Dissertation/Dissertation chapter

Mathematics

A dissertation in mathematics is a significant component of postgraduate education where students have to work independently and produce a coherent piece of text in which they need to describe the results of their investigation. It’s a challenging and time-consuming project which requires thorough understanding of complex mathematical concepts and producing coherent research outcome.

To help you cope with this complex academic paper, we’ve created this guide which covers the essential steps in writing a dissertation in mathematics and addresses the most common difficulties that students face during the writing process. You will learn how to structure your dissertation and write its chapters. We will provide you with editing and proofreading tips and give you a list of 15 interesting dissertation topics. If you struggle with writing any other academic project, on our website, you will find full guides to all writing assignments out there.

Dissertation in mathematics is an individual project where students investigate and study a specific area of mathematical research or an application of advanced mathematical techniques and write a report on their findings. When writing a dissertation in mathematics, you will learn to understand complex mathematical texts, work on open-ended problems, and communicate mathematical reasoning and ideas clearly.

In your dissertation, you need to report what was previously known on a particular topic and contribute to some field of mathematics. A Masters dissertation can be either expository that explains some results that are already known or research-based that includes a new theorem. A PhD dissertation in mathematics is based on original mathematical research and includes at least one previously unknown substantial theorem.

The length of a dissertation varies with the topic and typically is between 50 to 80 pages for Masters’ dissertation. Keep in mind that quality is more important than quantity. Mathematical sentences contain notations, figures, and equations that are difficult to type if you use a typical word-processing program. That’s why dissertations in mathematics should normally be produced in TEX or LaTEX which were specifically designed for the production of technical and scientific documentation. If you are going to type your dissertation in mathematics yourself, you should dedicate a certain amount of time to become familiar with this software.

And keep in mind that you will need several drafts because it’s practically impossible to concentrate on getting all aspects of your paper (layout, mathematics, grammar, spelling etc.) correct at once. Writing is a process and no one is capable of producing a well-organized, clear, and error-free exposition the first time he/she sits down to write. Typically, good papers are thoroughly rewritten at least once or twice and the key sections often require 3 to 5 major revisions.

Tired of all the guides and never-ending instructions?

A dissertation in mathematics reports results of your investigation and should meet the following standards:

- It should follow a standard format for scientific communication and include a title, introduction, results, discussion, conclusions, and bibliography. You need to follow conventions of mathematical writing and citations.
- Your final paper should be correct and logically coherent. You should clearly explain which contributions are your own and which you have taken from other scholarly sources. You also have to report negative results.
- The text of your dissertation should be clear, interesting, and readable. You should write it with your readers in mind and ensure that even non-experts could be able to make sense of what you write.
- Precision is crucial. Every mathematical statement in your dissertation must have a precise mathematical meaning. You must define every term and every symbol and use them according to their definition. You must also justify every mathematical conclusion that you reach in your paper.
- If you write something (ideas, facts, proofs, mathematical results) which you obtained from a published source, you must refer to such a source in the text and list it in your bibliography to avoid plagiarism.

The first step in writing a dissertation in mathematics is choosing a topic. The dissertation project typically relates to the research interests of your supervisor so you should speak to members of the staff about possible dissertation topics and find out who works in the areas that you are interested in.

If you can’t think of something to write about, here is a list of topic suggestions for your inspiration.

- Representations of Finite Hecke Algebras for Complex Reflection Groups;
- Homotopy Type Theory: Computing Homotopy Groups of Spheres;
- Geometric Quantisation of Symplectic Manifolds;
- Cohomology of Abelian Varieties;
- Geometric Class Field Theory;
- Application of Theorems of Grushko and Stallings for Classification of Groups up to Quasi-isometry;
- Evolution Equations: Problems Arising in Population Dynamics;
- Low-rank Approximations in Multiple Variables;
- Stochastic Gradient Descent Methods;
- Application of Discrete Mathematical Techniques in Cellular Automation Models in Biology;
- Fluctuation Theory of Levy Processes;
- Random Matrices and Free Probability;
- Intersection Theory of Singular Varieties;
- Linear Programming Methods in Combinatorics;
- Mathematical Modelling in Metal Forming Processes.

Dissertations in mathematics typically give an extended analysis of a particular topic and report on a research project or study. A dissertation commonly consists of multiple chapters:

**Abstract**of approximately 300 words summarizes the aims, scope, and the conclusions of a dissertation.**Introduction**introduces a specific research topic, its scope, and significance, gives a review of relevant literature, and briefly outlines the logical structure of your dissertation.**Body**of the paper should be structured into several chapters that contain proofs and details of numerical implementations.**Conclusion**restates the main arguments, tells about the consequences, and provides suggestions for future work.

