writing thesis mathematics

Writing Your Thesis

The thesis should be the heart of your graduate school career. It will certainly be the most involved and difficult thing you do while in grad school.

Of course, before writing the thesis, one needs to have research to report. To make things easier on yourself, it’s a good idea to record your results as you work. Don’t rely on your memory to save you when you need to write everything down in your thesis! While you needn’t have everything written in final draft, having a detailed account of your research progress is a great idea. When you start your research, you and your advisor should try to establish a goal for your thesis as soon as possible. Performing research without a goal can be very difficult and even more frustrating.

When one does mathematical research, one rarely knows exactly where they are going. Gaining mathematical intuition comes from lots of hard work, not simply being very smart. A tried and true method for doing research is to do lots of examples, and make simplifying assumptions when needed. Before you can prove a theorem, you need a conjecture; these aren’t going to just fall in your lap! The idea is that after seeing enough examples, one can make a general conjecture and then hopefully prove it.

It’s a good idea to find out who else in the community (both in and out of the department) thinks about your field. You may find it useful to contact these people from time to time. This serves multiple purposes: you’ll lessen the chance of duplicating someone else’s research; you’ll find multiple sources of advice. While your advisor will likely be the single biggest source of help in writing your thesis, they needn’t be your only source. Talking to many people about your work will give you several different perspectives on the same thing. Seeing the same thing in different ways can be invaluable in understanding something.

When you have enough results such that you and your advisor are satisfied, you need to organize your work into one coherent document. This can be a highly non-trivial task! Make sure that your problem is stated clearly, along with why it is important, and how you solved it. Your thesis shouldn’t simply be a list of definitions, theorems, and proofs; there should be quite a bit of prose to explain the mathematical ambiance of your work. What is the motivation for even thinking about this problem? The more people that find your research interesting, the better.

Please refer to this manual for guidelines on formatting your thesis:  http://grad.ucsd.edu/_files/academics/BlueBook%202017-18%20updated%204.13.18.pdf

Defending Your Thesis

Setting a time to defend your dissertation can be frustrating. Contact your committee members well in advance in order to check availability and schedule a date/time.

You would think that finding a time for 6 people to meet would be an easy task. However, it can be exceedingly difficult. You may need to be very flexible and accommodating in order to make things work. You may also need to be persistent about asking if you have a non-responsive committee member.

Please carefully review these guidelines regarding committee attendance:

Department  Policy on Graduate Examination  Format:

Effective Fall 2022, the default format of a graduate examination in the Mathematics Department is  in person , i.e.,  all the committee members and the student are physically present in the same room for a scheduled examination . (This is set by the Division of GEPA.) However, when an unexpected situation arises and affects a committee member’s ability to participate in the examination synchronously, and when the student agrees, a remote or hybrid examination is allowed and can be decided by the committee chair or co-chairs. The following guidelines should be followed to arrange a remote or hybrid, synchronous examination:

• In forming the committee, the student needs to provide different examination options, in person, remote, or hybrid, to potential faculty committee members, and based on the conversation, the student can decide whether or not they want the faculty member on their committee. If such conversation did not take place, and if an unexpected situation arises, the faculty committee member can request remote examination, and can be released from the committee duty should the student refuse the request.
• In general, the graduate student is not allowed to opt for a remote examination unless there are extenuating circumstances, such as illness, travel difficulties related to visa problems, or a graduation deadline. Under such circumstances, the committee chair can decide to reschedule an in-person examination, or have a remote or hybrid examination.
• According to the Division of GEPA, there must be sufficient expertise among present members to examine the student. If a committee member must be absent for the scheduled exam, it is permissible for one absent committee member to examine the candidate on a separate date. The committee chair, or one co-chair, must participate synchronously in the scheduled exam.

Make sure to inform the PhD staff advisor in advance if any of your committee members will not be physically present.

During this scheduling phase, you also want to schedule your “Preliminary Appointment” with Graduate Division:  https://gradforms.ucsd.edu/calendar/index.php  – this appointment is optional but highly recommended! The purpose of this appointment is for them to check the margins and the formatting of your dissertation. While the above information should get you through this part without any problem, sometimes there are minor issues that arise and must be confronted (for example, published work that shows up in your dissertation has some extra requirements associated to it). The meeting should last about 30 minutes and you’ll receive a couple questionnaires to complete before your final appointment. You will also be required to schedule a Final Appointment with Graduate Division – allow at least a few days between your defense and your final appointment in order to finalize department paperwork.

