how to write a math thesis

• Monday, September 15, 2014

How to Use Mathematical Equations in Your Thesis

By Rozita Rashtchi, PhD Student Systems & Computer Engineering

It is almost impossible to write a thesis in a scientific or engineering discipline without mathematical expressions. These expressions may contain mathematical functions, symbols, parameters, numbers, units or any combination of them.  For those of you who are writing your thesis, here are some brief guidelines on how to write these mathematical expressions

Keep in mind that your discipline may have slightly different or additional requirements, so always doublecheck with your supervisor.

Formatting Mathematical Equations

A mathematical expression usually contains several parameters.  All the parameters in the thesis have to be defined on their first appearance. However, they shouldn’t be the first word of any sentence. The following tips on formatting equations can help you in writing your mathematical expressions.

• Mathematical parameters should be written in italics, (not only in the equation itself but everywhere in the thesis), but not units, numbers and mathematical functions like logarithms.
• Use the same font size for mathematical expressions as the font size of the text.
• The equations that are written on a separate line are either indented or centred, but done consistently throughout the thesis.
• Some equations are long and tend to be multi-line equations. If an equation breaks to multiple lines, align them properly and use only one number for the whole equation as far to the right as possible.
• Provide some vertical space above and below the equations that are displayed in separate lines.
• Equations are regarded as being a part of the text, and are punctuated accordingly. If it is the end of a sentence, there should be a period at the end of the equation; otherwise put a comma at the end of it.
• Do not capitalize or indent “where” or “with” when listing variables following an equation.

There are two types of equations in a thesis: those which are going to be referred and those which are not. The first type of equation should be numbered consecutively (or chapter-based) through the whole thesis, whereas the second type can be either included in the running text if they are short and don’t break the flow of the sentence, or in a separate line without an equation number, if they are long and more complex. For numbering the first type of equations, i.e., those which are going to be referred somewhere in the thesis, here are some useful tips:

• The equation numbers should be enclosed by parentheses and placed at the right-hand side of the page.
• If you are referring to an equation in the text, use eq. (3) (or other abbreviations). If it is the first word in a sentence, capitalize it, i.e., Eq. (3). It is also pretty normal to refer to an equation in the text with only its number within parentheses, like “(3)”. In this case, if it is the first word in the sentence, spell out “Equation” before the number, like “Equation (3)”.

In a scientific or engineering thesis, it is very common to present the result of experiments, measurements or simulations. In all these cases, you need to provide numbers with their units. The following tips can help you to write them properly:

• The number and its unit should be followed on the same line.
• There should be a space between the number and the unit. There should also be a separator (space or /) between units when they appear together (as a product or division). However, there should be no space between the letters of units like millimeter “mm”.
• There shouldn’t be a space between a number and the percent, angular degree, angular minute, or angular second symbols (e.g. 35%, 90°, 15′, 4″).
• Use proper characters (sometimes special characters) for mathematical symbols.
• Units that accompany numbers should be abbreviated, like “10 m” but they should be spell out when they do not accompany numbers, like “a few meters”.

Writing mathematical expressions without using an appropriate editor is difficult and sometimes impossible. Also, the result of not using it nearly always looks unprofessional. Microsoft Word comes with an equation editor which makes it easy to write almost any kind of equation. The other possibility is to use LaTex which is a document preparation system developed specially for mathematical writing.

For more information on LaTex formatting check out this document .

Other articles in this series can be found in our Grad Student Blog section.

Monday, September 15, 2014 in Grad Student blogs , News Share: Twitter , Facebook

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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 2023 , the deadline for the first draft is Friday, November 17 and the final submission is due to the thesis supervisor and the undergraduate coordinator on Friday, December 1.

For students graduating in May 2024 , the deadline for the first draft is Friday, April 19 and the final submission is due to the thesis supervisor and the undergraduate coordinator on Friday, May 3.

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.

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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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.

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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  Master’s Thesis Proposal and Advisor Form.pdf , 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.pdf .

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COLLEGE OF LIBERAL ARTS AND SCIENCES

Department of Mathematics

Thesis formatting, introduction.

If you write a Ph.D. thesis, you have to follow the  specifications  of the graduate school.

Additional information and answers to frequently asked questions can be found the graduate school’s Doctoral Degree Programs page.

A thesis in mathematics is invariably written in some form of LaTeX. It is usually quite a large LaTeX project, so it should not be your first attempt at typesetting. This implies that you should have typed a couple articles, quizzes, homeworks or exams in LaTeX before embarking on this. Below, two options have been created that conform to the guidelines set forth by the graduate school. The first is a minimal option that builds on the familiar book class that comes standard with any LaTeX distribution, while the second is a class file to be used in place of the book class, but contains extra content not suitable for all users. You may choose the option that works best in your case.

