1. Overview

1.1 Objectives for the Student.

Independent Study and Creativity. The primary goal of the course Projects in Mathematics is to give the student a taste of what it is like to study independently and express creativity to a larger extent. This must include either reading material in the literature outside of the standard textbook presentation, or using known mathematics to make some mathematical modelling, or exploring topics of history of mathematics.

Exposure to Mathematical Concepts Outside of the Standard Curriculum. This project course is an opportunity for a student to see some areas of mathematics not covered in her/his coursework. This may mean a more advanced continuation of a completed course, or something in a completely different direction, but in any case there should be negligible overlap of subject material with courses offered by the Department.

Improved Ability to Communicate Mathematics. The student is expected to conclude the course MATH 490 Projects in Mathematics with an oral presentation, written report, and a poster, if possible. In all cases, the goal is that the student learn how to express mathematical concepts so as to be clearly understood by others. In particular, the student is expected to learn a standard mathematical typesetting program such as TEX or LATEX and demonstrate mastery of this program in the written report. Templates and documentation for these programs can be downloaded from the course web page.

1.2 Student Eligibility. It is mandatory that all students that graduate the Department of Mathematics take at least one project course. There is a choice of two project courses: MATH 490 Projects in Mathematics and MATH 491 Senior Project I. All students that are not eligible for the MATH 491 Senior Project course should take the course MATH 490 Projects in Mathematics. We list here the requirements for MATH 491 Senior Project I: a GPA of at least 3.00 out of 4.0, at least three elective mathematics courses with a passing grade, one project-related mathematics course with a grade of at least B, approval of the project proposal by the course committee.

2. Types of Subjects

There are three types of subjects acceptable for a project. However, each type should be centred around at least one theorem for which a clear proof should be provided, because this is a project in mathematics.

Pure Mathematics. This type of project refers to one important theorem, or a group of theorems, the preliminary results and all concepts, that can be found in articles or books, outside the curriculum.

Mathematical Modelling. This type of project refers to one or more applications of some important results in mathematics. For this type, students can provide numerical applications as well, performed by some programming environmets like Python, MATLAB, or the like.

History of Mathematics. An important mathematician and a few important results obtained by him/her.

3. Organisation

3.1 Before the Semester. In the term prior to enrolment in the course MATH 490 students may explore her/his standing and decide (see the criteria of eligibility for taking MATH 491 explained before) which course is suitable for her/him. Once the decision for taking the MATH 490 course is clarified, she/he may explore topics of mathematics that are possible subjects for her or his project. This can be done either by asking advise from some faculty members of the Department of Mathematics or by searching for possible subjects of pure mathematics, mathematical modelling, and history of mathematics.

3.2 During the Semester.

Choosing a Subject. The student is expected to choose a subject within the first two weeks of the semester. During the third week, each student will make a short presentation of the chosen subject for about ten minutes and answer questions for about five minutes.

Work. The student will perform independent study, organise, and write the project. Advise from the instructor should be sought at all times.

Midterm Presentation. Midway through the semester (week 7 or 8) all students enrolled in MATH 490 will make a short presentation on the progress and report difficulties, if any.

3.3 Conclusion of the Semester.

Final Presentation. At the end of the semester, each student enrolled in MATH 490 course will make an oral presentation summarising her or his work for about twenty five minutes and answering questions summarising his or her work. All students are required to attend all other students??? presentations. The material should be presented at a level accessi- ble to each student???s peers, while still touching on the highlights of the semester???s research. The presentations may be made using whiteboard and marker, slideshow projection, or a combination of the two.

Written Report. At the end of the semester, each student must provide a final draft of the written report. After the oral presentation, the student has one week to fine tune the final draft based on the comments of the course instructor and peers. At the end of this week, the final written report will be submitted to the course instructor for grading.

Poster. In addition, students enrolled in MATH 490 may create a poster summarising their findings, if they wish. The schedule for the poster is the same as for the written report: A (draft) computer file for the poster must be submitted together with the final written project, at least a week prior to the presentation. The student then has a week to make any corrections or adjustments, at which point the final version will be submitted for printing. Templates for making a poster in LATEX may be downloaded from the course web page.

4. Evaluation

4.1. Grading Scheme.

Midterm Evaluation: 20%. The course instructor will judge the midterm presentation. The following factors should be considered:

The student's understanding and mastery of the material listed in the project application and additional material considered in the project, if the case.

The student's progress and increased mathematical maturity.

The organisation of study and research.

Project: 40%. Each student's project will be evaluated by the course instructor.

The following factors should be considered:

Mathematical quality.

- The student should clearly state and prove all lemmas, propositions, and theorems, as well as provide an over-arching narrative.

- The student must explain the motivation for the project, develop any necessary background, and suggest future directions of inquiry.

- If the project is of mathematical modelling type, the student should clearly explain the algorithms on which the programming is based.

