PhD Mathematics Thesis Writing Help | Proposal, Synopsis, Paper & Publication Support

1) PhD Maths Topic Selection

Choosing the right PhD topic in Mathematics is the foundation of your entire research journey. A good topic is not only a personal interest area but also a subject that can contribute new knowledge to the mathematical community. Since mathematics covers a vast range of branches — algebra, analysis, geometry, number theory, statistics, applied mathematics, and mathematical modelling — the decision requires careful planning.

Why Topic Selection is Important?

• Defines Research Path: The topic sets the direction for your proposal, methodology, proofs, and publications.
• Determines Feasibility: A practical topic ensures you can complete your thesis within 3–5 years.
• Establishes Originality: A unique topic demonstrates your scholarly contribution and value.
• Boosts Career: A relevant topic aligned with modern research increases teaching, postdoc, and publication opportunities.

Checklist for Selecting a Topic in Mathematics

1. Relevance: Does the topic address a current mathematical problem?
2. Originality: Does it fill a gap in existing research or propose a new theorem/proof?
3. Scope: Is the problem neither too broad nor too narrow?
4. Resources: Are computational tools, data, and references available?
5. Feasibility: Can the research be completed within available time and resources?
6. Supervisor Fit: Does your university have faculty experts in the chosen area?

Popular Areas for PhD Topics in Mathematics

• Pure Mathematics: Number Theory, Algebraic Topology, Differential Geometry.
• Applied Mathematics: Mathematical Modelling, Operations Research, Optimization Techniques.
• Computational Mathematics: Algorithms, Machine Learning Models, Numerical Analysis.
• Statistics: Probability Theory, Biostatistics, Stochastic Processes.
• Cryptography: Applications of Number Theory in Cybersecurity.
• Graph Theory: Networks, Algorithms, and Social Graphs.
• Mathematical Physics: Quantum Mechanics, String Theory, Relativity Models.

Steps to Finalize Your Topic

• Start with your academic interests (e.g., algebra, statistics, cryptography).
• Read latest PhD theses and journal papers in those areas.
• Identify gaps — what questions are still unanswered?
• Discuss your ideas with professors and research peers.
• Check availability of computational tools (like MATLAB, Python, Mathematica).
• Refine your idea into a precise research question or hypothesis.

Examples of Specific PhD Topics in Mathematics

• “Graph Theory Applications in Social Network Analysis.”
• “Number Theory Approaches in Cryptography and Data Security.”
• “Numerical Methods for Solving Partial Differential Equations in Physics.”
• “Algebraic Geometry and Its Applications in Coding Theory.”
• “Mathematical Modelling of Climate Change and Environmental Systems.”

2) PhD Maths Research Proposal

A PhD Research Proposal in Mathematics is the formal document that explains what you plan to research, why it is important, and how you will carry it out. It is the first major step in your doctoral journey and often decides whether your application will be accepted. A strong proposal shows that you understand your chosen area, are aware of current research trends, and have a realistic plan for contributing new results to mathematics.

Purpose of a Research Proposal

• Demonstrates originality: The proposal must show that your research problem is not already solved and will add new knowledge.
• Defines scope: It explains what exactly you will study and what lies outside your work.
• Provides methodology: It details whether you will use proofs, modelling, simulations, or computational experiments.
• Convinces evaluators: It builds confidence that your project is feasible within the PhD timeline.

Structure of a PhD Maths Research Proposal

1. Title: Should be clear, specific, and measurable. Example: “Graph Theoretic Approaches to Network Security Problems.”
2. Introduction: Short background on your chosen field and its importance. Example: Why algebraic number theory is crucial in cryptography.
3. Research Problem: Define the exact gap or open problem. Example: “Current numerical methods for nonlinear PDEs are computationally expensive; I propose a new efficient algorithm.”
4. Objectives: 3–5 clear goals.
   • Develop new mathematical models.
   • Design efficient proof or algorithm.
   • Apply results to real-world data.
5. Research Questions: Convert objectives into questions. Example: “Can a new iterative method reduce error margins in PDE simulations?”
6. Literature Review (short): Mention a few landmark studies and where they fall short.
7. Methodology: Explain whether you will use theoretical proofs, computational simulations, or applied mathematical models.
8. Expected Outcomes: Define your contribution — a new theorem, an improved algorithm, or an applied solution.
9. Timeline: Year 1 (review + formulation), Year 2 (proof/modelling), Year 3 (analysis + thesis writing).
10. References: Add 5–10 recent journal articles or textbooks relevant to your topic.

Sample Research Proposal Topics in Mathematics

• “New Optimization Techniques in Operations Research Using Graph Theory.”
• “Algebraic Geometry Applications in Cryptographic Systems.”
• “Numerical Analysis of Nonlinear Partial Differential Equations in Fluid Mechanics.”
• “Probabilistic Modelling for Risk Management in Financial Mathematics.”
• “Machine Learning and Mathematical Modelling for Predictive Analytics.”