Below are a few of the well-done thesis variants from our department.

• Thesis class by Waseet Kazmi – 2023
• Thesis class by Ben Salisbury – 2012
• Thesis class by Marc Corluy – 2000

After unzipping put files into  one  directory. Some browsers (e.g. Safari on the Macs in the department) will put the files on your desktop. In this case create a new folder and put them all in there. In any case, make this move before you start compiling, because LaTeX generates a multitude of files when compiling and your desktop will turn into a (more?) disorderly mess.

Also make sure that the extensions are preserved. Some browsers have a tendency to slap on “.txt” at the end.

Open a text editor or a dedicated LaTeX editor,  such as TeXShop  (installed on department Macs).  When you are typing LaTeX code, it is usually clearer to use a fixed width font so that you have a clear view of your indentations and matrices (should there be any). In TeXShop, a good choice is “Courier Bold 14pts”. This font also has different characters for “|” and lowercase L. This is particularly important if you are trying to typeset tables where both “|” and lowercase L are used in the declaration.

You will now have to open some of the files that you downloaded and edit them. Here are descriptions of the set of files prepared by Ben Salisbury, listed (as best as possible) in order of priority.

• macros.tex  This file is the heart of your personalization. If you have been using LaTeX for your other manuscripts and have developed your own set of user-defined commands, then you should put them in this file. If you are importing text into one of the later files from an older LaTeX file with predefined commands, you will want your personal macros to be loaded already before fumbling with LaTeX error messages.

so those who prefer to use, say,

• thesis.tex  Lines 8 through 17 of this file require the user to input their personal data; i.e., name, degrees, year of graduation, advisory committee, and title. Further down in the file, you’ll notice the command \input{ch1.tex} . This imports the text from  ch1.tex  to the current position in this file. By copying the command and changing ch1 to ch2 (and so on), you will be able to link all chapters of your work to this file. Essentially, this file is the glue that holds the whole project together. This is also the file on which the TeX engine is run to obtain the desired output.
• abstract.tex ,  acknowledgments.tex , and  ch1.tex  should be self-explanatory. Of course, you should have more than one chapter to their thesis. The challenge of creating LaTeX files for the subsequent chapters is left to the user. Good luck!
• thesis.bib  This is a standard BibTeX file to be used as the source for your bibliography output. This may take some time to learn, so the file comes preloaded with a sample  .bib  file. Your best bet is find some literature on BibTeX to tackle this beast. Of course, one could simply replace the bibligraphy declarations in the  thesis.tex  with a manual bibliography, as Marc Corluy does in his setup below. To each his own.
• frontmatter.tex  Simply put: don’t touch it. If you need to make changes here, you should probably consider using Marc Corluy’s template (unless you are familiar enough to make the appropriate changes).

Here are comments on the files prepared by Marc Corluy.

• bibliography.tex  is not your first concern. The references that are listed in there are there to give you an idea about the exact format in which a book or article should be entered into this file.
• chapter1.tex ,  chapter2.tex , and  introchapter.tex  are almost empty; they are merely there to show you how to use titles and sections.
• The file  definitions.tex  you can completely ignore if you want to. It contains (re)definitions of some symbols and a fairly long list of basic mathematical symbols and functions. It can be handy to get a certain feel for how to use “def” in LaTeX and you can also add whatever you end up using often in your thesis.
• Most of the static data (title of the thesis, your previous degrees, your advisors, acknowledgment etc.) are entered in  front_matter.tex . The comments in this file will explain you what to put where.
• If you want to explain some of the notation that you use in your thesis, you should do so in notationpage.tex . It is technically part of the front matter of the thesis, but it is a separate file because this is probably the only part of the front matter data that changes as the thesis evolves.
• You will probably not change much to  settings.tex . This file loads a lot of the extensions to LaTeX and defines the elements of the  amsthm.sty  package, as is explained in the comments in  settings.tex .
• As you can guess,  thesis.tex  is the file where it all comes together. The includeonly command will allow you to compile only parts of your thesis if you prefer to do so. Note that you should compile the thesis twice to make sure that all the labels are set correctly.
• In principle, you should never change anything to  uconnthesis.cls .

Write your thesis. Good Luck.

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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

How to Effectively Write a Mathematics Research Paper

Mathematics research papers are different from standard academic research papers in important ways, but not so different that they require an entirely separate set of guidelines. Mathematical papers rely heavily on logic and a specific type of language, including symbols and regimented notation. There are two basic structures of mathematical research papers: formal and informal exposition .

