, writing in a concise and succinct manner is well worth the effort because it contributes to producing work that is clear, well organised, and direct.

2. Use language precisely and correctly. In mathematics more than any other subject one needs to be careful of word choice. Theorems must be stated carefully and unambiguously, arguments must be logical and clear, and exposition must convey what the writer intended. A mathematician is unlikely to use technical terms, such as “differentiable” or “Hausdorff”, incor- rectly. However, when writing English prose it is common to be more sloppy

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as one tries to convey technical ideas with what is often imprecise language. Moreover, one should be careful of using words that may have unintended meanings. For instance, referring to an example of a ring as “simple” to mean that it is easy to understand could be misconstrued as meaning “the ring has no ideals”. Similarly, English words such as “complex”, “trivial”, and “natural” can be misinterpreted.

3. Organise your paper in an order that makes the exposition clear. In par- ticular, this will not usually be the order of discovery. Often when proving theorems one will first obtain a collection of results, and then later prove a theorem or create a theory which encapsulates these results as special cases. When writing up these results, one may want to first prove a general the- orem and then obtain the special cases as corollaries. On the other hand, it may be appropriate to begin with a few specific examples which identify the important concepts and motivate the more general work to follow. (In either case, however, one would not want to prove a specific theorem first, and then a more general theorem later, since this would result in unnecessary repetition.)

4. Write a good introduction. Most people who read a mathematics paper will only read the introduction and skim theorems. Furthermore, when a reviewer reads a proposal for a grant or fellowship, it is the introduction which will have the most influence on the reviewer’s/marker’s opinion of your work. Consequently, you should put a great deal of time and effort into writing an effective introduction. Introduce the problem you are working on, motivate the solution, clearly state (or summarise) your results, and explain the significance of the solution. In addition, do your best to connect your work to the work of others, to other areas of mathematics, and to various applications. Remember that in an introduction you are often trying to “sell” your work and convince others of its importance. Also be aware that many people write the introduction to a paper after they have finished the body. This gives them the advantage of knowing exactly what will be done in the paper as they compose the introduction.

5. Write with the reader in mind. Identify an audience, and write with an awareness of that audience. As you write a mathematics paper remember that, unlike you, the reader has not been thinking intensely about the ma- terial for an extended period of time. Therefore, provide the reader with references, include useful comments, and give additional explication so that someone unfamiliar with the work can follow it. In addition, when writing a proposal for a grant or fellowship, remember that the reader will be a mathe-

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matician that may not be in your area, or at the very least may be unfamiliar with the specifics of the subject you work on. Finally, be especially care- ful when writing for non-mathematicians. It is all too easy to assume your reader knows more mathematics than they actually do. If you are going to talk about continuous functions, keep in mind that most people don’t even know what a function is, much less understand the concept of continuity, so you will have to explain these ideas if you wish to use them. Also be careful of English language that is commonly used in mathematics, but is usually unfamiliar to the layman. For example, terms such as “if and only if”, “con- trapositive”, or “nontrivial” are used so often in mathematics that one often forgets that a non-mathematician may not know what they mean.

6. Make sure your writing flows. Avoid writing a succession of loose sen- tences. Particularly when writing proofs, it is easy to become so engrossed in the mathematics that one forgets to pay attention to English style. The result is often a proof that reads “. . . and then . . . and then . . . and then . . .”. Try to use a variety of words in proofs, such as “therefore”, “consequently”, “it follows that”, “we see”, “hence”, or “thus”.

7. Listen to criticism and learn from it. It is natural to become attached to your writing and even to be proud of it. Consequently, it is often dif- ficult to receive criticism without taking it personally. Keep in mind that criticism and the comments of others are often your most valuable tools for improving your writing. Often you have invested so much in what you have written that you are incapable of putting yourself in the position of the unini- tiated reader. Therefore, feedback from others can be an indispensable tool for making your writing clearer, identifying portions that are confusing, and anticipating your readers’ reactions.

6.2.2 Punctuation and language

1. Do not use common blackboard abbreviations. For example, write “if and only if” rather than “iff”, and “without loss of generality” rather than “WLOG”. This also applies to symbols such as 8 and 9. Unless one is writing a paper in mathematical logic, one should write out “for all” and “there exist(s)”.

2. Punctuate equations and mathematical symbols. Mathematical expres- sions are no different than the words they represent, and they should be punctuated accordingly. This applies even to displayed equations so that, for example, if a displayed equation is at the end of a sentence it should end with a period.

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3. Do not use contractions in formal writing. Thus words such as “don’t”, “can’t”, “I’m”, and “we’ve” should be written out.

4. Use the first person plural when writing mathematics papers. It is a convention in the mathematical community to use the first person plural, or “we”, when writing papers. This choice has many advantages. It conveys the active and participatory nature of the project, making readers feel involved as they work through the paper. In addition, it is what most people are used to hearing in mathematics classes or talks, and therefore has a familiar ca- dence which is less likely to cause distraction. Furthermore, it avoids many of the problems encountered with other choices, such as the pretension of the first person singular “I”, or the awkward sentences that arise with the third person singular “one”.

5. Write all necessary hypotheses in statements of theorems. A person should be able to open your text to any theorem, read it, and know what you are talking about without having to refer to earlier portions of the document. In fact, it is likely that this is how most people will use your papers. If at all possible, make the statements of your theorems completely self-contained (as much as possible) so that the reader does not have to look throughout your text to decipher notation or find definitions of special terminology. 6. Use Latin abbreviations correctly. The following table summarises the meanings of some commonly used Latin abbreviations:

Abbreviation Latin term English translation i.e. id est that is e.g. exempli gratia for example cf. confer compare n.b. nota bene note well (or just note) q.v. quod vide which see viz. videlicet namely et al. et alii and others

In particular, the abbreviations i.e. and e.g. are often mistakenly interchanged, and cf. is often misused to mean “see” when it actually means “compare”. Also note that there is no period after “et” since it is not an abbreviation.

7. Do not start a sentence with a variable or symbol. Although it is perhaps technically correct, it is considered bad style to do so. Usually this can be avoided by simply rewording the sentence; e.g., rather than “n points are on the interior” one would write “The interior contains n points”. (There are, of course, some exceptions to this rule. In particular, most people would con-

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sider it acceptable to start a sentence with a mathematical term that contains a symbol, especially if it is an uppercase symbol. For example, one should feel free to start a sentence with the word C⇤-algebra or the word K-theory.)

6.3 How to write a good project

1. Give answers to challenging questions. Markers are more likely to be impressed by your solution to harder exercises. (As a guide, “Exercise 14.8 from a book is likely to be harder than Exercise 1.1.”)

