what is thesis math

• Skip to main content
• Skip to primary site menu

Department of Mathematics

Senior theses.

An undergraduate thesis is a singly-authored mathematics document, usually between 10 and 80 pages, on some topic in mathematics. The thesis is typically a mixture of exposition of known mathematics and an account of your own research.  

To write an undergraduate thesis, you need to find a faculty advisor who will sponsor your project. The advisor will almost surely be a faculty member of the pure math department, though on occasion we have accepted theses written by people with applied math advisors. In these rare cases, the theses have been essentially pure math theses.

Explore Brown University

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.

9500 Gilman Drive, La Jolla, CA 92093-0112

(858) 534-3590

Guidelines for writing a thesis

These guidelines are intended for students writing a thesis or project report for a  Third Year Project Course ,  Honours year  or  Postgraduate Coursework Project . Postgraduate research students should see  Information about Research Theses  for postgraduate research students.

Before you start your Honours or Project year, you should speak to members of staff about possible thesis topics. Find out who works in the areas that you are interested in and who you find it easy to talk mathematics with. If at all possible, settle on a topic and supervisor before the start of the first semester of your Honours or Project year.

Most students see their supervisor about once a week, although this is usually open to negotiation between the student and the supervisor. Even if you haven't done much between visits it is a good idea to have a regular chat so that your supervisor can keep track of how you are going. You can expect your supervisor to:

• Help you select - and modify - your topic.
• Direct you to useful references on your topic.
• Help explain difficult points.
• Provide feedback on the direction of your research.
• Read and comment on drafts of your thesis.
• Help prepare you for your talk.
• Give general course advice.

Your thesis or project report is an overview of what you have been studying in your Honours or Project year. Write it as if you were trying to explain the area of mathematics or statistics that you have been looking at to a fellow student.

• Include an introduction that explains what the project is all about, and what its contents are. (It is sometimes better to leave writing this part to the end!) For many reports, a conclusion or summary is appropriate.
• Your thesis should be a coherent, self-contained piece of work.
• Your writing should conform to the highest standards of English. Aim at clarity, precision and correct grammar. Start sentences with capital letters and end them with full-stops. Don't start sentences with a symbol.
• Take great care with bibliographic referencing. Wherever some material has an external source, this should be clear to the reader. Don't just write in the introduction: 'This report contains material from [1],[2] and [3]' - give the references for the material wherever it is used. Don't gratuitously pad your reference list with references that are not referred to in the text. Check current journals for acceptable referencing styles.
• Be careful not to plagiarise. What constitutes plagiarism is perhaps a little different in mathematics and statistics compared to some other subjects since there is a limit to how different you may be able to make a proof (at least in its basic structure). We do, however, expect the report to be written in your own words. A basic rule is: if you put a fact or an idea in your report which is not your own, the reader should be able to tell where you got this fact or idea.
• The University has  policies on academic honesty and plagiarism  which all students should familiarise themselves with.

Generally, mathematics reports and theses are almost always typed in LaTeX. If you are going to type it yourself, you should allow a certain amount of time to become familiar with this software. Indeed, starting to learn LaTeX well before you actually want to write is a very good idea.

You should not underestimate the time it takes to produce a polished document. You will almost certainly need several drafts. It is very difficult to concentrate on getting the mathematics, spelling, grammar, layout, etc., all correct at once. Try getting another student to proofread what you have written - from their different viewpoint they may pick up on lots of things that you can't see.

P R Halmos (1970) in  How to write mathematics, Enseignement Math.  ((2) 16, 123-152) has the following advice: "The basic problem in writing mathematics is the same as in writing biology, writing a novel, or writing directions for assembling a harpsichord: the problem is to communicate an idea. To do so, and to do it clearly:

• you must have something to say (i.e., some ideas), and you must have someone to say it to (i.e., an audience)
• you must organize what you want to say, and you must arrange it in the order you want it said in
• you must write it, rewrite it, and re-rewrite it several times
• and you must be willing to think hard about and work hard on mechanical details such as diction, notation, and punctuation.

That's all there is to it."