Structure and Style

Formal Exposition

The author must start with an outline that develops the logical structure of the paper. Each hypothesis and deduction should flow in an orderly and linear fashion using formal definitions and notation. The author should not repeat a proof or substitute words or phrases that differ from the definitions already established within the paper. The theorem-proof format, definitions, and logic fall under this style.

Informal Exposition

Informal exposition complements the formal exposition by providing the reasoning behind the theorems and proofs. Figures, proofs, equations, and mathematical sentences do not necessarily speak for themselves within a mathematics research paper . Authors will need to demonstrate why their hypotheses and deductions are valid and how they came to prove this. Analogies and examples fall under this style.

Conventions of Mathematics

Clarity is essential for writing an effective mathematics research paper. This means adhering to strong rules of logic, clear definitions, theorems and equations that are physically set apart from the surrounding text, and using math symbols and notation following the conventions of mathematical language. Each area incorporates detailed guidelines to assist the authors.

Related: Do you have questions on language, grammar, or manuscript drafting? Get personalized answers on the FREE Q&A Forum!

Logic is the framework upon which every good mathematics research paper is built. Each theorem or equation must flow logically.

Definitions

In order for the reader to understand the author’s work, definitions for terms and notations used throughout the paper must be set at the beginning of the paper. It is more effective to include this within the Introduction section of the paper rather than having a stand-alone section of definitions.

Theorems and Equations

Theorems and equations should be physically separated from the surrounding text. They will be used as reference points throughout, so they should have a well-defined beginning and end.

Math Symbols and Notations

Math symbols and notations are standardized within the mathematics literature. Deviation from these standards will cause confusion amongst readers. Therefore, the author should adhere to the guidelines for equations, units, and mathematical notation, available from various resources .

Protocols for mathematics writing get very specific – fonts, punctuation, examples, footnotes, sentences, paragraphs, and the title, all have detailed constraints and conventions applied to their usage. The American Mathematical Society is a good resource for additional guidelines.

LaTeX and Wolfram

Mathematical sentences contain equations, figures, and notations that are difficult to typeset using a typical word-processing program. Both LaTeX and Wolfram have expert typesetting capabilities to assist authors in writing.

LaTeX is highly recommended for researchers whose papers constitute mathematical figures and notation. It produces professional-looking documents and authentically represents mathematical language.

Wolfram Language & System Documentation Center’s Mathematica has sophisticated and convenient mathematical typesetting technology that produces professional-looking documents.

The main differences between the two systems are due to cost and accessibility. LaTeX is freely available, whereas Wolfram is not. In addition, any updates in Mathematica will come with an additional charge. LaTeX is an open-source system, but Mathematica is closed-source.

Good Writing and Logical Constructions

Regardless of the document preparation system selected, publication of a mathematics paper is similar to the publication of any academic research in that it requires good writing. Authors must apply a strict, logical construct when writing a mathematics research paper.

There are resources that provide very specific guidelines related to following sections to write and publish a mathematics research paper.

• Concept of a math paper
• Title, acknowledgment, and list of authors
• Introduction
• Body of the work
• Conclusion, appendix, and references
• Publication of a math paper
• Preprint archive
• Choice of the journal, submission
• Publication

The critical elements of a mathematics research paper are good writing and a logical construct that allows the reader to follow a clear path to the author’s conclusions.

Good advice. For me, writing an essay on mathematics was very difficult. I did not have enough time and knowledge to write a quality essay. I worked a lot in the library and read many articles on the Internet. I studied information about essay writing. But I couldn’t finish the essay in full. I had to look for professional writers on the subject of mathematics. He helped me finish a few paragraphs. The work was delivered on time and on an excellent assessment.

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Preparation of the Final Thesis

The mathematics degree programs conclude with a Bachelor's or Master's thesis, in which independent work on a mathematical topic is to be demonstrated. The Examination Board has compiled the most important requirements for theses and some assessment criteria as guidelines.

• Please read the document carefully before you register your thesis.
• When registering the thesis, you confirm with your signature that you have taken note of the requirements in the document.

The teacher training programs also end with a final thesis. This can be completed in the subject mathematics.

Rules for Theses in the Subject-Specific Study Programs

Here you will find the regulations for registering and submitting final theses that apply to both mathematics programs. Specific deadlines and rules are listed under the respective degree program.

As a rule, students find a supervisor for their thesis on their own initiative.