2. Give your own examples after definitions.

3. Generalise results.

4. Deviate from the standard texts. Do not just follow the definition, the- orem, proof sequence given in a particular textbook. Work out what is important about what you want to say and decide for yourself what should be a lemma, a theorem, a definition.

5. Collect from a variety of sources. What is the point of just rewriting every sentence from the standard text book?

6. Use consistent notation. Different books use different notation. If you replace the notation in the correct places, then you demonstrate under- standing.

7. Tell us what you have added: E.g., “In [6] Smith sketches a proof that every grundle is rationally elliptic, here I shall give the details”. Another example is “Jones’ proof in [12] is incomplete as she asks the reader at two points why a statement is true. I have provided the answers”.

8. If you have to take something verbatim from a source, then quote the source and demonstrate that you understand it by giving a pertinent discussion or a good example.

9. Make sure the account is clear (e.g., words in theorems are defined) and logical (e.g., the definitions come before theorem).

10. When analysing data in Statistics, take care to describe exactly what sta- tistical methods are being used and how they enhance the understand- ing and interpretation of the data.

11. Use appendices, for example for original source code of numerical work.

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12. Use a number of sources from journals.

A key point is that you should demonstrate that you understand what you have written. You do not want the marker to think that everything has just been copied without any thought. Markers are more likely to reward inde- pendence and students being proactive, but that should not stop you from asking for help and advice. Moreover, having fewer meetings with the su- pervisor or not seeking their advice should not be used as a criterion for independence.

References [1] Kevin Houston, How to Think Like a Mathematician: A Companion to Undergraduate Math-

ematics, Cambridge University Press (2009).

[2] S. Krantz, A Primer of Mathematical Writing, AMS, Providence, Rhode Island (1997).

[3] N. E. Steenrod, P. R. Halmos, M. M. Schiffer and J.-R. Dieudonné, How to Write Mathe- matics, AMS, Providence, Rhode Island (1973).

[4] L. Gillman, Writing Mathematics Well: A Manual for Authors, The Mathematical Associ- ation of America, Washington, D.C. (1987).

[5] N. J. Higham, Handbook of Writing for the Mathematical Sciences, SIAM, Philadelphia, Pennsylvania (1993).

[6] American Mathematical Society, A Manual for Authors of Mathematical Papers, 8th ed., pamphlet, 20 pp, AMS (1984).

[7] E. Strunk (jr.) and E. B. White, Elements of style, The Penguin Press, New York (2005).

[8] The Chicago Manual of Style: The Essential Guide for Writers, Editors, and Publishers (16th edition), Chicago University Press, Chicago - London (2010).

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7 List of project topics and supervisors for 2018/19

In this section we present a list of project topics, proposed by the supervisors available for 2018/19 academic year. You are also encouraged to propose your own topics to any of the potential supervisors from this list. You will have to convince your potential supervisor that you have a suitably thought- out plan of what you are going to do, which is at a suitable level. If necessary, you can discuss your choice with one of the project coordinators, who may be able to suggest appropriate supervisors for the topics which interest you.

Scroll through the next pages!

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7.1 List of projects with Dr Chris Antonopoulos

Dr Chris Antonopoulos Room: STEM 5.8 E-mail: [email protected] Ext. tel.: 3018

Research interests: • Nonlinear dynamics • Dynamical systems • Chaos theory • Mathematical modelling • Complex systems and complex networks

All projects can be chosen as one-term projects.

1. Strange attractors - The Lorenz attractor and chaotic dynamics An attractor is a set of numerical values toward which a dynamical system tends to evolve, for a wide variety of initial conditions. System values that get close enough to the attractor remain close even if initially are slightly disturbed. The attractor is a region in an N-dimensional space. In physical systems, the N dimensions may be, for example, 2 or 3 positional coordinates for each of one or more physical entities; in economic systems, they may be separate variables such as the inflation rate and the unemployment rate. If the evolving variable is 2- or 3-dimensional, the attractor of the dynamical system can be geometrically represented in 2 or 3 dimensions. An attractor can be a point, a finite set of points, a curve, a manifold, or even a compli- cated set with a fractal structure known as a strange attractor. If the variable is scalar, the attractor is a subset of the real number line. Describing the at- tractors of chaotic dynamical systems has been one of the achievements of chaos theory.

The Lorenz system was developed by Edward Lorenz in 1963 as a sim- plified mathematical model for atmospheric convection and it is a system of 3 coupled nonlinear ordinary differential equations. It is notable for having chaotic solutions for certain parameter values and initial conditions. Partic- ularly, the Lorenz attractor is a set of chaotic solutions of the Lorenz system

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Figure 1: The Lorenz attractor

which, when plotted, resemble a butterfly or figure eight such as the one in Fig. 1.

The project will be a fascinating journey into the world of dynamical sys- tems, chaos theory and strange attractors. It has theoretical and computa- tional aspects, however, it is more likely to contain programming.

Suitable for a group of 2 - 3 students working in collaboration.

References [1] C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Dover Pub-

lications, 1982.

[2] Chaos, and What to do About it?: http://chaosbook.org/

2. Mathematical modelling of complex systems: The brain

A plethora of phenomena in nature can be effectively described by networks, among them the structure in the brain. The study of the function of the brain is of primordial importance in neuroscience. Neuroscientists have used tools for the analysis of complex networks that help us realise even more deeply the functionality and structure of the brain. It was found that many aspects of brain network structures are typical of a wide range of non-neural or non- biological complex networks. One of the main findings in neuroscience is the modular organisation of the brain, which in turn implies an inherent parallel nature of brain computations. Modular processors must be sufficiently iso- lated and dynamically differentiated to achieve independent computations, but also globally connected to be integrated in coherent functions. It has

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Figure 2: Studying the brain using mathematical modelling

been revealed that the cortical network is a hierarchical and clustered net- work with a complex connectivity. A possible network description for this modular organisation is that brain networks may be small-world structured with properties like many other complex networks. This viewpoint has been driven by the systematic finding of small-world topology in a wide range of human brain networks derived from structural, functional, and diffusion tensor MRI studies.

In this project, you will be introduced to the beautiful world of complex systems and the brain, and you will learn how to study the brain using math- ematical modelling, networks and dynamical systems theory (see Figs. 2 and 3). This project has theoretical and computational aspects. However, it is more likely to contain programming.

Suitable for a group of 2 - 3 students working in collaboration.