His other advice includes:

• Say something: "To have something to say is by far the most important ingredient of good exposition---so much so that if the idea is important enough, the work has a chance to be immortal even if it is confusingly misorganized and awkwardly expressed..... To get by one the first principle alone is, however, only rarely possible and never desirable."
• Audience: "The second principle of good writing is to write for someone. When you decide to write something, ask yourself who it is that you want to reach." Your broad audience will be fellow Masters and Honours students, who may not be experts in your thesis topic. "The author must anticipate and avoid the reader's difficulties. As he(/she) writes, he(/she) must keep trying to imagine what in the words being written may tend to mislead the reader, and what will set him(/her) right."
• Organise: "The main contribution that an expository writer can make is to organize and arrange the material so as to minimize the resistance and maximize the insight of the reader and keep him(/her) on the track with no unintended distractions". 
• Think about the alphabet: "Once you have some kind of plan of organization, an outline, which may not be a fine one but is the best you can do, you are almost ready to start writing. The only other thing I would recommend that you do first is to invest an hour or two of thought in the alphabet; you'll find it saves many headaches later. The letters that are used to denote the concepts you'll discuss are worthy of thought and careful design. A good, consistent notation can be a tremendous help".
• Write in spirals: "The best way to start writing, perhaps the only way, is to write on the spiral plan. According to the spiral plan the chapters get written in the order 1,2,1,2,3,1,2,3,4 etc. You think you know how to write Chapter 1, but after you've done it and gone on to Chapter 2, you'll realize that you could have done a better job on Chapter 2 if you had done Chapter 1 differently. There is no help for it but to go back, do Chapter 1 differently, do a better job on Chapter 2, and then dive into Chapter 3... Chapter 3 will show up the weaknesses of Chapters 1 and 2".
• Write good English: "Good English style implies correct grammar, correct choice of words, correct punctuation, and, perhaps above all, common sense."

More information on how to write mathematics:

• Lee, K. A guide to writing mathematics
• Lee, K. Some notes on writing mathematics 
• Jackson, M. Some notes on writing in mathematics
• Reiter, A. Writing a research paper in mathematics
• Honours thesis
• Postgraduate Coursework Project
• Third Year Project Courses

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.

Quick links

• Directories

Undergraduate Honors & Senior Thesis

Students excelling in their major coursework and interested in pure math should consider Departmental Honors. Departmental Honors means you will graduate “With Distinction” as opposed to College Honors which is “With Honors”.  The most important component of graduating with Departmental Honors is researching and writing a Senior Thesis.

Requirements for Departmental Honors:

• Must complete a B.S. Mathematics Degree.
• Must satisfactorily complete at least one three-quarter sequence 402-3-4, 424-5-6, or 441-2-3; or two two-quarter sequences from this list. Exceptions must be approved by the chair of the Departmental Honors Committee.
• Must earn a GPA of 3.5 or better in Math coursework completed at the UW.
• Must write a senior thesis (earn a numerical grade for MATH 496).
• Must have a 3.3 minimum cumulative GPA at UW.

Please note: If you are not interested in the College Honors or Departmental Honors in Mathematics, you may still write a Senior Thesis. The process is the same as above, but it does not need to be approved by the Honors Committee.

Research credit (Math 498) may be available with faculty permission.

Beginning of your final year at the UW : think about a thesis topic and seek out a faculty supervisor.  Read below for more details about selecting a topic.

First week of classes the quarter before you expect to graduate: submit a thesis proposal form to the Dept. Honors Committee.  The form is online here: Math Dept. Honors Thesis Proposal Form

Last day of your final quarter: Once your advisor approves the thesis, email it to [email protected] and cc your faculty advisor.  You may also wish to upload it to the University Libraries archive .

Nature of the thesis

The senior thesis shall be an expository account of a topic in pure or applied mathematics related to the student’s area of interest. (Original results or proofs are welcome but are definitely not expected.) The thesis must contain some nontrivial mathematical arguments. (E.g., a non-technical essay on “fractals in nature” would not be acceptable.) The thesis should normally be about 20 to 30 pages in length (double spaced, Times New Roman 12pt font, 1” margins). These figures are guidelines, not rigid requirements. The topic should be something that cannot simply be read out of a standard textbook. Writing the thesis should involve:

• obtaining material from the periodical literature, or
• consulting several books and synthesizing material from them, or
• reading an account of a topic in a book that is substantially more advanced than the student’s regular coursework, digesting it, and putting it into readable form.

Choosing a topic

Finding a topic is the students’ responsibility, although consultation with faculty members is encouraged. The topic must be approved by a faculty member of the Mathematics Department who will supervise the work (the “supervisor”) and by the chair of the Departmental Honors Committee. A Senior Thesis Topic Proposal form can be found at the link above, and should be filled out by the student with the supervisor's support (the Dept. will check in with your supervisor). The topic proposal must be submitted to the chair of the Departmental Honors Committee no later than the end of the first week of classes the quarter preceding the quarter in which the student expects to graduate. Exceptions to this deadline may be granted only by the chair of the Departmental Honors Committee. Students contemplating writing a thesis are strongly encouraged to start thinking about a topic in the autumn quarter of their senior year.