You may structure the body of the work like this:

- Introduction
- Proof of Theorem 1
- Proof of Theorem 2
- Numerical Examples
- Extensions

A dissertation in mathematics is based on independent mathematical research which differs from most researches in other fields of knowledge. To get your PhD in math, you need to create at least one new theorem and become a world’s expert in your niche. That’s not easy. Typically, it takes about a year and a half of hard work reading papers and attacking different math problems before you find one that you can prove.

So first, you need to choose an advisor, select a topic, and work hard on your research reading papers, making conjectures, and proving lemmas. While reading, you should analyze examples, and memorize important vocabulary. You should understand that you won’t be able to prove your theorem by making one big step. You’ll need work hard and add lemma after lemma, write down your research results, and keep them organized in your notebook. You may need a couple of years to prove your theorem and you never know if you are following the right or the wrong path for a proof until you actually find it. Research will take up most of your time but you’ll also need to meet regularly with your supervisor and attend all relevant seminars.

When you start writing your dissertation, you should provide definitions for terms and notations that you will use throughout the paper so that your readers can understand your work. The best way to do it is to provide explanations of the key terms in the introduction section. Your introduction should also clearly describe the problem and put it into the context.

When writing the main chapters of your dissertation, you should split lengthy proofs in several steps or several lemmas, taking care of the logical flow. When writing a proof, you must start with the hypothesis and use other mathematical truths such as definitions, axioms, computations or theorems to arrive at the desired conclusion. Write your proof like a manual in a natural step-by-step order. You need to provide all logical steps but there is no need to explain obvious arguments. It’s important to provide clear arguments and justify every step. Don’t just add a parenthetical remark, telling readers that a certain equation was used. You should provide step-by-step explanations in full sentences. When you finish a long argument, you should summarize it. You should also mark the end of your proof.

Don’t forget to link ideas in different chapters using transition words and phrases. This way, you will integrate your work and see your dissertation as one whole in your mind. It’s important to number equations, as well as theorems, lemmas, etc. by section to help your readers follow your argument. You should also include numerical results, examples, computations, comparisons, and interpretations.

Don’t try to write an abstract of your dissertation in mathematics first because you’ll only waste your time. It’s the most difficult part of any academic paper and it should be written last in spite of the fact that it appears at the very beginning of a document. You should try to condense the content of your dissertation into one paragraph and ensure that your readers will be able to understand your message. Abstract provides readers with a quick overview of your dissertation and its organization. It should express your hypothesis and your key points in the proof. Besides, your abstract should suggest applications o implications of the research you discuss in the paper.

You must properly cite all sources of the results used in your dissertation that are not your own because if you state a definition, theorem or an idea and don’t give reference to it, you tell your audience that it is something absolutely new and make a claim that you have created it. And if it’s not true, you commit plagiarism. Every time you use a person’s words or even ideas, you need to give them credit. This includes direct citing and paraphrasing. All major citation styles require using in-text citations and an entry in a reference or bibliography list.

When referring to your bibliography in text of your dissertation in mathematics, you need to include the item into the square brackets e.g. [45] and then provide more specific information such as theorem number and page number.

*e.g. E. Snow [46, Proposition 4.1, p.3] gave a good example of …*

When you present a technical material word for word, such as a definition or a statement of theorem, you don’t need to use quotation marks if you provide an appropriate reference.

As you write each chapter of your dissertation in math and finish it, you should read it for understanding and pay attention to the logical flow and sense of continuity. Then, you should read it again and this time, you need to pay attention to how comprehensible it is to ensure that your audience will understand your arguments. Finally, you should read it once more and pay attention to language issues like spelling, grammar, and punctuation. Following these three stages, you will be able to evaluate each chapter for its structure, content, and style.

When you complete all chapters, you should read the dissertation again and this time, check overall understanding, coherence, clarity, precision, and presentation. Ask your supervisor to read and criticize your early draft of your dissertation. Keep in mind that the more you polish up your math dissertation, the better your chances of getting good mark for it.

A good idea is to do a final revision of the text of your dissertation in math on a printed copy instead of doing it on the computer. Printing your math dissertation out and revising it on hard copy can help you feel as if you are reviewing someone else’s paper. It’s a useful technique because it will help you overcome the writers’ blindness and be more objective so that you will be able to spot more mistakes, typos or inconsistencies.

If you take the right approach and take full responsibility for your project from the very beginning, you will be able to write a competent dissertation in mathematics and make a significant contribution to this field of study.