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Figure 3: Complex topology in the brain (picture taken from [“Network Neuroscience The- ory of Human Intelligence”, Barbey, Aron K., Trends in Cognitive Sciences, 22, 1, 2018])

References [1] H. Sayama, Introduction to the Modelling and Analysis of Complex Systems, OPEN SUNY

Textbooks, Milne Library, 2015.

[2] C. G. Antonopoulos, S. Srivastava, SEdS Pinto, M. S. Baptista MS, Do Brain Networks Evolve by Maximizing Their Information Flow Capacity?, PLOS Computational Biology 11 (2015), 8, e1004372.

3. Network inference from time-series data using information theory tools

We understand a complex system as a system with many interacting com- ponents whose aggregated behaviour is nonlinear and undetermined from the behaviour of the individual components. If we now consider these com- ponents as nodes of a network, and the underlying physical interaction be- tween any two nodes as links, one way to understand these complex sys- tems is by studying its topological structure, namely, the network connectiv- ity. In natural complex systems, the connectivity of the components is often unknown or is difficult to detect by physical methods due to large system- sizes. Hence, it is of interest to infer the network structure that represents the physical interaction between time-series collected from the dynamics of the nodes.

In this project, you will be introduced to the beautiful world of com- plex systems and you will learn how to study dynamical systems and use information-based methodologies to infer the structure of complex systems from time-series data (see Figs. 4 and 5). This project has some theoretical and computational aspects, and the relative proportions of the two can be tai- lored according to the students’ taste. However, it is more likely to contain programming.

Suitable for a group of 2 - 3 students working in collaboration.

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Figure 4: Different network topologies to construct complex systems

Figure 5: Network inference using information-dynamical methods for a network of nodes with neural dynamics

References [1] H. Sayama, Introduction to the Modelling and Analysis of Complex Systems, OPEN SUNY

Textbooks, Milne Library, 2015

[2] E. Bianco-Martinez, N. Rubido, Ch. G. Antonopoulos, M. S. Baptista, Successful Net- work Inference from Time-series Data Using Mutual Information Rate, Chaos: An Interdisci- plinary Journal of Nonlinear Science, 26 (2016) 043102.

4. Complex statistics and diffusion in nonlinear disordered particle chains

The absence of diffusion in disordered media, often called Anderson local- isation, is a general phenomenon that applies to the transport of different types of classical or quantum waves. An interesting question is what hap- pens to the diffusion if nonlinearity is introduced. Many studies so far have focused on the evolution of an initially localised wave packet showing that it spreads subdiffusively for moderate nonlinearities; while for stronger ones, a substantial part of it remains self-trapped. Currently, a greatly debatable problem concerns the long-time spreading of the wave packet.

You will learn about q-exponential statistics to study the existence of pos- sible connections between regimes of “weak” and “strong” chaos and sub- diffusive motion in the presence of nonlinearity and disorder in the Klein -

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10-4

10-3

10-2

10-1

-20 -10 0 10 20

P(sM (j))

sM (j)

a)q1=0.993

10-4

10-3

10-2

10-1

-30 -20 -10 0 10 20 30

P(sM (j))

sM (j)

b)q29=1.22

Figure 6: Numerically computed probability distributions functions for different observ- ables in the system

Gordon system (see Fig. 6). This project has some theoretical and computa- tional aspects. However, it is more likely to contain programming.

Suitable for a group of 2 - 3 students working in collaboration.

References [1] Ch. G. Antonopoulos, T. Bountis, Ch. Skokos, L. Drossos, Complex Statistics and Diffusion

in Nonlinear Disordered Particle Chains, Chaos: An Interdisciplinary Journal of Nonlinear Science, 24 (2014) 024405.

[2] Ch. Skokos and S. Flach, Spreading of Wave Packets in Disordered Systems with Tunable Nonlinearity, Physical Review E, 82 (2010), 016208.

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7.2 List of projects with Dr Dan Brawn

Dr Dan Brawn Room: STEM 5.39 E-mail: [email protected] Ext. tel.: 3620

Research interests: • Statistical and data science

1. Exploring Generalised Linear Models (GLMs)

This is a statistics project in which you will explore the formulation and prac- tice of Generalised Linear Models (GLMs) . It is only offered as one term (spring term only) Capstone Project (MA830). The project will have two stages:

Stage 1. A prep stage where you will self-study and introduce yourself to theory and practice of GLMs using the book ”An introduction to Generalized Linear Models” by Dobson and Barnett, 4th Edition, Pub. 2018. CRC. The old 3rd edition will also serve just as well. This study should be done in the autumn term to cover the book up to section 9 but excluding chapter 6 and some other parts. This task represents between 100 to 200 pages of study depending on your choices of model for the report itself. There will be no teaching as such but we shall have group meetings in the autumn term and I will offer limited support at this time. The goal is to comprehend the basic theory prior to starting work on the actual project in the spring term.

Stage 2: The project itself will largely be of your choice on this topic. One suggestion would be a literature review of a popular GLM such as the Lo- gistic model or you may choose to write your project using GLMs applied to a data set of your own choice. I will offer further advice, support and suggestions at this stage. You will be using R to implement the GLMs.

Essential prerequisites from first and second year modules.

• Very good multivariate calculus;

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• Good linear algebra;

• Very good general statistical background;

• Basic R skills;

• Good report writing skills.

You are welcome to email me with queries before deciding to adopt this project. Please note that the prior skills listed as pre-requisites above will all be needed for this project.

Up to 3 students can take this project.

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7.3 List of projects with Prof. Edd Codling

Prof. Edward Codling Room: STEM 5.11 E-mail: [email protected] Ext. tel.: 4567

Research interests: • Mathematical biology • Analysis of animal movement and behaviour • Population dynamics • Natural resource management (fisheries)

1. Analysis of dairy cow movement and behaviour

Background Since 2013 we have been using wireless tracking devices (Fig- ure 7) to collect a range of movement and behavioural data sets from dairy cows in various farm locations. Our aim is to determine the links between the health and welfare status of individual cows (and the herd as a whole) and their observed behaviour. This research has huge ‘real-world’ potential impact as there is a great demand for early identification of disease and other health problems in farmed animals, for both financial and welfare reasons. In particular, we are directly investigating how diseases such as lameness, mastitis (infection of the udder) and ketosis (metabolic disease) affect dairy cow behaviour, and whether cows suffering these diseases can be identified by changes in their behaviour (Figure 8). The ultimate aim of the project is to create an automated ‘early warning’ system that farmers can use to auto- matically track their animals and be alerted to potential health and welfare problems as they arise. We are working directly with colleagues at Writtle University College and other universities, as well as local dairy farmers and the manufacturer of the wireless sensors used to track the cows. Project This will be an exciting and challenging project that will allow stu- dents to get directly involved with an ongoing research project with real- world impacts and outputs. Students will be expected to work on open-