Writing the thesis

The student must register for Math 496 (Honors Senior Thesis) during the last quarter of thesis work. The student may receive three credit hours of W-course credit for writing the thesis. Normally, the students will register for a reading course (Math 498) with the supervisor during the preceding quarter (s). The student will receive three hours of credit for each of these courses, but in exceptional cases, with the approval of the supervisor, the number or credit hours may be increased. The supervisor may allow the student to replace Math 498 with a suitable topics course; however, it is still expected that the student will meet periodically with the supervisor.

There is no specific required thesis template for an undergraduate thesis.  Some students may choose to use a modified version of the graduate thesis templates, but this is not required.

Approval of thesis

The student shall submit a draft of the thesis to the supervisor for comments and criticisms, and then shall submit a final version with appropriate revisions. The supervisor shall read the thesis and certify its acceptability with respect to both content and exposition. In order to ensure sufficient time for these things, the student must submit the first draft no later than three weeks before the last day classes of the quarter in which the student expects to graduate, and the final draft no later than the last day of classes. Exceptions to these deadlines may be granted only by the chair of the Departmental Honors Committee.

Once the thesis has been approved by your faculty supervisor, you will need to email the document to [email protected] (required) as well as submit it to the ResearchWorks archive , part of the University Libraries (optional but strongly recommended).  Submission to the archive will allow your thesis to be included in the dissemination and preservation of scholarly work.  Your thesis will be made publicly available.

Interdisciplinary theses

Theses which are concerned with the application of some part of mathematics to some others field are acceptable, as long as they contain some substantial mathematics. In exceptional cases the student may wish to work most closely with a faculty member in another department in preparing the thesis. However, in such cases the thesis topic and the thesis itself must still be approved by a member of the Mathematics Department.

•   Calendar
•   Linkedin
•   News Feed
•   Alumni Update

• Senior Thesis

A thesis is a more ambitious undertaking than a project. Most thesis writers within Applied Mathematics spend two semesters on their thesis work, beginning in the fall of senior year.  Students typically enroll in Applied Mathematics 91r or 99r (or Economics 985, if appropriate) during each semester of their senior year.  AM 99r is graded on a satisfactory/unsatisfactory basis.  Some concentrators will have completed their programs of study before beginning a thesis; in situations where this is necessary, students may take AM 91r for letter-graded credit, for inclusion in Breadth section (v) of the plan of study.  In the spring semester, the thesis itself may serve as the substantial paper on which the letter grade is based.  Econ 985 is also letter-graded, and may be included in the Breadth section of the plan of study in place of AM 91r.

Another, somewhat uncommon option, is that a project that meets the honors modeling requirement (either through Applied Mathematics 115 or 91r) can be extended to a thesis with about one semester of work.  Obviously the more time that is spent on the thesis, the more substantial the outcome, but students are encouraged to write a thesis in whatever time they have. It is an invaluable academic experience.

The thesis should make substantive use of mathematical, statistical or computational modeling,  though the level of sophistication will vary as appropriate to the particular problem context.  It is expected that conscientious attention will be paid to the explanatory power of mathematical modeling of the phenomena under study, going beyond data analysis to work to elucidate questions of mechanism and causation rather than mere correlation. Models should be designed to yield both understanding and testable predictions. A thesis with a suitable modeling component will automatically satisfy the English honors modeling requirement; however a thesis won't satisfy modeling Breadth section (v) unless the student also takes AM 91r or Econ 985.

Economics 985 thesis seminars are reserved for students who are writing on an economics topic. These seminars are full courses for letter-graded credit which involve additional activities beyond preparation of a thesis. They are open to Applied Mathematics concentrators with suitable background and interests.

Students wishing to enroll in AM 99r or 91r should follow the application instructions on my.harvard.

Thesis Timeline

The timeline below is for students graduating in May. The thesis deadline for May 2025 graduates is Friday, March 28, 2025 at 2:00PM. For off-cycle students, a similar timeline applies, offset by one semester. The thesis due date for March 2025 graduates is Friday, November 22, 2024 at 2:00PM. Late theses are not accepted.

Mid to late August:

Students often find a thesis supervisor by this time, and work with their supervisor to identify a thesis problem. Students may enroll in Econ 985 (strongly recommended when relevant), AM 91r, or AM 99r to block out space in their schedule for the thesis.

Early December:

All fourth year concentrators are contacted by the Office of Academic Programs. Those planning to submit a senior thesis are requested to supply certain information. This is the first formal interaction with the concentration about the thesis.

Mid-January:

A tentative thesis title approved by the thesis supervisor is required by the concentration.