Tips for Writing a Good Maths Proposal

• Keep the problem statement precise and mathematical.
• Show that you are aware of existing research and identify exact gaps.
• Don’t make the scope too broad; focus on one key problem.
• Demonstrate both theoretical understanding and practical application.
• Proofread carefully — mathematical writing should be clear and logical.

3) PhD Maths Synopsis Writing

A PhD synopsis in Mathematics is a concise summary of your entire research plan. While the research proposal goes into detailed explanations, the synopsis is a shorter version (around 2,000–3,000 words) that highlights the key aspects of your planned study. It is usually submitted to the university’s research committee for approval before you start the full thesis work. A strong synopsis shows that your work is well-planned, feasible, and relevant.

Purpose of a Synopsis

• To give a clear roadmap of your research in a short format.
• To present your research problem, objectives, and methods clearly.
• To convince the committee that your research is original and achievable.
• To check feasibility in terms of time, resources, and scope.

Structure of a PhD Maths Synopsis

1. Title of the Thesis: Specific and precise. Example: “A Study of Numerical Algorithms for Nonlinear PDEs in Fluid Dynamics.”
2. Introduction: Background of the area and importance of the problem.
3. Research Problem: Define the gap in current mathematical research. Example: “Existing optimization techniques are limited for large-scale graphs; my study proposes improved polynomial-time methods.”
4. Objectives: List 3–4 clear, measurable goals. Example:
   • Develop a new proof technique for graph coloring problems.
   • Propose efficient algorithms for solving nonlinear equations.
   • Apply mathematical models in real-world systems like cryptography or networks.
5. Research Questions: Convert objectives into specific questions. Example: “Can iterative methods improve convergence speed for high-dimensional optimization?”
6. Review of Literature (Brief): Summarise 4–5 key works and highlight the research gap.
7. Methodology: Explain your approach —
   • Theoretical proof-based analysis.
   • Computational methods (MATLAB, Python, Mathematica, Maple).
   • Mathematical modelling for applied problems.
8. Expected Outcomes: Define the new results you aim to achieve — a theorem, algorithm, or application.
9. References: A short list of important works (books, journals, conference papers).
10. Timeline: Simple schedule for 3 years (Year 1: Review + Formulation; Year 2: Proof/Computation; Year 3: Thesis Draft + Paper Writing).

Sample Synopsis Titles in Mathematics

• “Graph Theoretic Models for Optimizing Social Network Analysis.”
• “Efficient Numerical Methods for Solving Nonlinear PDEs in Applied Physics.”
• “Probabilistic Modelling in Financial Mathematics: Risk Prediction Frameworks.”
• “Applications of Algebraic Topology in Data Science.”
• “Mathematical Modelling of Infectious Diseases Using Differential Equations.”

Tips for Writing a Strong Synopsis

• Keep it short and focused (2,000–3,000 words max).
• Use clear and simple language, especially while explaining abstract concepts.
• Follow your university’s format strictly (font, spacing, citation style).
• Ensure that objectives, methods, and outcomes are linked logically.
• Check the plagiarism level before submission (keep below 10%).

4) PhD Maths Thesis Introduction

The Introduction chapter of a PhD in Mathematics is the foundation of the entire thesis. It explains what you are studying, why it is important, and how it connects to existing research. Unlike other chapters, the introduction does not give proofs or detailed solutions but builds the context, motivation, and research problem.

Purpose of the Introduction

• Provide Background: Introduce the mathematical area — Algebra, Analysis, Geometry, Differential Equations, Topology, Graph Theory, or Applied Mathematics.
• State the Problem: Explain the exact problem you want to solve or study.
• Justify Importance: Show why this problem matters — theoretical contribution, applications in science, engineering, economics, or computer science.
• Define Scope: Clearly mention what your study will include and exclude.
• Set Objectives: Link the background to the aims of your research.

Structure of a Good Introduction

1. General Background: Briefly discuss the broad area of mathematics where your research fits. Example: “Partial Differential Equations play a crucial role in describing fluid flow, heat conduction, and quantum mechanics.”
2. Literature Gap: Show what has already been studied and where the gap lies. Example: “Existing iterative methods are computationally expensive for high-dimensional PDEs.”
3. Research Problem: State your main problem in clear terms. Example: “This research focuses on developing faster numerical algorithms for nonlinear PDEs with stability guarantees.”
4. Research Objectives: Outline 3–4 precise aims. Example:
   • To design and analyze new numerical schemes for nonlinear PDEs.
   • To prove convergence theorems for the proposed algorithms.
   • To apply the developed models in real-world problems such as fluid dynamics.
5. Significance of the Study: Explain the theoretical contribution (new theorems, proofs, algorithms) and applied importance (engineering, cryptography, finance, data science).
6. Thesis Structure: Give a short description of upcoming chapters (Review, Methodology, Results, etc.).