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Figure 7: Wireless sensor used to track the position and level of activity of individual dairy cows. Each sensor generates 28 different measurements every 8 seconds for up to 6 months, leading to enormous data sets with billions of data points.

ended research problems that have not yet been solved. In previous years, students working on this project have contributed to novel research publica- tions and there will be the opportunity to do this again if results are of suf- ficient quality. The project is ideally suited to those students who can work both independently and as part of a wider research team, and who are not put off by open-ended work, problem solving, and learning a number of new techniques and tools. The project will require students to undertake coding (using the R language) and undertake statistical analysis of existing data sets, as well as undertaking extensive literature review into animal health and an- imal behaviour, and mathematical and statistical tools for analysis. Note: This project is only available for the full year (MA831) because of the signif- icant time required to get fully familiar with the data and statistical analysis tools that will be used. Up to 5 students can take this project.

2. Mathematical Biology

Background Mathematical Biology is the study of biological problems us- ing mathematical tools such as differential equations. Topics cover a wide variety of applications including population dynamics of predator and prey species, the development of tumours, biological invasions, and the develop- ment of animal coat patterns (Figure 9). Project In this project, students will work through existing notes (provided by the supervisor) or selected chapters in textbooks and will explore one or more topics in Mathematical Biology. Students will need to work inde- pendently and develop the topic in their chosen direction using a range of sources (textbooks, notes, journal papers). A strong background in calculus and differential equations is required. Note: Available for up to 5 students. Full year or spring term only (MA830/31).

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Figure 8: Daily space-use intensity plots for a healthy cow (A) and a lame cow (B). Note that the lame cow only uses the right-hand side of the barn and shows much less exploratory behaviour, spending much of the day in a single location. Such behavioural differences may allow us to predict the health status of cows within the herd in an automated way, helping farmers and improving cow welfare.

Figure 9: Animal coat patterns (a-c) from mathematical simulations, and (d-g) real big cats (image taken from Murray, 1993). The patterns arise from a reaction-diffusion system be- cause of a phenomenon known as a Turing instability (first studied by Alan Turing in 1953). The complexity of the pattern depends on the domain size and explains why the patterns are simpler at the end of the tail (smaller domain).

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7.4 List of projects with Dr Georgi Grahovski

Dr Georgi Grahovski Room: STEM 5.37 E-mail: [email protected] Ext. tel.: 3033

Research interests: • Mathematical physics • Integrable systems (continuous and discrete) • Theory of solitons. Nonlinear waves • Lie groups and Lie algebras. Symmetries • Analysis of PDEs. Spectral theory

1. Integrable systems. Theory of solitons

The so-called solitons are localised solutions of nonlinear partial difference equations having the remarkable property to interact completely elastically. Although there are no direct methods for solving such equations (there is no analogue of the Cauchy existence theorem) the most powerful technique de- veloped for such models is the Inverse Scattering Method (ISM). This method is based on the fact, that one can relate to a given PDE of this class a scattering problem for a suitably-chosen linear operator.

In theory of solitons nonlinear integrable equations are usually represented as a compatibility condition of a linear system called the zero curvature rep- resentation (also known as Lax or Zakharov-Shabat representations). Vari- ous analytic methods of investigation of soliton equations (like the inverse scattering method, algebra-geometric integration, asymptotic analysis, etc.) are based on this representation.

Another indispensable feature of integrable systems is that they possess Bäcklund-Darboux transformations. These special transformations are often used to generate new solutions from the known ones.

It is a characteristic feature of soliton (integrable) partial differential equa- tions that they appear not separately but are always organised in hierarchies of commuting flows.

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Depending on student’s preferences, half-year or full-year projects can be picked up in one of the following directions:

(1A) Inverse scattering method: it is a method which allows one to solve the given PDEs via linearisation in the spectral space (the corresponding scattering data evolve in time by linear ODEs) and then coming back using an inverse spectral transform. Thus the ISM allows one to reduce the Cauchy problem for nonlinear PDEs to a set of linear problems with unique solution.

(1B) Bäcklund and Darboux transformations and their geometry: these trans- formations were discovered in the XIX century in studying surfaces with constant negative curvature. In 1970s it was realised that these surfaces were described by integrable nonlinear PDEs and the interest towards such transformation was revived. Such transformations can also gener- ate integrable discrete equations (see the next project).

(1C) Solitons in Biological systems: These are mainly one-dimensional mod- els. Such models deal with, e.g., super-coiling of the DNA molecule, or in the case of the rod (for polymers) they study the thin elastic properties and the propagation of nonlinear excitations along the polymer.

The projects are accessible for third year students, who have done MA201, MA203 and MA210 in their second year and taking MA302, MA303 and MA323 in their third year.

References [1] M. J. Ablowitz, P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scat-

tering, Cambridge University Press (1991).

[2] A. C. Scott, Nonlinear Science: Emergence and Dynamics of Coherent Structures, Oxford University Press (2003).

[3] C. Rogers, W. K. Schief, Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory, Cambridge University Press (2002).

[4] V. S. Gerdjikov, G. Vilasi, A. B. Yanovski, Integrable Hamiltonian Hierarchies: Spectral and Geometric methods, Springer-Verlag (2008).

[5] http://maths-magic.ac.uk/course.php?id=241

[6] http://maths-magic.ac.uk/course.php?id=300

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2. Discrete integrable systems and their symmetries

In theory of solitons the problem is to discretize an integrable differential equation preserving its integrability. Various approaches to this problem be- gan to be discussed in the soliton literature starting from the mid-1970s. The basic idea is to discretize the zero curvature representation of the smooth sys- tem, i.e., to find proper discrete analogues of the corresponding linear prob- lems. This idea appeared first in Ablowitz-Ladik (1975) and a semi-discrete (differential-difference) analogue of the nonlinear Schrodinger equation.

Some of these developments go back to results from the 19th century and even earlier, but it is fair to say that in combining classical results (from Leib- niz, Bernoulli, Gauss, Lagrange, etc.), via the turn of the XIX/XX century (Poincarè, Birkhoff, Norlund, etc.) with the results of the modern age, the study of integrability in discrete systems forms at the present time the most promising route towards a general theory of difference equations and dis- crete systems.