Early February:

The student should provide the name and contact information for a recommended second reader, together with assurance that this individual has agreed to serve. Thesis readers are expected to be teaching faculty members of the Faculty of Arts and Sciences or SEAS. Exceptions to this requirement must be first approved by the Directors, Associate Director, or Assistant Director of Undergraduate Studies. For AM/Economics students writing a thesis on a mathematical economics topic for the March thesis deadline, the second reader will be chosen by the Economics Department. For AM/Economics students writing for the November deadline, the student should recommend the second reader.

On the thesis due date:

Thesis due at 2pm. Late theses are not accepted. Electronic copies in PDF format should be delivered by the student to the two readers and to [email protected] (which will forward to the Directors of Undergraduate Studies, Associate and Assistant Director of Undergraduate Studies) on or before that date and time. An electronic copy should also be submitted via the SEAS  online submission tool  on or before that date. SEAS will keep this electronic copy as a non-circulating backup and will use it to print a physical copy of the thesis to be deposited in the Harvard University Archives. During this online submission process, the student will also have the option to make the electronic copy publicly available via DASH, Harvard’s open-access repository for scholarly work.

Contemporaneously, the two readers will receive a rating sheet to be returned to the Office of Academic Programs before the beginning of the Reading Period, together with their copy of the thesis and any remarks to be transmitted to the student.

The Office of Academic Programs will send readers' comments to the student in late May, after the degree meeting to decide honors recommendations.

Thesis Readers

The thesis is evaluated by two readers, whose roles are further delineated below.  The first reader is the thesis adviser.  The second and reader is recommended by the student and adviser, who should secure the agreement of the individual concerned to serve in this capacity.  The reader must be approved by the Directors, Associate Director, or Assistant Director of Undergraduate Studies.  The second reader is normally are teaching members of the Faculty of Arts and Sciences, but other faculty members or comparable professionals will usually be approved, after being apprised of the responsibilities they are assuming.   For theses in mathematical economics, the choice of the second reader is made in cooperation with the Economics department.  The student and thesis adviser will be notified of the designated second reader by mid-March.

The roles of the thesis adviser and of the outside reader are somewhat different.  Ideally, the adviser is a collaborator and the outside reader is an informed critics.  It is customary for the adviser's report to comment not only on the document itself but also on the background and context of the entire effort, elucidating the overall accomplishments of the student.  The supervisor may choose to comment on a draft of the thesis before the final document is submitted, time permitting.  The outside reader is being asked to evaluate the thesis actually produced, as a prospective scientific contribution — both as to content and presentation.  The reader may choose to discuss their evaluation with the student, after the fact, should that prove to be mutually convenient.

The thesis should contain an informative abstract separate from the body of the thesis.  At the degree meeting, the Committee on Undergraduate Studies in Applied Mathematics will review the thesis, the reports from the two readers and the student’s academic record. The readers (and student) are told to assume that the Committee consists of technical professionals who are not necessarily conversant with the subject matter of the thesis so their reports should reflect this audience.

The length of the thesis should be as long as it needs to be to make the arguments made, but no longer!

Thesis Examples

The most recent thesis examples across all of SEAS can be found on the Harvard DASH (Digital Access to Scholarship at Harvard) repository . Search the FAS Theses and Dissertations collection for "applied mathematics" to find dozens of examples.

Note: Additional samples of old theses can be found in McKay Library. Theses awarded Hoopes' Prizes can be found in Lamont Library.

Recent thesis titles

Theses submitted in 2024.

Theses submitted in 2023

Senior Thesis Submission Information for A.B. Programs

Senior A.B. theses are submitted to SEAS and made accessible via the Harvard University Archives and optionally via  DASH  (Digital Access to Scholarship at Harvard), Harvard's open-access repository for scholarly work.

In addition to submitting to the department and thesis advisors & readers, each SEAS senior thesis writer will use an online submission system to submit an electronic copy of their senior thesis to SEAS; this electronic copy will be kept at SEAS as a non-circulating backup. Please note that the thesis won't be published until close to or after the degree date. During this submission process, the student will also have the option to make the electronic copy publicly available via DASH.  Basic document information (e.g., author name, thesis title, degree date, abstract) will also be collected via the submission system; this document information will be available in  HOLLIS , the Harvard Library catalog, and DASH (though the thesis itself will be available in DASH only if the student opts to allow this). Students can also make code or data for senior thesis work available. They can do this by posting the data to the Harvard  Dataverse  or including the code as a supplementary file in the DASH repository when submitting their thesis in the SEAS online submission system.

Whether or not a student opts to make the thesis available through DASH, SEAS will provide an electronic record copy of the thesis to the Harvard University Archives. The Archives may make this record copy of the thesis accessible to researchers in the Archives reading room via a secure workstation or by providing a paper copy for use only in the reading room.  Per University policy , for a period of five years after the acceptance of a thesis, the Archives will require an author’s written permission before permitting researchers to create or request a copy of any thesis in whole or in part. Students who wish to place additional restrictions on the record copy in the Archives must contact the Archives  directly, independent of the online submission system. 