• Every professor of mathematics in Bonn can of course assign topics for theses.
• Many other doctoral lecturers at Bonn Mathematics have been appointed by the Examination Board to supervise theses. You can enquire about this with the person concerned.
• One of the two reviewers must always be a professor of mathematics at Bonn University.
• If you have not found a supervisor yourself, you can also have one assigned to you by the Examination Board . In this case, please contact the Bachelor-Master Office Mathematics.

The thesis must be registered using the form Registration of the Thesis .The form must be signed by both you and the supervisor of the thesis.

• The form must state the topic of the thesis and your first supervisor.
• You also confirm that you have taken note of the requirements for the final theses of your degree program.
• At the same time, you will be registered for the seminar accompanying your thesis.
• The registration form must be submitted to the Bachelor-Master Office Mathematics immediately after the topic has been assigned and within four weeks of the supervisor's signature.
• If the registration form is received in February or August, the thesis can still be assessed in the semester in which it is submitted.
• For the accompanying thesis seminar you earn 6 credit points. It is registered for the semester in which your thesis is due for submission.
• During the time you are working on your thesis, there are usually three presentations in the thesis seminar on the topic of the thesis and the results achieved.
• The examination of the thesis seminar consists of a graded (final) presentation, which should be held shortly before or shortly after the submission of the thesis. 
• The day on which you give the graded presentation is an examination date and is therefore relevant for your degree.
• Therefore, please make sure that the presentation takes place before the end of the semester in which you wish to graduate.
• The deadline for submitting your thesis is calculated from the signature date of your supervisor plus the working time.
• You can see the submission date of your thesis in BASIS.
• The deadline for submitting your thesis is strict. If you submit the thesis after this date, it will be failed.
• Please note that you are responsible for meeting the submission deadline. Your supervisors are not necessarily aware of your submission deadline, and in no case could they change the deadline for you, as this is set by the examination regulations.
• In the event of illness , the deadline can be extended by up to six weeks.

The thesis must be submitted on time in the required number of copies together with the form Submission of the Thesis at the Bachelor-Master Office Mathematics . On the submission form, you confirm that you have written the paper independently and have not used any sources or aids other than those specified and that you have indicated any quotations.

Requirements:

• DIN A4, printed on both sides, with cover page
• in a bound version (no spiral bindings please!)
• If programming code or similar is available, it must be attached to each copy of the work on a CD or a USB stick that is as flat as possible (do not send it by e-mail). The CD or stick must be glued to the last page of a copy.
• A single copy of the submission form must be submitted separately.

Options for submitting the thesis:

• personal delivery during office hours
• personal delivery after making an appointment outside office hours
• by post to the Bachelor-Master Office Mathematics The date of the postmark is decisive for the submission.

The thesis is evaluated by two assessors.

• The first assessor is the person who provided the topic of the thesis.
• The second assessor must be proposed by you when you submit your thesis. You are therefore responsible for finding a suitable second reviewer. On request, the supervisor can of course help you with this.
• Both reviewers must be noted on the title page of the thesis (see templates for the title page).

As a rule, the candidate is notified of the thesis' evaluation six to eight weeks after the submission date.

Bachelor's Thesis (B.Sc. Mathematik)

• The topic of the Bachelor's thesis is usually assigned towards the end of the fifth semester.
• You need to have earned at leat 90 credit points in order to register the Bachelor's thesis.

The working period of a Bachelor's thesis is five months.

• The Bachelor's thesis is estimated to require a workload of 360 hours.
• Therefore it earns 12 credit points.

The text part of the Bachelor's thesis must be between 5 and 50 pages long.

• Deviations from this require the approval of the Examination Board.
• In this case, please obtain the consent of both your advisors.
• Then send an application by e-mail to the Bachelor-Master Office Mathematics .

The language of the Bachelor's degree program is German. You can write your Bachelor's thesis in English if

• your supervisor agrees with it and
• the thesis contains a summary in German.

There is a coursework for the Bachelor's thesis seminar, the training in subject-specific literature research . It will be noted in BASIS when you register your Bachelor's thesis and it is a prerequisite for passing the Bachelor's thesis seminar.

The training courses are offered by the University and State Library in the form of a one-off two-hour course. The course imparts knowledge that is very useful for academic work, especially when writing a Bachelor's thesis, for example the use of the relevant academic online archives.

• The courses are held in the MNL departmental library in the training room on the 1st floor.
• Usually three dates are offered every semester, each on a Monday or Wednesday from 16.15 to 18.00 hrs.
• You can register for a date via an online form.

After you have taken part in a course, we will enter the coursework as passed in BASIS.

• You must complete this training shortly before or during the time you are working on your Bachelor's thesis.