Later on various attempts to construct fully discrete (partial difference) in- tegrable equations were made and various realizations based on the bilinear method, algebro-geometric integration, integral equations, R-matrices, and Lagrangian mechanics were developed. The development of this field led to a progress in various branches of mathematics.

Depending on student’s preferences, half-year or full-year projects can be picked up in one of the following directions:

• Ordinary difference equations (ODEs) and partial difference equations (PDEs).

• Integrable discrete-time systems (mappings).

• Symmetries of discrete equations.

• Discrete and “difference” geometry.

The projects are accessible for third year students, who have done MA201, MA203 and MA210 in their second year and taking MA302, MA303 and MA323 in their third year.

References [1] J. Hietarinta, N. Joshi and Frank W. Nijhoff, Discrete systems and integrability, Cambridge

texts in Applied Mathematics, Cambridge University Press (2016).

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[2] A. I. Bobenko, Y. B. Suris, Discrete Differential Geometry: Integrable structure, AMS Provi- dence (2006).

[3] R. Yamilov, Symmetries as integrability criteria for differential difference equations, J. Phys. A: Math. Gen. 39 (2006) R541–R623.

[4] D. Levi and P. Winternitz, Continuous symmetries of difference equations, J. Phys. A: Math. Gen. 39 (2006) R1–R63.

[5] http://maths-magic.ac.uk/course.php?id=241

[6] http://maths-magic.ac.uk/course.php?id=309

3. Topical problem solving project

The purpose of this type of projects is to study a subject which is not covered in your list of taught modules or is supplementary to it. It is similar in style and fashion to the Problem solving project, offered by Prof. Peter M Higgins. However, this will be a topical focused project.

Depending on student’s preferences, topical problem solving projects can be offered in the following areas in Mathematics:

• Geometry of curves and surfaces.

• Functional analysis.

• Lie groups and Lie algebras.

• Advanced complex analysis (Conformal mappings, Riemann surfaces, Elliptic functions, Hyperbolic geometry). This option will be available in the Spring term only.

• Groups and symmetries in Physics.

• Topics in Mathematical physics: Integral equations and calculus of vari- ations.

and Theoretical/Mathematical physics:

• Classical Mechanics (Lagrangian and Hamiltonian mechanics, Oscilla- tions and waves, Rigid body motion, Canonical theory of perturbations, Celestial mechanics).

• Classical electrodynamics and special relativity.

• Advanced quantum mechanics.

• General relativity, gravitation and cosmology.

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MA830/831 Capstone Project Handbook

• Statistical physics (including Classical and quantum statistical Mechan- ics, Thermodynamics and Kinetics).

• Quantum field theory (this option is restricted to students who have achieved first class in their first two years).

This project is available in both options: for one term (MA830) and two terms (MA831). For most of the topics above, prerequisites are the main maths modules from the second year: MA201, MA203 and MA206, and sometimes MA210. For the projects related to Quantum mechanics, Statistical physics and Quantum field theory MA225 is also a prerequisite.

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7.5 List of projects with Dr Haslifah Hashim

Dr Haslifah Hashim Room: STEM 5.9 E-mail: [email protected] Ext. tel.: 3025

Research interests: • Life and general insurance • Pensions • Financial mathematics • Enterprise risk management • Forensic economics • Islamic finance and takaful

Please take a few minutes to explore the research topics below. The list is not exhaustive; if you think of a different topic in all areas of actuarial science, please send me an email.

1. Underinsurance in the UK

Underinsurance is a global issue. According to Aviva’s Family Finances Re- port, 61% of families have no life insurance and 89% do not have any Crit- ical Illness Cover (CIC). People who do not hold adequate life cover leave substantial risks to the financial wellbeing of their dependents. Underinsur- ance can also lead to a reliance on government benefits, since it brings about greater likelihood of financial difficulty for dependents and hence more ben- efit claims. However, there is little consensus on what constitutes adequate life insurance coverage.

In this project, we will investigate the current underinsurance situation in the UK for life insurances. We will determine the monetary amount of the life insurance protection gap in the UK at a household level through an analytical study.

2. Value-at-Risk

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Value at risk (VaR) is a statistical technique used to measure and quantify the level of financial risk within a financial institution i.e. bank and insurance company, over a specific time frame. VaR can be estimated either paramet- rically, for example, variance-covariance VaR or delta-gamma VaR or non- parametrically, for examples, historical simulation VaR and Monte Carlo ap- proaches. In this project, we will review performances of several methods in calculating VaR

3. Measuring working-life expectancy techniques

The project will involve a review of the traditionally used and modern sta- tistical techniques for the analysis of life tables with a view to measure ac- curately the duration of future employment time. The students are also ex- pected to carry out numerical study, which is to adapt the technique of the life table to a measurement of the work-life span.

4. Living to and beyond 100 (longevity study)

Based on the predictions by the Government Actuary’s Department, the num- ber of pensioners living to and beyond 100 years old will reach 1.2 million by 2074. This project will look at the actuarial and risk management impli- cations of future extensions of life expectancies; considerations can involve insurance/underwriting, pension and social and economy issues.

5. The price volatility of Bitcoin

Even if you don’t know what Bitcoin is, you have probably heard of it. Cre- ated in 2009, the digital currency of bitcoin is a relatively new phenomenon. ‘Bitcoin’ appears in the Future Actuary magazine in Summer 2014. Due to the limited amount of previous bitcoin research within the area of price volatility, this project will examine the volatility of Bitcoin using GARCH- family models.

6. Modelling the adequacy of life insurance purchases

“How much life insurance does a person need?” This is a common ques- tion, but the answer can be quite complicated. This has been at the core of the dilemma of life insurance agents and financial advisers. They can merely guess using a simple multiple of current income or some of them turn to tools like life insurance or financial calculators that used scientific methods. The fact is, not all life insurance agents and financial advisers scientifically evalu- ate the life insurance cover needed for their clients and hence a professional basis has been lacking in life insurance.

7. Other topics:

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• Asset-Liability Management • Economics • Non-life insurance • Option pricing and hedging • Islamic finance and takaful (Islamic insurance)

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7.6 List of projects with Dr Junlei Hu

Dr Junlei Hu Room: STEM 5.36 E-mail: [email protected] Ext. tel.: 6037

Research interests: • Risk management • Non-life insurance • Optimal insurance and reinsurance design (both theoretical and em-

pirical) • Robust optimisation with application to actuarial science • Numerical optimisation with application to actuarial science

1. Risk measure

Risk measure has been used as a popular tool of determining the amount of asset to be kept as a reserve by financial institutions such as banks and insurance companies. In this project, students will explore the literature of risk measure. Students will need to study various resources, e.g. journal papers and books, recommended by the supervisor. After some independent reading and learning, students will choose a direction to further explore the topic based on their own interest. There are two possible routes to complete the project:

1. The first route can be a literature review of one or two chosen risk mea- sure(s).

2. For students who have strong programming skills in R and Matlab, the second route can be a project involving an empirical study on the topic.