In addition, the following information is critical to you completing your thesis, defending it, and completing your PhD:

• The university requires that your committee members each have a good readable draft of your dissertation at least FOUR WEEKS before your final defense.
• It  is your responsibility  to make arrangements with each committee member for the date and time of your defense.  Room reservations should be made at the Front Desk (in person or email to  [email protected])
• The  Final Report  form must have the original signatures of all members of the doctoral committee; the  Final Report  must also be signed by the program chair. (The  Final Report  form is initiated by the graduate coordinator and signatures are obtained from each faculty member through DocuSign.). Proxy signatures are not accepted.
• After your examination, committee chair emails PhD staff advisor confirming the passing of the defense. PhD staff advisor prepares Final Report through DocuSign.
• The final version of the thesis must conform to procedures outlined in the " Preparation and Submission Manual for Doctoral Dissertations and Master's Theses "
• The student submits the final approved dissertation to the Graduate Division  at the final document review  (the  Final Report  form is routed electronically from the program’s graduate coordinator via DocuSign). Final approval and acceptance of the dissertation by the Dean of the Graduate Division (on behalf of the University Archivist and Graduate Council) represents the final step in the completion of all requirements for the doctoral degree.

A few other suggestions:

About a week before you defend, you should send an email to your committee to remind them that your defense is coming, and you might even want to send a day-before or day-of reminder.

You should discuss the details of your defense with your advisor, but it’s basically a 50-minute talk where you highlight the main results of your dissertation. The audience is usually your committee plus a few graduate students.

Once Graduate Division has signed off on your thesis, it is time to submit your thesis online to Proquest/UMI. When you do this, they give you an option to purchase bound copies of your thesis from them. This is not particularly appealing for three reasons:

• They are rather pricey, about $40-$60 per copy
• They will print it exactly as you submitted it, according to Graduate Division standards: double-spaced, 8.5×11, etc, which doesn’t make for an attractive book. (How many of the math books on your shelf are 8.5×11 double-spaced?)

Fortunately, another option is available: self-publishing services. Originally these were intended for authors who had written a book, but couldn’t find a publisher for it, so they’d have it printed at their own expense. Nowadays, there are online sites filling this market, where you submit your manuscript and design the book yourself through their site. They can print on demand, so there is no minimum number of copies to order, and they can be quite inexpensive. A former graduate student, Nate Eldredge, chose to go with Lulu, so this article will describe that service.

You can begin by creating an account on Lulu’s site, which is pretty self-explanatory. They have several different book types available. I decided to go with a 6×9 “casewrap hardcover”, which is a pretty standard size and style for a book. If you have a yellow Springer book on your shelf, that’s a pretty good facsimile of what we’re talking about here.

The main issue, then, is reformatting the thesis into a 6×9 format. Fortunately, LaTeX makes this pretty easy. Pretty much, you just need to swich from the UCSD thesis class to the standard LaTeX book class and make a few other changes. Here is a modified version of the UCSD thesis template, modified to fit this format. Nate put comments in various places indicating the relevant changes and choices he made. In several places he took advantage of the fact that he no longer had to conform to OGS’s awkward requirements to make the thesis more “book-like” and remove some things that wouldn’t appear in a book. It shouldn’t take you more than an hour or two to convert your thesis file, depending how fastidious you are. (If you don’t want to go to this trouble, Lulu will also print 8.5×11 books. You could use your existing PDF without change. It may not look as pretty, but it will still be cheaper than UMI.)

Note that you should check carefully for overfull \hbox’es when you compile the thesis, because changing the paper size may have caused things to run outside the margins or off the page. You may have to manually break up long equations or reword paragraphs. Also, the book class will insert several apparently blank pages; these relate to the fact that the book will be printed double-sided, and guarantee that certain things always appear on the left- or right-hand side of a spread. If you want a book-like effect, you should not try to defeat this.

Once you’ve generated an appropriate 6×9 PDF file and uploaded it to Lulu, you can design a cover for it. They have a couple of different interfaces. For his thesis, Nate created a pretty simple cover with a UCSDish blue color scheme, and the abstract and a graduation photo on the back cover.

When you are all finished, Lulu creates a page where you or anyone else can buy copies of the book. (You have the option of keeping this private, so that only people you share it with can find it.) Then you can buy as many copies as you want to keep or give away, and you can also send the link to your parents if they want to buy lots of copies for all the relatives. (In this case, Lulu’s “revenue” option may be useful, where you select an amount to add to the price of the book, which Lulu passes along to you after each sale. The page remains up indefinitely if you want more copies later.

If you want to see what a finished product looks like, Nate Eldredge’s thesis Lulu page is located at http://www.lulu.com/content/7559872.

The book turned out quite nice looking, with quality and appearance comparable to commercially published math books. And they were only $15.46 per copy (plus tax and shipping). Overall that is a vast improvement over UMI.

