Senior Project I MATH 491 and Senior Project II MATH 492

Course Guide

1. Overview

1.1 Objectives for the Student

Introduction to Mathematical Research. The primary goal of the Senior Project courses is to give the student a taste of what it is like to study independently and do mathematical research. This must include reading material in the literature outside of the standard textbook presentation and may go further to proving original results. The particular topic of a Senior Project serves to guide the direction of research, but the most important thing is that some sense of independent study is gained.

Exposure to Mathematical Concepts Outside of the Standard Curriculum. The Senior Project is an opportunity for a student to see some areas of mathematics not covered in his or her 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 Senior Project with an oral presentation, written report, and (for MATH 492) poster summary of the work. 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 L A TEX and demonstrate mastery of this program in the written report. Templates and documentation for these programs can be found at [...]

1.2 Student Eligibility. The Senior Project courses are open to seniors and advanced juniors. Any interested student must demonstrate by completed coursework a solid mathematical background. Before the start of the semester, students must apply to be enrolled in a Senior Project. In order to be accepted for the course MATH 491, a student should have 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, and approval of the project proposal by the course committee.

2. Organization

2.1 Before the Semester.

Topic Selection and Application. In the term prior to enrollment in the Senior Project, any interested student must apply for admission. The application should indicate the following:

- The proposed topic of study, including references that will be read.

- The proposed faculty advisor.

- The student's relevant background (e.g., completed courses in related fields).

The application form is meant to be filled by the student with the cooperation of the advisor and, before sent to the course coordinator, should be approved by the advisor. Briefly, this is a document that will be examined by the Project Courses Committee that will decide if the student can register to the required course. The decision of the committee is based on the following criteria:

1. Quality of the Project (50%): The project should refer to material that is not taught in any of the courses in the curriculum, with a subject that is at a scientific level significantly higher than the courses taught but not extremely difficult and such that it is achievable to a regular honour student.

2. Potential of the Student (30%): In order to qualify for a senior project, a student should have taken sufficiently many mathematics elective courses, usually at least three, coded MATH 3** and MATH 4** or higher, with good grades.

3. Others (20%): In exceptional cases, other criteria can be applied, for example for a student that could not take sufficiently many mathematics elective courses but that shows a trustable potential of research.

Description of the Project: A description of the project should be brief, in a discursive manner, and addressed to a wide audience, not necessarily specialists. We suggest the following structure to be applied:

1. Brief description of the Project: At the beginning, a paragraph that presents the significance and importance of the subject, put in a perspective that shows how the student will benefit from it. Then, a short description of the topics to be studied or investigated (if the case). Finally, some possible applications and/or other objectives that might be touched.

2. List of the resources to be used: At this item, it is expected that there should be a bibliography that will be used, articles, monographs, reports, etc., and, if the case, computer resources. The bibliography should be written following the standard of mathematical publications.

Application Forms:

MATH 491 Senior Project I: Libre Office Form Adobe Acrobat Form

MATH 492 Senior Project II: Libre Office Form Adobe Acrobat Form

2.2 During the Semester.

Advisor Meeting. The student is expected to meet with the project advisor for at least one hour each week. The structure of these meetings is largely a matter for the student and advisor to agree on, but at the very least there should be regular reports on the student???s progress in terms of reading the literature, internalization of concepts and proofs, and drafts of the final written report.

Midterm Interview. In addition, midway through the semester (week 7 or 8) all students enrolled in a Senior Project will meet together with a representative of the coordinating committee. The midterm interview is an opportunity for the student to report on the progress of the project and make a good preparation for the oral presentation at the end of the semester.

Each student has 15 minutes of presentation and 5 minutes to answer questions from the audience. We suggest the following structure of the presentation:

- The first 5 minutes can be a review of the background of the topic, an explanation of what the project is about, and the goals that are supposed to be touched. The presentation should be targeted to a wide audience, not necessarily to specialists.

- The middle 5 minutes can be a presentation of what has been done during the first half of the semester, the literature that was explored so far, the results that have been understood, and other achievements.