One-term project to be taken in spring term.

2. Optimal insurance

People and businesses buy insurances as a protection from financial losses. In order to receive this service, the insurance buyer has to pay a price, also known as the insurance premium, which usually increases with the amount

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of insurance cover. Therefore, if the buyer wants to receive enough insurance cover and to be cost efficient at the same time, there is a question to ask: how much insurance should be purchased? One way to approach this question is to use the optimal insurance theory, which has become one of the most popular research topics in the field of Actuarial Science since 1960.

In this project, students will explore the literature of optimal insurance by reading various resources recommended by the supervisor, e.g. journal papers and books. As in the Risk Measure project, there are two possible routes to complete the project:

1. The first route can be a literature review of existing optimal insurance models. Students may choose their own focus of the literature when study the project.

2. The second route is suitable for students who have strong programming skills in R and Matlab. Students will build a toy model of a chosen opti- mal insurance problem and solve it numerically.

One-term project to be taken in spring term.

3. Optimal reinsurance

Reinsurance is often used by insurance companies as a means of protection from financial losses. Like in the optimal insurance problem, the insurance company has to decide how much reinsurance should be purchased, so that the insurance company can have enough financial protection and remain cost efficient at the same time. In this project, students will learn to build a toy model for the optimal reinsurance problem and solve it numerically using R or Matlab. Students will use various settings with the model, e.g. employ independent/dependent risks, and observe their effects on the outcome of reinsurance.

Students who wish to take this project are expected to have good knowl- edge on R or Matlab and on Monte Carlo simulations. The project is also suitable for students who only have basic knowledge on R or Matlab and on Monte Carlo simulations but are able to pick up new skills quickly.

This project can be taken as either a one-term (spring term) or a two-term (autumn + spring term) project and can be taken by up to 2 students.

4. Optimal capital allocation

Capital allocation refers to the process of distributing the total available fi- nancial resources to various constituent parts of the business, e.g. different

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lines of insurance business. It is a management goal to optimise the alloca- tion of capital in a way that it generates wealth as efficiently as possible. In this project, students will learn to build a toy model for the optimal capital allocation problem and solve it numerically using R or Matlab. Students will use various settings with the model, e.g. employ independent/dependent risks, and observe their effects on the outcome of capital allocation.

Students who wish to take this project are expected to have good knowl- edge on R or Matlab and on Monte Carlo simulations. The project is also suitable for students who only have basic knowledge on R or Matlab and on Monte Carlo simulations but are able to pick up new skills quickly.

This project can be taken as either a one-term (spring term) or a two-term (autumn + spring term) project and can be taken by up to 2 students.

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7.7 List of projects with Dr Vanni Noferini

Dr Vanni Noferini Room: STEM 5.40 E-mail: [email protected] Ext. tel.: 3032

Research interests: • Root-finding • Matrix polynomials • Matrix functions • Canonical forms • Complex network analysis

1. Newton’s fractals

In numerical analysis, Netwon’s method, sometimes called Newton-Raphson method, is a technique to approximate numerically the solutions of an equa- tion. It is an iterative method: at each step, some manipulations are per- formed on a certain number (the current “guess” of a solution of the equa- tion), obtaining the next element of a sequence.

Let p(z) be a polynomial with complex coefficients. If one applies New- ton’s method to the equation p(z) = 0, then the behaviour of the generated sequences depends on their starting points. Usually, the sequence will con- verge to one of the roots, but it is possible that they cycle indefinitely, and even more unexpected things may happen. Therefore, given a complex num- ber z0 and a polynomial p(z), one can ask the question: what happens if we apply Netwon’s method to p(z) = 0 with starting guess z0? Partitioning the complex plane accordingly generates certain regions called “Fatou sets” and “Julia sets”, thus yielding a fascinating connection between this numerical technique and theory of fractals.

The proposed project involves exploring further theory behind the above ideas, and developing some computer software to experiment and generate fascinating pictures:

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Figure 10: Newton’s fractal

This project has both theoretical and computational aspects, and the ratio of the two can be tailored to the student’s taste. However, its nature is mainly computational, and it will definitely involve some programming.

Maximum number of students who can take this project: 3. If more than one student takes the project, group meetings are possible in the first part of the project. However, both the writing of the dissertation and the numerical experiments are strictly individual.

2. Google’s PageRank

When we want to know more about something and have an internet connec- tion at hand, we are all very used to just typing a few key words on a search engine, and we almost instantaneously receive a list of relevant web pages, ordered with some criterion.

In the Western world, the most popular search engine is Google. One of the reasons for Google’s success is the great efficiency of its algorithm that orders web pages according to their relative importance. This algorithm is called PageRank, and was written as a research project by the company’s two founders, Sergey M. Brin and Lawrence (“Larry”) Page. In fact, the de- velopment of PageRank coincided with the very beginning of Google, and originated as a research project when Brin and Page were Ph.D. students at Stanford University in California, United States.

Whenever we enter a list of key words in Google’s search engine, PageR- ank is run and it lists all the web pages that match the query. The goal of this project is to understand the mathematics behind PageRank, which is based

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Figure 11: Google logo

on beautiful ideas from matrix theory, graph theory, and numerical linear algebra.

This project has both theoretical and computational aspects. Its nature is mainly theoretical, but numerical experiments can be included in the project according to the student’s taste.

Maximum number of students who can take this project: 2. If more than one student takes the project, group meetings are possible in the first part of the project. However, both the writing of the dissertation and the numerical experiments are strictly individual.

3. Polynomial rootfinding

Let p(x) be a polynomial of degree at most n with real or complex coefficients. One of the oldest mathematical problems that mankind has ever considered is the quest for the set of the solutions of the equation p(x) = 0. Exact meth- ods are known for n  2 (since classical times) and for n = 3, 4 (since the 16th century).

However, in the 18th and 19th century, mathematicians such as Abel, Galois and Ruffini proved that a general algebraic method for high degree (n � 5) polynomials cannot exist. Hence, devising reliable numerical meth- ods to approximate numerically the solutions is a very central theme in mod- ern mathematics. In the field of numerical analysis, the (vague) word “reli- able” can be translated into precise mathematical concepts such as backward stability.