The title page of your Bachelor's thesis must be agreed with your supervisor.

• In particular, make sure that you name the correct institute to which your supervisor belongs.
• Please use our LaTeX-template for the title page of your Bachelor's thesis.

The submission deadline for the Bachelor's thesis is 5 months after the date on which the supervisor signed the application.

• Three copies of the Bachelor's thesis must be submitted.

Master's Thesis (M.Sc. Mathematics)

• The topic of the Master's thesis is usually assigned towards the end of the second semester.
• You need to have earned at leat 30 credit points in order to register the Master's thesis.

The working period of a Master's thesis is twelve months.

• The Master's thesis is estimated to require a workload of 900 hours.
• Therefore it earns 30 credit points.

The text part of the Master's thesis must be between 10 and 100 pages long.

The language of the Master's degree program is English. You can write your Master's thesis in German if

• the thesis contains a summary in English.

The title page of your Master's thesis must be agreed with your supervisor.

• Please use our LaTeX-template for the title page of your Master's thesis.

The submission deadline for the Master's thesis is 12 months after the date on which the supervisor signed the application.

• Four copies of the Master's thesis must be submitted. One copy will be made available to the Mathematics Library.

If you would like to apply for a PhD position at BIGS Mathematics you should find out about the application process at the beginning of the third Master's semester.

Reimbursement of the Printing Costs for the Thesis

The printing costs for the required copies of your thesis can be reimbursed upon application. This also applies to theses in the teacher training programs that were written in mathematics.

• Please fill out the application form and hand it in at the Bachelor-Master Office Mathematics.
• The enclosed original receipt must show the number of copies printed.
• Please attach the receipts to the separately printed page 2 of the form using a glue stick.

Please submit the form promptly, as the original receipts are only valid for 6 months.

Examination Office Mathematics

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For everyone whose relationship with mathematics is distant or broken, Jo Boaler , a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start – to approach the subject with playfulness and curiosity, not anxiety or dread.

“Most people have only ever experienced what I call narrow mathematics – a set of procedures they need to follow, at speed,” Boaler says. “Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience.”

Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed , a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced How to Learn Math , the first massive open online course (MOOC) on mathematics education. She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users.

In her new book, Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics , Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls “ishing” a math problem can help students make better sense of the answer.

What do you mean by “math-ish” thinking?

It’s a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport – these are generally answered with what I call “ish” numbers, and that’s very different from the way we use and learn numbers in school.

In the book I share an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They’re given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2 – but the most common answer 13-year-olds gave was 19. The second most common was 21.

I’m not surprised, because when students learn fractions, they often don’t learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.

But don’t you also risk sending the message that mathematical precision isn’t important?

I’m not saying precision isn’t important. What I’m suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they’ll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.

Some people ask me, “Isn’t ‘ishing’ just estimating?” It is, but when we ask students to estimate, they often groan, thinking it’s yet another mathematical method. But when we ask them to “ish” a number, they're more willing to offer their thinking.

Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mindset. I think we all need a little more ish in our lives.

You also argue that mathematics should be taught in more visual ways. What do you mean by that?

For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things – all of that contributes to our understanding of how it works.

There’s an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik’s Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking.

Some years back we were interviewing students a year after they’d done that activity in our summer camp and asked what had stayed with them. One student said, “I’m in geometry class now, and I still remember that sugar cube, what it looked like and felt like.” His class had been asked to estimate the volume of their shoes, and he said he’d imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.

When we learn about cubes, most of us don’t get to see and manipulate them. When we learn about square roots, we don’t take squares and look at their diagonals. We just manipulate numbers.

I wonder if people consider the physical representations more appropriate for younger kids.

That’s the thing – elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it’s all symbolic. There’s a myth that there’s a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they're drawing, they're sketching, they're building models, and nobody says that's elementary mathematics.

There’s an example in the book where you’ve asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn’t it be easier to teach students one standard method?

A depiction of various ways to calculate 38 x 5, numerically and visually. | Courtesy Jo Boaler

That narrow, rigid version of mathematics where there’s only one right approach is what most students experience, and it’s a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they’re not developing number sense. They don’t learn how to use numbers flexibly in different situations. It also makes students who think differently believe there’s something wrong with them.

When we open mathematics to acknowledge the different ways a concept or problem can be viewed, we also open the subject to many more students. Mathematical diversity, to me, is a concept that includes both the value of diversity in people and the diverse ways we can see and learn mathematics. When we bring those forms of diversity together, it’s powerful. If we want to value different ways of thinking and problem-solving in the world, we need to embrace mathematical diversity.