Professionalism of writing.

- Both the report and poster should be clearly written in good, grammatically correct English.

- The report should follow the standards of a research paper, including having an abstract, bibliography, and clean separation of topics into sections.

- Literature citations should be precise, referring to the exact page or result number within a cited article from where a result is taken.

- The student should demonstrate a mastery of TEX typesetting with no major errors.

Oral presentation: 30%. The course instructor will judge the final oral presentation. The following factors should be considered:

- The student should clearly state the problem being considered and provide necessary motivation and background.

- The student should clearly present the main results of the semester's work, as well as outlining the ideas behind major outcomes.

- The student should demonstrate mastery of the subject matter by being able to answer questions posed by the jury. Examples include: giving more detail on a particular point, drawing connections with other mathematical fields, or speculation about how to attack open problems.

- The student should give a good, professional presentation. This includes white-board technique (clear handwriting and top-to-bottom, left-to-right flow instead of just randomly scribbling words on whatever bit of the board is closest at the moment, etc.) or good design of the slideshow (an absorbable amount of information on each slide, a reasonable number of slides, giving enough time for the audience to read each slide and follow along with the talk, not excessively referring to "this" as a point on the slide that the student should state directly, etc.), timing, and language skills. The student is strongly advised to hold a practice talk with peers before the formal presentation.

Cross Evaluation: 10%. Each student will evaluate all the other projects following a scheme of grades provided by the course instructor.

4.2 The Letter Grade. The overall grade for the semester is determined by the following rubric:

A [90,100]

A- [80,90)

B+ [75,80)

B [70,75)

B- [65,70)

C+ [60,65)

C [55,60)

C- [50,55)

D+ [45,50)

D [40,45)

F [0,40)

4.3 Plagiarism. Plagiarism is the uncredited reproduction of another's work or text being passed off as one's own. Examples of plagiarism include:

- Direct copying from a source without citation.
- Rewording a passage from a source without citation.
- Claiming, either implicitly or explicitly, that an idea due to another is one's own.

Any instance of plagiarism on the part of the student will result in an automatic final grade of F for the semester. Written reports will be tested by computer software for evidence of plagiarism.

4.4 Work. No matter whether a student has a single project or a group project, she/he is supposed to perform all the underlying activities by herself/himself. Although it is possible that some students ask and get some advise from some faculty members of the Department of Mathematics, or other departments of Bilkent University, this must be acknowledged in the project, clearly and with all possible details. In case there are reasons to believe that the project was done in collaboration with other people not mentioned in the project, this is considered to be an attempt of cheating and treated according to the university rules.

5. Semester Schedule.

The following is a rough timeline for when the major milestones of a senior project should be met:

Week 1 Course description, making project proposals, available topics, resources.

Week 2 Fundamentals of AMS-LaTeX typesetting.

Week 3 Presentations of project proposals.

Week 4 Remarkable mathematical results from articles or books.

Week 5 Mathematical modelling of real-life problems.

Week 6 Scientific software based on mathematical results.

Week 7 History of outstanding mathematical results and mathematicians.

Week 8 Midterm oral presentations.

Week 9 Midterm oral presentations.

Week 10 Overcoming project difficulties.

Week 11 Structures of mathematical writings: reports, articles, books.

Week 12 Plagiarism.

Week 13 Oral presentations.

Week 14 Oral presentations.

Every week, there will be at least one hour dedicated to advising by the instructor for answering questions, discussing the projects, evaluating difficulties, providing feedback, and other similar activities.

6. Mathematical Editing

Modern mathematical editing uses TEX, a programming environment designed by Donald Knuth during the 1980's, to produce high printing quality of articles and books of mathematics. The TEX evolved to very sophisticated versions: AMS-TEX, LATEX, LATEX2e, and AMS-LATEX. We recommend the usage of AMS-LATEX that has the most capabilities and high flexibility. The students should study independently and be able to write their project using one of these versions. There are different textbooks available for this, among which we recommend Guide to LATEX. 4th edition, by H. Kopka, P.W. Daly, Addison-Wesley 2004 that can be found in the bookshop.

You can use on any computer from the BCC the MikTeX environment by using the path:


and then a window will open. After a short configuration you will be able to type your LATEX file. Please note that there are at least three stages before producing your written pages:

- Editing your file (that with extension .tex).

- Compiling (building out) to produce your .dvi file that can be previewed.

- Viewing and printing, that is, producing your .pdf or .ps files that can be viewed and printed.

If you have a personal computer or a laptop, other distributions of the TEX are available: MikTeX is freely available from the Internet to be downloaded and installed under Windows operating system, teTeX is part of any Linux operating system, TeXShop is freely available for MacIntosh operating systems, etc. From the course web page you can download an Example file in AMS-LATEX, and the corresponding Example file in PDF.