Students interested in commercializing ideas in their theses may wish to consult Dr. Fawwaz Habbal , Senior Lecturer on Applied Physics, about patent protection. See Harvard's policy for information about ownership of software written as part of academic work.

In Applied Mathematics

• First-Year Exploration
• Areas of Application
• AM & Economics
• How to Declare
• Who are my Advisors?
• Secondary Field
• Research for Course Credit (AM 91R & AM 99R)
• AB/SM Information
• Peer Concentration Advisors (PCA) Program
• Student Organizations
• How to Apply
• PhD Timeline
• PhD Model Program (Course Guidelines)
• Oral Qualifying Examination
• Committee Meetings
• Committee on Higher Degrees
• Research Interest Comparison
• Collaborations
• Cross-Harvard Engagement
• Clubs & Organizations
• Centers & Initiatives
• Alumni Stories
• Graduate student stories
• Undergraduate Student Stories

Stack Exchange Network

Stack Exchange network consists of 183 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Q&A for work

Connect and share knowledge within a single location that is structured and easy to search.

What is expected from a mathematics student bachelor's thesis?

Well, I like maths, I like studying it and applying it. But I always feared the moment when I'd actually have to do a thesis. I'm in my third year of bachelor's studies. What I was wondering about is what kind of topic should I be looking for, for my thesis? It just is hard to me to see how I could actually research and contribute something to the field as a bachelor's student, as in something not trivial and that no one with much more education has come up with before.

From merely passing courses to coming up with something of mine is a huge leap for me, and I can't see how I can do something meaningful with the limited education I have. I'm feeling a bit like in maths, before say a phd there would be nothing new for me to come up with. I am interested in cryptography, so I was given by one of my teachers a paper on elliptic cryptography that seemed interesting and a nice starting point. But now I realise I don't know where to head.

• mathematics

• 9 The usual expectation in mathematics is that one give an original exposition of known material. This means that one digests and understands on one's own terms things already known, maybe filling them out with well chosen examples, and provides a coherent expository account. Only rarely does an undergraduate math thesis contain new research; the only student I've known who wrote a thesis containing real novelties later won a Fields Medal. It is the advisor's task to help you find a suitable and well delineated topic. –  Dan Fox Commented Jan 3, 2015 at 17:33
• 6 @DanFox, you should make your comment into an answer. –  Bill Barth Commented Jan 3, 2015 at 18:44

2 Answers 2

What is expected varies dramatically between different schools. In my experience, bachelor's projects in math can vary from trivial to nontrivial, and also from expository to original research, and the two scales are somewhat independent. I have supervised both trivial and nontrivial expository papers, as well as several undergraduate original research projects.

The concerns you have about being able to do research are common, but they are somewhat misplaced. You can almost certainly do more research than you believe you can do.

Here is an anecdote. In the 1980s, U.S. math departments began to offer "Research Opportunities for Undergraduates" (REUs), summer programs for undergrads to work on research. At the time, these were viewed very skeptically by many mathematicians. I have heard personal anecdotes that the NSF and NSA gave out a few initial grants on a purely experimental basis, not expecting to see very good results. But the projects flourished! Now there are many math REUs , and grants that allow schools to offer them are very competitive. The math community went from doubting that undergraduate research was possible to embracing it. Many math faculty are now required, or at least strongly encouraged, to mentor in undergraduate research if they expect to get tenure. See " Undergraduate Research in Mathematics Has Come of Age " from Notices of the AMS , 2014.

The key thing is to find a good project - which means finding a good advisor. Your advisor is responsible for finding a question that is accessible to you, for pointing out the background that you need to learn, and for mentoring you as you go. You still need to work on the research - we won't do it for you - but we don't expect you to operate as if you have already earned a PhD.

As the topic is still relevant to many others, please forgive my very late reply. (I sincerely hope that the person who posted this question did well and is in no need to read my response.)

Most bachelor theses are like the "appetizer" for a specific problem. It contains enough knowledge to get a good idea about the area, but it's kept simple so that "rookies" of that area still understand the content.

The most important questions should be covered. E.g:

(i) What am I talking about? -> Solving a PDE by approximations. (ii) Why? -> To approximate the dynamics in fluids (iii)Relevance of this thesis -> Explaining the advantages and disadvantages of my approach. (iv) How? -> Mathematical theory (v) How exactly? -> Explaining my code (vi) How good is my approach? -> Calculate accuracy (vii)Conclusion

As mathematics often tends to be, I sometimes feel overwhelmed with all the needed input. But as long as the chosen topic interests you (and yes, a good supervisor is very helpful), it's quite nice to deepen one's knowledge in a specific area.