The goal of this project is to understand the mathematics of polynomial rootfinding, to explore the history of the subject, and to understand the mod- ern challenges about the numerical solution of this problem.

The project involves an essential theoretical part. According to the stu- dent’s taste, it can be tailored to also involve a historical part or a compu- tational part. (It is possible for a particularly motivated student to include both, but in this case the project must be full year). Some programming is likely to be involved for the computational part.

Maximum number of students who can take this project: 4 (with the further

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Figure 12: Evariste Galois

constraint of at most 2 focusing on the historical aspects and at most 2 focusing on the computational aspects). If more than one student takes the project, group meetings are possible in the first part of the project. However, both the writing of the dissertation and the numerical experiments are strictly individual.

4. Evaluating betting strategies

Suppose that you have a set of estimated probabilities for a sport event, which might differ from the ones (reflected in odds) of your favourite book- maker. What is the best strategy that you should follow as a better? This project will involve assessing various short-term and long-term strategies and testing them against practical outcomes of sport events. Students will explore topics in applied probability and time series analysis, as well as de- velop programming skills. The project will be co-supervised by Dr Noferini and two other mathematicians who lecture at other UK universities: Dr Ste- fan Guettel (University of Manchester) and Dr Martin Lotz (University of Warwick).

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7.8 List of projects with Dr Christopher Saker

Dr Christopher Saker Room: STEM 5.14 E-mail: [email protected] Ext. tel.: 2961

Research interests: • Combinatorics on words • Semigroup theory • Mathematical education

For students interested in mathematics education or considering a career in teaching a project focusing on mathematics education provides a way to ex- plore these areas. Mathematics education projects will, in general, take one of the following two forms, However, I am open to discuss other ideas if you have them:

1. A placement based project – projects of this type involve a placement in a local school comprising of approximately 10 visits to undertake obser- vations of lessons. In addition to the school placement you will also be undertaking reading of appropriate educational literature and problem solving materials to better appreciate the ways in which people learn mathematics. The write up for this project takes the form of a reflec- tion on your placement with reference to the educational materials you have read and an investigation of the problem-solving materials and how people think when such problem solving.

2. A resource creation project – projects of this type will focus on the cre- ation of mathematical resources for use either at school or undergradu- ate level. These resources could, for example, be a set of e-assessment problems created in a package like Numbas to accompany an under- graduate module, a set of interactive resources created in a package like GeoGebra to help students to better visualise and understand one or more areas of mathematics, or a series of problem solving resources with

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accompanying explanatory videos to support students preparing for the transition from school to university. For projects of this type in general no prior knowledge of package like Numbas of GeoGebra is requires.

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7.9 List of projects with Prof. Abdel Salhi

Prof. Abdellah Salhi Room: STEM 5.34 E-mail: [email protected] Ext. tel.: 3022

Research interests: • Optimisation (continuous, discrete) • Numerical mathematics (matrix algebra, systems of linear equations,

eigenproblems, Lanczos algorithms) • Algorithm design, application and analysis (nature-inspired algo-

rithms, the plant propagation/strawberry algorithms in particular) • Origami (modelling errors in origami constructions, serious maths

with origami) • Smart logistics (data generation and exploitation to improve the dis-

tribution of goods)

1. Error modelling in Origami construction

Given a square piece of paper, assuming that it is not stretchable, and has no thickness, a fold or crease can be made in a number of ways. There are 7 axioms which govern making such folds. They are known as the Husita- Justin 1-fold axioms.

Making creases according to any of these axioms is never accurate. In other words, errors will always be made. Moreover, since Origami objects often involve a series of consecutive folds, it is clear that starting from a piece of paper with one or more folds in it, which have errors in them, making a further fold/crease may add to the errors already accumulated. But, there is also the possibility that some errors, based on certain folds and the way they follow each other in a sequence, may actually council each other, resulting potentially in a more accurate Origami object.

The project, therefore, concerns 1. the representation of these errors for some or all of these axioms in math-

ematical terms;

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2. the evaluation of individual errors in terms of the area of the paper in which a fold resides given an initial error epsilon;

3. the classification of folds in relation to their succession and whether their errors accumulate or council when applied in a sequence;

4. the evolution of the error of an Origami object from start to finish (this is perhaps too ambitious).

Note: This project can be taken by up to two students.

2. An optimisation approach to Fantasy Football Management

Consider the problem of choosing a football team of eleven players plus 3 substitutes out of a squad of 25 players. Players are scored over a number of attributes such as current fitness, latest performance, overall performance, Home/Away game, opposition, availability (different from fitness, could be International duty, suspension etc. . . ). Others may be added.

The Manager’s selection problem looks like a linear integer program with the objective:

1. Select the best team out of the squad, subject to constraints some of which are:

1. Only 14 players will be selected; 2. Only available players will be selected, etc.

The problem can be made more complex by aiming it at aiding a Manager that participates in the Fantasy Premier League (FPL). In this set up, other conditions are imposed such as players have a price tag, and all managers start with a budget of £100M. Only 3 players can be bought from any team. There are restrictions on the number of defenders, forwards etc. . . to be used. The problem becomes harder as it now, potentially, has 2 objectives:

1. Select the best team; 2. Select the cheapest team;

It also has more constraints and costs in terms of the number of players bought every gameweek.

The project involves, first of all, solving the simpler version based on score tables constructed using information found in Football Professional Web- sites. Once this version of the problem is fully analysed, the student will then tackle the 2nd version. Existing optimisation software will be used. The optimality of the solutions found will be measured against Managers’ teams of the FPL.

Although, the problem is really stochastic in nature since, for instance, a player may become unavailable after a training session, we will only be

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concerned with the deterministic version unless the student has enough time and desire to look at the more realistic stochastic version within the allowed time. Note: This project can be taken by up to two students.

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7.10 List of projects with Dr Hadi Susanto

Dr Hadi Susanto Room: STEM 5.12 E-mail: [email protected] Ext. tel.: 2689

Research interests: • Dynamical systems • Applied analysis • Ordinary and partial differential equations • Nonlinear waves • Stability theory • Pattern formation

1. Solitary wave theory

Waves are around us, from sound that propagates in the air to sun light that travels through the empty space. In addition to continuous waves (as the aforementioned examples), there are localised waves, such as the tsunamis or the stadium waves created by football fans. The project is to study localised waves and their characteristics in various natural systems.

