The purpose of the senior capstone is to engage each senior in the exploration and communication of mathematical ideas beyond material covered in courses taken, or to be taken, by the student.

### Basic Components

- The student independently studies a topic of interest.
- The student writes a paper on the topic.

### Requirements and Important Dates

- 1. Students must choose a topic.
- 2. Student must submit a formal proposal describing the proposed topic for the paper.
- 3. Students must submit weekly progress reports to and meet regularly with their faculty consultant. The specific timing of these meetings will be arranged between the student and the faculty consultant.
- 4. Submit a complete draft of the paper.
- 5. Submit the final paper.
- Evaluation
- Second Chance

#### 1. Students must choose a topic.

As this is the capstone experience for the major, we expect the work to build on earlier intermediate and advanced work in the discipline. Thus, the senior capstone topic will ordinarily come from the students' area of focus in the major. Topics from outside the area of emphasis require permission from the department.

The shelves of the Finkbeiner Reading Room and the stacks in the library are obvious places to look for ideas. Expository journals such as the American Mathematical Monthly, The College Math Journal, and the American Statistician are also good sources.

The student's topic should be of suitable difficulty — not too easy and not too hard. If in doubt, the student should consult with faculty members about this. Once the student has a topic and a few print resources (books or articles) in mind, they should approach one or more faculty members to consult on the suitability of the proposed topic and to discuss ideas for the proposal.

Students should have topic ideas and some resources in hand no later than the first week of the fall semester. (Classes begin Thursday, August 24, 2023.) We encourage students to begin thinking about possible topics in the *spring of their junior year*.

#### 2. Student must submit a formal proposal describing the proposed topic for the paper.

Even though the proposal itself is fairly short, the amount of work that goes into it is much greater. Writing a good proposal requires a reasonably good understanding of the major ideas and technical material involved in the topic. Before attempting to write the proposal:

- The student needs to understand the general motivation behind the topic.
- The student needs to identify the end-goal(s) for the senior capstone.
- The student needs to make sure they have the necessary background and appropriate resources for completing the topic.
- The student needs to be fairly sure the project is doable in the given timeline.

In other words, the student cannot write a reasonable proposal without studying their major resources enough to get a good sense of the topic, its difficulty, and its feasibility. All of this requires a substantial amount of reading and background work prior to writing the proposal. Therefore the preliminary work needs to begin several weeks in advance of this first deadline.

In addition to extensive reading, the student should be consulting with one or more faculty members who will be able to give guidance on end goals and feasibility. It will ordinarily take several discussions with faculty members to really narrow down the goals of the project.

Do not underestimate the importance of the preparation step or the amount of work that goes into it. This is a big part of the overall senior capstone project and a big deadline.

The proposal must include:

- a description of the topic: this is more than the mere title of the topic. The description should include motivation and describe the basic ideas underlying the topic. It should also state specific end-goals and the nature of the mathematical material that is needed and will be covered. The description should also explain how the topic fits the students’ background and interests in mathematics. The student should also be careful to choose a topic that has substantial content and that is also doable in the timeline given for the senior capstone.
- an annotated bibliography saying generally what is in each source and how the student expects the source to contribute to the eventual paper, (these annotations need not be extensive; keep them general and short, but writing them will require the student to have done some reading in the source.)
- a list of courses the student has had that will support his or her study of the topic, and
- a summary of the discussions the student has had with faculty members concerning the topic.

**Deadline for the Proposal: **Sunday, Sept. 3, 2023, by 5 p.m.**Departmental Response by: **Friday, Sept. 8, 2023

Upon acceptance of the student's proposal, the department will appoint a faculty consultant for the student's senior capstone.

#### 3. Student must submit weekly progress reports.

Students must submit weekly progress reports to and meet regularly with their faculty consultant. The specific form of the updates and the timing of these meetings will be arranged between the student and the faculty consultant.

#### 4. Submit a complete draft of the paper.

The paper should include all major definitions, examples, theorems (along with complete proofs), and any other critical content. The draft will be submitted via the capstone Moodle page.

**Deadline for Draft:** Friday, Oct. 20, 2023, by 5 p.m.**Faculty consultant will respond:** Monday, Oct. 30, 2023

#### 5. Submit the final paper.

General instructions:

- The paper will be 12-15 pages in length. It will be typeset in LaTeX 12-point article format.
- The paper should be self-contained. The necessary background, basic definitions, and useful examples must be explicitly set out in the paper so that the faculty consultant can understand it without consulting other sources. The paper
*may*, however, assume the general background covered in the core courses in mathematics and (if necessary) the material covered in other math courses the student has taken at Kenyon. - The paper should be written so that it is understandable to fellow math majors; thus we
*strongly*encourage seniors to read and comment on each others' papers before turning them in to the department. - Diagrams and calculations that are central to the understanding of the paper should be included in the body of the paper,
*not*relegated to an appendix at the end. It may be appropriate to put as appendices ancillary materials that fill gaps in the development if their absence from the body of the paper does not interrupt the flow of the reading. - The student will be expected to work out the details that are not explicitly set out in the references, so as to put together a complete picture of the topic.
- The student will be expected to coordinate and systematize material from various sources into a coherent whole. this includes standardizing the notation, which often varies from one source to another.
- The paper will be graded on the correctness, completeness, and clarity of the writing. As part of this, of course, the student is expected to use correct English and mathematical grammar throughout.

**Deadline for Paper: **Friday, Nov. 17, 2023. Submit via Moodle by 5 p.m.

#### Evaluation

The department will evaluate each senior capstone paper(the result being either distinction, pass or failure) based on the following criteria:

- Meeting deadlines
- General acceptability of the work submitted at each stage
- The paper, which itself will be rated on the following, before getting an overall score:
- Accessibility
- Mathematical/statistical depth
- Mathematical/statistical correctness
- Evidence of understanding
- Originality and independence
- Organization
- Style (citation, grammar, etc.)

The student's work will be rated as exemplary, acceptable, marginally acceptable, or unacceptable in each category.

An evaluation of "unacceptable" on any component of the senior capstone may be considered grounds for failure*.*

To achieve distinction on the senior capstone, each component of the evaluation must be judged “exemplary.”

**A Sampling of Papers from Years Past**

Enumeration Problems, With Applications in Music (2022) (Read-Only PDF)

Nonparametric Alternatives to Least-Squares (2022) (Read-Only PDF)

Senior Capstone (2022) (Read-Only PDF)

What Can We Hear About the Shape of a Drum? (2022) (Read-Only PDF)

#### Second Chance

College policy provides each student who fails a senior capstone the first time with a second chance. If the senior capstone paper was found to be unacceptable, the student will be required to write a paper on a topic chosen by the department. The final version of this second paper will be submitted to the department during the next to the last week of classes in the spring semester. Other (preliminary) deadlines will be imposed by the department in consultation with the student.