Also, Nate uploaded the template as a Lulu project. It can be found at http://www.lulu.com/content/7686303.

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Senior Thesis

This page is for Undergraduate Senior Theses.  For Ph.D. Theses, see here .

A senior thesis is required by the Mathematics concentration to be a candidate for graduation with the distinction of High or Highest honors in Mathematics. See the document ‘ Honors in Mathematics ’ for more information about honors recommendations and about finding a topic and advisor for your thesis. With regards to topics and advisors: The document ‘ Faculty research areas ’ lists the research interests of current members of the Math Department.

So that Math Department senior theses can more easily benefit other undergraduate, we would like to exhibit more senior theses online (while all theses are available through Harvard University Archives, it would be more convenient to have them online). It is absolutely voluntary, but if you decide to give us your permission, please send an electronic version of your thesis to cindy@math. The format can be in order of preference: DVI, PS, PDF. In the case of submitting a DVI format, make sure to include all EPS figures. You can also submit Latex or MS word source files.

If you are looking for information and advice from students and faculty about writing a senior thesis, look at this document. It was compiled from comments of students and faculty in preparation for, and during, an information session. Let Wes Cain ([email protected]) know if you have any questions not addressed in the document.

• Senior Thesis Guidelines

A senior thesis can form a valuable part of a student's experience in the  Mathematics Major . It is intended to allow students to cover significant areas of mathematics not covered in course work, or not covered there in sufficient depth. The work should be independent and creative. It can involve the solution of a serious mathematics problem, or it can be an expository work, or variants of these. Both the process of doing independent research and mathematics exposition, as well as the finished written product and optional oral presentation, can have a lasting positive impact on a student's educational and professional future.

Supervision

Supervision by a qualified member of the field of mathematics at Cornell is the normal requirement for a senior thesis. Other arrangements are possible, however, provided they are made with the assistance of the student's major advisor, and with the approval of the Mathematics Major Committee.

Finding a supervisor/Encouraging students.  

It should be emphasized that both the writing and the supervising of a senior thesis are optional activities, both for students and faculty. Students interested in doing this will need to find a suitable supervisor — perhaps with the aid of their major advisor or another faculty member whom they know. Advisors and other faculty who encounter students whom they think would benefit from this activity are invited to mention this option to them and assist them in finding a supervisor.

Standard venues for senior theses . 

One obvious way in which a senior thesis can be produced is through an independent research course (MATH 4900); another way is through an REU experience, either at Cornell or elsewhere. (If the REU work was accomplished or initiated elsewhere, a "local expert" will still be needed to supervise or "vouch for" the work as a senior thesis.) In yet a third way, a student may present a faculty member with a solution or partial solution to an interesting problem. In such cases, this could form the core of a senior thesis. Faculty are invited to encourage such work from their students.

Public Lecture

A public lecture in which the results of the senior thesis are presented is welcome but optional. This should be arranged by the thesis supervisor in conjunction with the undergraduate coordinator and adequately advertised. Department faculty and graduate students are encouraged to attend these presentations.

Submission Deadlines

The supervisor must approve the student's thesis. The student will submit a completed first draft of the thesis to the thesis supervisor. If the supervisor asks the student to make changes, the student will have two weeks to do so and submit a PDF copy of the thesis in final form. The thesis will be posted on the department's web site.

For students graduating in December 2024 , the deadline for the first draft is Friday, November 22 and the final submission is due to the thesis supervisor and the undergraduate coordinator on Friday, December 6.

For students graduating in May 2025 , the deadline for the first draft is Friday, April 18 and the final submission is due to the thesis supervisor and the undergraduate coordinator on Friday, May 2.

Format of the Thesis

Ideally, the final document should be TeXed or prepared in some equivalent technical document preparation system. The document must have large left margins (one and one-half inches or slightly larger). The title page should contain:

The student's name and graduating class.

The title of the senior thesis.

The name of the faculty supervisor. (If there is more than one supervisor, list both. If one of the supervisors is not in the Mathematics Department, list the department and institution.)

The date of completion of the thesis.

This information will be used to produce a standard frontispiece page, which will be added to the document in its library copies.

Judgment as to the merit of a senior thesis will be based largely on the recommendation of the faculty member supervising the thesis. The Mathematics Major Committee will use this recommendation both in its determination of honors and in its decision on whether to place the thesis in our permanent library collection.

The senior thesis will automatically be considered by the Mathematics Major Committee as one of the ingredients for deciding on an  honors  designation for the student. Students may receive honors without a thesis and are not guaranteed honors with one. However, an excellent senior thesis combined with an otherwise excellent record can elevate the level of honors awarded.

Library Collection

Meritorious senior theses will be catalogued, bound, and stored in the Mathematics Library.

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