- The last 5 minutes can be dedicated to what remains to be done, a brief presentation of the landmarks to be seen, and a short discussion of difficulties, if any.

This interview:

- is not a detailed presentation of the subjects, proofs, etc.;

- is not graded in anyway;

- might be a useful experience, in view of the oral presentation, on how to organize and present the material in a time efficient manner;

- might be an opportunity of reflection whether the project is on the right way and the student will be successful in the end.

2.3 Conclusion of the Semester.

Final Presentation, Written Report, and Poster. At the end of the semester, each student enrolled in a Senior Project course will make an oral presentation summarizing his or her work. All students are required to attend all other students' presentations on pain of receiving an FX grade. The material should be presented at a level accessible 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.

If a student???s project is particularly strong, the presentation may take place in one of the Department???s seminar series (Algebra, Algebraic Geometry, Analysis, Topology, Applied Mathematics, etc.) with permission of the seminar organizer; in this case, the full 50-minute time slot may be used. Those students who do not present in a departmental seminar will have a separate presentation time set in the beginning of the final exam period. The length of such a presentation will be determined each semester, depending on the number of students enrolled in a Senior Project, but will be at least half an hour.

At least one week prior to the student's presentation, the advisor must be given three (3) copies of the final draft of the written report, which will then be distributed to the oral presentation jury members. After submitting the final draft, the student has one week to fine tune the final draft based on the comments of the advisor and jury members. At the end of this week, the final written report will be submitted to the course coordinator for grading.

In addition, students enrolled in MATH 492 will create a poster summarizing their findings. 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 L a TeX may be found at [...].

3. Evaluation

3.1 Grading Scheme.

Advisor's evaluation: 50%. The advisor is the faculty member who will have worked most closely with the student over the semester, and accordingly the advisor's sense of the student's performance weighs most heavily. The following factors should be considered:

- The student's understanding and mastery of the material.

- The student's progress and increased mathematical maturity.

- The student's originality in proving results and drawing novel connections.

- The student's regular attendance in weekly meetings. (Failure to make at least 75% of the weekly meetings will result in an FX grade).

- Coordinator's evaluation: 20%. Each student will be evaluated by a member of the organizing committee who is not the student's advisor. This coordinator will grade the student's midterm status update, final written report, and (in the case of MATH 492) poster. 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.

- 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. Moreover, the student should demonstrate a mastery of TeX typesetting with no major errors.

- Oral presentation: 30%. A panel of three faculty members, ideally distinct from both the student's advisor and coordinator-evaluator, 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 proofs.

- 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 whiteboard 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 the advisor before the formal presentation.

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

3.2. 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.

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

Before the start of the semester: The student meets with a prospective advisor. Together, they fill out the details of a course application.

Week 1: All enrolled students meet together with one of the course coordinators to review the rules and give brief introductions of their projects to their peers.

Week 7 or 8: All enrolled students have a mid-semester joint meeting with one of the course coordinators to indicate their progress. Students will give a short oral presentation of where they stand at this point.

Week 14 or 15: Any presentation taking place in a seminar must have the final draft of the written report submitted a week before, most likely Week 15; all other presentations will have their final drafts submitted in week 15. Final corrections are then to be made in the week following. Week 15 or Finals Week 1: Presentations are given together with PDF submissions of the final report (and poster, if applicable).

Of course, the student is expected to meet regularly with the advisor every week throughout the semester, on top of the above main events.

4. Mathematical Editing

Modern mathematical editing uses TeX, a programming media 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:

Programs>Applications>TeXnicCenter>TeXnicCenter

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.

• Printing, that is, producing your .pdf or .ps files that can be 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. Here you can download an Example file in AMS-LaTeX, and the corredsponding Example file in PDF.

Course Coordinators: Aurelian Gheondea, Özgün Ünlü, Gökhan Yıldırım.

Page maintained by Aurelian Gheondea. Last update October 12^th, 2021.