Hopefully I could help someone with my experience.

I wish everyone success and passion!

You must log in to answer this question.

Not the answer you're looking for browse other questions tagged thesis mathematics bachelor ..

• Featured on Meta
• Site maintenance - Mon, Sept 16 2024, 21:00 UTC to Tue, Sept 17 2024, 2:00...
• User activation: Learnings and opportunities
• Join Stack Overflow’s CEO and me for the first Stack IRL Community Event in...

Hot Network Questions

• Color nested bonds
• How to prevent a bash script from running repeatedly at the start of the terminal
• Proving commutator of velocity differentiation and total time differentiation is position differentiation
• Is it really a "space walk" (EVA proper) if you don't get your feet wet (in space)?
• In Photoshop, when saving as PNG, why is the size of my output file bigger when I have more invisible layers in the original file?
• Color an item in an enumerated list (continued; not a duplicate)
• Copyright Fair Use: Is using the phrase "Courtesy of" legally acceptable when no permission has been given?
• Was Willy Wonka correct when he accused Charlie of stealing Fizzy Lifting Drinks?
• Do I have to use a new background that's been republished under the 2024 rules?
• Why do I often see bunches of medical helicopters hovering in clusters in various locations
• Rocky Mountains Elevation Cutout
• How to reply to a revise and resubmit review, saying is all good?
• Сhanging borders of shared polygon shapefile features in QGIS
• For a bike with "Forged alloy crank with 44T compact disc chainring", can the chainring be removed?
• Looking for a short story on chess, maybe published in Playboy decades ago?
• Keyboard shortcuts for running last command with changes?
• How did people know that the war against the mimics was over?
• What is the oldest open math problem outside of number theory?
• Input impedance of an op amp when sharing bias resistor across multiple op amps
• Could Prop be the top universe?
• Custom PCB with Esp32-S3 isn't recognised by Device Manager for every board ordered
• On the history of algae classification
• mmrm R package : No optimizer led to a successful model fit
• What unintended side effects may arise from making bards count as both arcane and divine spellcasters?

• Sustainability

Mathematics

Senior thesis information.

Both senior thesis and senior seminar satisfy the Bates W3 writing requirement and highlight mathematical research, writing, presentation, and group collaboration. Senior thesis is a good choice for students wanting to improve all these, with special emphasis on mathematical research on a topic chosen by the student. Senior theses also involve significant amounts of writing, presentations, and check-ins with other math thesis writers.

To ensure that each senior thesis writer has an enriching experience, the math department limits how many theses each faculty member advises, typically to no more than two theses per semester per advisor. To help the department determine senior thesis advisors, each junior math major who would like to write a senior thesis completes a request form by NOON on the last day of Winter Semester classes of the junior year, that is, by 12:00pm (noon) on Friday, April 12, 2024 . Some details:

• The request form seeks background information on the student, the student’s preferences regarding senior thesis, the student’s reasoning behind their preferences, and a description of the proposed senior thesis project. The project description should include enough information to show that the student has given their topic serious thought and that the project is feasible, given the student’s background and given the amount of time the student has to do the research.
• The math department strongly advises juniors to discuss senior thesis topics and ideas with faculty members before writing a request. The request form asks whether you have had such discussions.
• Students should plan to work at least 12 hours per week on thesis, and at least 15 hours per week if pursuing an Honors thesis.
• The math department meets to consider all senior thesis and senior seminar proposals. The department chair typically notifies students of the results of the meeting during Short Term.
• The mathematics department keeps copies of past senior theses in our lounge in Hathorn 209. We encourage prospective senior thesis writers to look through these past theses as part of deciding whether to write a thesis: past theses provide topic ideas, writing structures, and a sense of the scope of a senior thesis.

Types of thesis

• One-semester thesis: A one-semester thesis may be either in the fall (MATH 457) or winter (MATH 458). One-semester theses are due by the Friday of the final examination period of the semester in which the student is writing their thesis.
• Two-semester thesis: Two-semester theses (MATH 457 and MATH 458) not in the Honors Program are due by the last day of classes of the winter semester.
• Honors thesis: Honors theses (MATH 457 and MATH 458) are always two-semester theses and follow the procedures and deadlines of the Honors Program . While all capstone experiences expect students to demonstrate mathematical reading skills and ability to communicate mathematics, a thesis earning Honors in Mathematics is distinguished by an exceptional level of achievement in these areas. Students preferring to write an Honors thesis state this preference at the time of their senior thesis proposal. The Department then decides which students to nominate for the Honors Program, based on the thesis work presented at the end of the first semester.
• Double thesis with another major: A double thesis is a single year-long project that satisfies the thesis requirements of both mathematics and another department, and as such, requires a significant amount of mathematics. A student writing a double thesis signs up for their math thesis in one semester (either MATH 457 or MATH 458) and the other department thesis in the other semester. The math department requires the student to present a talk or poster in the “math semester.” A student who applies thesis course credit to another major may not apply that same credit to the Mathematics Major. The Department will not approve a proposal for a one-semester double thesis.