2. Language death, X-Men, and Zombies are related

Language death is a process where the level of linguistic competence that speakers possess of a language decreases, eventually resulting in no native speakers of the language. It is generally because of the presence of a second language that is seen to be more superior or attractive than the first language. In the Marvel Universe, X-Men are in competition with a group of villains led by Magneto. In the Earthy world with zombies, zombies have an immense appetite for human flesh and hence their aim is to kill people, while the liv- ing tries to eliminate the zombies. Even though they are different systems, there is a strong similarity, i.e. there is one component acting as a ’predator’ and the other as a ’prey’. The project is to study and model these processes mathematically.

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7.11 List of projects with Dr Alexei Vernitski

Dr Alexei Vernitski Room: STEM 5.15 E-mail: [email protected] Ext. tel.: 3024

Research interests: • Knot theory, especially combinatorial and algebraic techniques of

studying knot diagrams • Theoretical computer science, including applications of discrete

mathematics to computer science • Applying mindset theory and the “mathematical mindsets” ap-

proach to teaching mathematics

I see your final-year project as your chance to improve your CV and to get a good project mark. If you want to be supervised by me, come to me to talk and we shall discuss your project plans according to these criteria. As a result of this discussion, we shall either choose a project topic for you or I shall recommend a more suitable supervisor to you. I cannot give you specific topics in advance, without discussing your plans and interests with you, but I can describe what areas I prefer to cover in my projects.

Area 1: Computer programming I am interested in information technology in general and in computer programming in particular. If you consider doing a project related to IT, I may be a suitable supervisor for you.

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Area 2: Teaching mathematics at the university level I am interested in developing mathematical activities which can help students to better grasp abstract concepts and facts of university mathe- matics. There is research behind it, which you are welcome to read, and then you can suggest some activities which can complement or replace traditional ways of teaching mathematics.

Area 3: Theoretical mathematics Within theoretical mathematics, currently I am mostly interested in knot theory and its connections with algebra. You can write a project describ- ing some facts of knot theory. Also, if you want, some computer pro- gramming can be done within this project.

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7.12 List of projects with Dr Gerald Williams

Dr Gerald Williams Room: STEM 5.16 E-mail: [email protected] Ext. tel.: 3035

Research interests: • Combinatorial, computational, geometric, cohomological aspects of

infinite group theory • Algebraic number theory and connections with linear algebra (circu-

lant matrices)

1. Strongly regular graphs Strongly regular graphs are regular graphs with conditions on the number of common neighbours of pairs of adjacent and non-adjacent vertices. Promi- nent small examples include the Petersen graph and the Clebsch graph. In this project you will survey the state of knowledge of strongly regular graphs, stating and proving theorems that strongly regular graphs must satisfy, and give examples of important strongly regular graphs. You will bring older surveys up to date and highlight recent developments and state key open problems.

References

[1] Xavier L. Hubaut, Strongly regular graphs, Discrete Mathematics 13 (1975) 351-381.

[2] Norman Biggs, Algebraic graph theory, Cambridge University Press (1974).

2. Circulant graphs Circulant graphs are a type of graphs that admit a cyclic symmetry. The def- inition is that they have an adjacency matrix that is a circulant matrix (ie one

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where each row is obtained from the one above by cyclically permuting the columns). In this project you will give examples of circulant graphs, study key algebraic invariants to distinguish circulants graphs and study graph theoretic properties relating to colouring problems, connectedness, and pla- narity.

References

[1] A. K. Lal and A. Satyanarayana Reddy, Non-singular circulant graphs and digraphs, Electron. J. Linear Algebra 26 (2013), 248-257.

[2] Clemens Heuberger, On planarity and colorability of circulant graphs, Dis- crete Math. 268 (2003), No.1-3, 153-169.

[3] Gerald Williams, Smith forms for adjacency matrices of circulant graphs, Lin- ear Algebra Appl. 443 (2014), 21-33.

3. Automata, Languages, and Turing machines

Turing machines are the abstract machines that describe theoretical structure of computers. It is believed that any ‘real world computation’ can be carried out by a Turing machine. It is known that there are certain problems that are algorithmically undecidable, in the sense that they cannot be answered by Turing Machines. Finite state automata are a less powerful kind of ab- stract machine, but are easier to understand. Certain computations cannot be carried out using a finite-state automaton but can be with a Turing ma- chine. The kinds of problems that can be expressed by different machines can be expressed formally using the so-called Chomsky hierarchy of formal languages. In this project you will build on what your learned in MA182 to study theory of finite-state automata and Turing machines, and related prob- lems in formal language theory. Your work will involve stating and proving theorems and developing examples that illustrate theory.

References

[1] J. M. Howie, Automata and Languages, Clarendon Press, Oxford (1991).

[2] Richard Johnsonbaugh, Discrete mathematics, Pearson Publishers (2007).

4. Fibonacci numbers

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The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . arise from a recurrence re- lation (each term is the sum of the previous 2) and they arise in diverse ar- eas of mathematics, and there are close connections with the golden ratio (1 +

p 5)/2. There are also interesting connections in nature, such as the

number of seeds on the head of a sunflower, or the number ancestors that a bee has.

In this project you will explore the mathematical theory of Fibonacci num- bers (and related numbers, such as the Lucas numbers). The project can be taken in several directions and it will for you to propose a detailed plan of what your aspects of Fibonacci numbers your project will address.

References [1] Ralph Grimaldi, Fibonacci and Catalan Numbers: An Introduction, Blackwell J. Wiley &

Sons, Hoboken (2012).

[2] Nicolai N. Vorobiev, Fibonacci numbers, Birkhäuser, Basel (2002).

[3] Ron Knott’s web site on the Fibonacci numbers: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/

fib.html

5. Circulant matrices

An n ⇥ n circulant matrix is one whose first row is (a1, a2, . . . , an) whose sec- ond row is (a2, a3, . . . , an, a1), whose third row is (a3, a4, . . . , a1, a2), and so on. These matrices occur in diverse areas of mathematics and have applications in electronic engineering. In particular, they can be used for finding roots of low degree polynomials.

In this project you will survey one or two problems in theory of circulant matrices, such as calculations of determinants, or how circulant matrices can be used to find roots of polynomials. It is expected that experiments in the computer algebra package Maple will form part of the investigations. The project requires an understanding of linear algebra (MA114, MA201) and fluency in Maple.

References [1] D. Kalman and J. E. White, Polynomial equations and circulant matrices, Amer. Math.

Monthly 108 (2001), 821–840.

[2] F. Geller, I. Kra, S. Popescu and S. Simanca, On Circulant Matrices: http://www.math.sunysb.edu/˜sorin/

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[3] A. Wyn-Jones, Circulants: A reference work on the algebraic theory of circulants with some applications to number theory, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.164.

4784.

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