Completing the thesis

• Students turn in their thesis to their advisor, in a format determined by the advisor, and students give the department chair a final printed copy of the thesis to be placed on permanent display in the mathematics lounge.
• one-semester thesis students present a poster or a talk;
• two-semester non-honors thesis students present a talk in Fall Semester and a poster or a talk in Winter Semester;
• Honors thesis students present a talk in Fall Semester and give their Honors defense during a College-designated Honors defense time period;
• when there is a choice of a poster or a talk, this decision is to be made with the thesis advisor.

Future Students

Majors and minors, course schedules, request info, application requirements, faculty directory, student profile.

Have a language expert improve your writing

Run a free plagiarism check in 10 minutes, generate accurate citations for free.

• Knowledge Base
• How to Write a Thesis Statement | 4 Steps & Examples

How to Write a Thesis Statement | 4 Steps & Examples

Published on January 11, 2019 by Shona McCombes . Revised on August 15, 2023 by Eoghan Ryan.

A thesis statement is a sentence that sums up the central point of your paper or essay . It usually comes near the end of your introduction .

Your thesis will look a bit different depending on the type of essay you’re writing. But the thesis statement should always clearly state the main idea you want to get across. Everything else in your essay should relate back to this idea.

You can write your thesis statement by following four simple steps:

• Start with a question
• Write your initial answer
• Develop your answer
• Refine your thesis statement

Instantly correct all language mistakes in your text

Upload your document to correct all your mistakes in minutes

Table of contents

What is a thesis statement, placement of the thesis statement, step 1: start with a question, step 2: write your initial answer, step 3: develop your answer, step 4: refine your thesis statement, types of thesis statements, other interesting articles, frequently asked questions about thesis statements.

A thesis statement summarizes the central points of your essay. It is a signpost telling the reader what the essay will argue and why.

The best thesis statements are:

• Concise: A good thesis statement is short and sweet—don’t use more words than necessary. State your point clearly and directly in one or two sentences.
• Contentious: Your thesis shouldn’t be a simple statement of fact that everyone already knows. A good thesis statement is a claim that requires further evidence or analysis to back it up.
• Coherent: Everything mentioned in your thesis statement must be supported and explained in the rest of your paper.

Here's why students love Scribbr's proofreading services

Discover proofreading & editing

The thesis statement generally appears at the end of your essay introduction or research paper introduction .

The spread of the internet has had a world-changing effect, not least on the world of education. The use of the internet in academic contexts and among young people more generally is hotly debated. For many who did not grow up with this technology, its effects seem alarming and potentially harmful. This concern, while understandable, is misguided. The negatives of internet use are outweighed by its many benefits for education: the internet facilitates easier access to information, exposure to different perspectives, and a flexible learning environment for both students and teachers.

You should come up with an initial thesis, sometimes called a working thesis , early in the writing process . As soon as you’ve decided on your essay topic , you need to work out what you want to say about it—a clear thesis will give your essay direction and structure.

You might already have a question in your assignment, but if not, try to come up with your own. What would you like to find out or decide about your topic?

For example, you might ask:

After some initial research, you can formulate a tentative answer to this question. At this stage it can be simple, and it should guide the research process and writing process .

Now you need to consider why this is your answer and how you will convince your reader to agree with you. As you read more about your topic and begin writing, your answer should get more detailed.

In your essay about the internet and education, the thesis states your position and sketches out the key arguments you’ll use to support it.

The negatives of internet use are outweighed by its many benefits for education because it facilitates easier access to information.

In your essay about braille, the thesis statement summarizes the key historical development that you’ll explain.

The invention of braille in the 19th century transformed the lives of blind people, allowing them to participate more actively in public life.

A strong thesis statement should tell the reader:

• Why you hold this position
• What they’ll learn from your essay
• The key points of your argument or narrative

The final thesis statement doesn’t just state your position, but summarizes your overall argument or the entire topic you’re going to explain. To strengthen a weak thesis statement, it can help to consider the broader context of your topic.

These examples are more specific and show that you’ll explore your topic in depth.

Your thesis statement should match the goals of your essay, which vary depending on the type of essay you’re writing:

• In an argumentative essay , your thesis statement should take a strong position. Your aim in the essay is to convince your reader of this thesis based on evidence and logical reasoning.
• In an expository essay , you’ll aim to explain the facts of a topic or process. Your thesis statement doesn’t have to include a strong opinion in this case, but it should clearly state the central point you want to make, and mention the key elements you’ll explain.

If you want to know more about AI tools , college essays , or fallacies make sure to check out some of our other articles with explanations and examples or go directly to our tools!

• Ad hominem fallacy
• Post hoc fallacy
• Appeal to authority fallacy
• False cause fallacy
• Sunk cost fallacy

College essays

• Choosing Essay Topic
• Write a College Essay
• Write a Diversity Essay
• College Essay Format & Structure
• Comparing and Contrasting in an Essay

 (AI) Tools

• Grammar Checker
• Paraphrasing Tool
• Text Summarizer
• AI Detector
• Plagiarism Checker
• Citation Generator

A thesis statement is a sentence that sums up the central point of your paper or essay . Everything else you write should relate to this key idea.

The thesis statement is essential in any academic essay or research paper for two main reasons:

• It gives your writing direction and focus.
• It gives the reader a concise summary of your main point.

Without a clear thesis statement, an essay can end up rambling and unfocused, leaving your reader unsure of exactly what you want to say.

Follow these four steps to come up with a thesis statement :

• Ask a question about your topic .
• Write your initial answer.
• Develop your answer by including reasons.
• Refine your answer, adding more detail and nuance.

The thesis statement should be placed at the end of your essay introduction .

Cite this Scribbr article

If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.

McCombes, S. (2023, August 15). How to Write a Thesis Statement | 4 Steps & Examples. Scribbr. Retrieved September 11, 2024, from https://www.scribbr.com/academic-essay/thesis-statement/

Is this article helpful?

Shona McCombes

Other students also liked, how to write an essay introduction | 4 steps & examples, how to write topic sentences | 4 steps, examples & purpose, academic paragraph structure | step-by-step guide & examples, get unlimited documents corrected.

✔ Free APA citation check included ✔ Unlimited document corrections ✔ Specialized in correcting academic texts

ScholarWorks

Home > A&S > Math > Math Undergraduate Theses

Mathematics Undergraduate Theses

Theses from 2019 2019.

The Name Tag Problem , Christian Carley

The Hyperreals: Do You Prefer Non-Standard Analysis Over Standard Analysis? , Chloe Munroe

Theses from 2018 2018

A Convolutional Neural Network Model for Species Classification of Camera Trap Images , Annie Casey

Pythagorean Theorem Area Proofs , Rachel Morley

Euclidian Geometry: Proposed Lesson Plans to Teach Throughout a One Semester Course , Joseph Willert

Theses from 2017 2017

An Exploration of the Chromatic Polynomial , Amanda Aydelotte

Complementary Coffee Cups , Brandon Sams

Theses from 2016 2016

Nonlinear Integral Equations and Their Solutions , Caleb Richards

Principles and Analysis of Approximation Techniques , Evan Smith

Theses from 2014 2014

An Introductory Look at Deterministic Chaos , Kenneth Coiteux

A Brief Encounter with Linear Codes , Brent El-Bakri

Axioms of Set Theory and Equivalents of Axiom of Choice , Farighon Abdul Rahim

• Collections
• Disciplines
• SelectedWorks Gallery
• Albertsons Library
• Division of Research
• Graduate College

Advanced Search

• Notify me via email or RSS

Author Corner

• Department of Mathematics

Home | About | FAQ | My Account | Accessibility Statement

Privacy Copyright

• Ph.D. in Mathematics
• Ph.D. in Atmosphere Ocean Science
• M.S. at Graduate School of Arts & Science
• M.S. at Tandon School of Engineering
• Written Exams
• PhD Oral Exams
• PhD Dissertation Defense

General Information

Students who have earned a GPA of 3.7 or higher and taken at least 18 credits in the program have the option to write a Master's thesis under the supervision of a Mathematics faculty member. In certain cases involving interdisciplinary research, a second advisor outside the Department of Mathematics may be approved by the Director of Graduate Studies.

All students must submit the  Thesis Proposal and Advisor form , outlining the research plan for the thesis which has been approved by the thesis advisor, to the Program Administrator at least four months prior to the graduation date. The completed Master's thesis must be approved by two readers -- the thesis advisor and a second reader. At least one of the readers must be a full-time Courant Mathematics faculty member.

You can find more detailed information in the  Thesis Guidelines FAQ .

Dissertations

Most Harvard PhD dissertations from 2012 forward are available online in DASH , Harvard’s central open-access repository and are linked below. Many older dissertations can be found on ProQuest Dissertation and Theses Search which many university libraries subscribe to.