Sample Introduction Excerpt (Short)

“Mathematics has always been central to scientific discovery, providing the language for physics, engineering, economics, and computer science. Among its branches, Partial Differential Equations (PDEs) have shaped our understanding of natural phenomena. Despite significant progress, challenges remain in solving nonlinear PDEs efficiently. This thesis proposes novel iterative algorithms with improved convergence and stability, bridging the gap between theoretical mathematics and real-world applications.”

Tips for Writing the Introduction

• Start with general background, then move to the specific research problem.
• Use simple but precise mathematical language.
• Always link the problem with practical or theoretical importance.
• Keep the introduction chapter 8–12 pages (around 3,000–5,000 words).
• End with a roadmap of the thesis — what each chapter will cover.

5) PhD Maths Thesis Literature Review

A literature review in Mathematics is not just a summary of textbooks or research papers. It is a critical survey of what has already been studied, what the major theorems and methods are, and what gaps still exist. The aim is to connect your research with existing mathematical knowledge.

Purpose of Literature Review

• Identify Research Gap: Find out what has already been solved and what remains unsolved.
• Build Theoretical Framework: Show the theories, models, or proof techniques relevant to your study.
• Avoid Duplication: Ensure your research is not just repeating existing work.
• Guide Methodology: The review suggests which mathematical tools or approaches are most effective.

Steps for Writing a Strong Literature Review

1. Collect Sources: Books, journal articles, PhD theses, and conference proceedings (MathSciNet, Scopus, Springer, Elsevier, AMS journals).
2. Organise by Theme: Group works under Algebra, Analysis, Geometry, Applied Mathematics, Probability, etc.
3. Summarise Key Contributions: Highlight the main results (theorems, models, proofs) and their limitations.
4. Compare and Criticise: Show how different authors approached similar problems.
5. Identify Gaps: Mention open problems or unsolved areas where your research will contribute.

Sample Structure of a Maths Literature Review

• Algebra: Past studies on Group Theory, Ring Theory, and their applications in cryptography.
• Analysis: Work on Differential Equations, Functional Analysis, and Approximation Theory.
• Applied Mathematics: Numerical methods for PDEs, optimisation problems, and real-world applications.
• Probability & Statistics: Models for random processes, stochastic calculus, and machine learning links.
• Recent Trends: Mathematical modelling in climate science, quantum computing, and data science.

Example (Short Excerpt)

“Several methods have been developed for solving nonlinear differential equations. Early works by Euler and Fourier provided analytical techniques, while modern approaches focus on iterative numerical algorithms. Recent research (Smith, 2020; Gupta, 2022) highlights computational efficiency but struggles with stability in higher dimensions. This thesis addresses this gap by proposing improved algorithms with provable convergence.”

Tips for Effective Writing

• Use mathematical precision but avoid unnecessary technical jargon in the review section.
• Include citations in APA / MLA / AMS style as required by your university.
• End each subsection with a clear gap statement — “Thus, there is a need for …”.
• Keep the chapter 25–40 pages, depending on thesis requirements.

6) PhD Maths Thesis Research Methodology

The Research Methodology of a PhD in Mathematics explains how you will conduct your study. Unlike other subjects, mathematics research focuses on logical reasoning, proofs, models, and problem-solving techniques. This section guides examiners on the exact steps you will take to answer your research questions.

Key Elements of Methodology in Mathematics

• Research Design: Whether the study is theoretical (developing new theorems/proofs), applied (real-world problems using models), or computational (using simulations and algorithms).
• Problem Definition: A clear statement of the mathematical problem or hypothesis.
• Tools and Techniques: Use of algebraic methods, differential equations, optimisation, probability theory, graph theory, numerical analysis, or other frameworks relevant to your topic.
• Data/Inputs: For applied maths, data may come from experiments, surveys, or simulations. For pure maths, the “data” are existing theorems, definitions, and logical rules.
• Proof Strategy: Inductive proofs, deductive reasoning, construction of counterexamples, or mathematical modelling approaches.
• Validation: Checking results through known theorems, comparison with existing models, or computational verification.

Example of a Methodology in Simple Words

If your research is on optimisation problems, your methodology may include:

• Defining the optimisation problem in mathematical form.
• Reviewing earlier solution methods (linear programming, dynamic programming, etc.).
• Developing a new algorithm or proof to improve accuracy or speed.
• Testing the algorithm with numerical examples or computer simulations.
• Comparing results with existing methods to show improvements.

Why Methodology is Important?

• It shows your research is systematic and logical.
• It proves your approach is valid and original.
• It helps examiners understand how your conclusions are achieved.

7) PhD Maths Data Collection & Analysis

In Mathematics research, Data Collection and Analysis depends on the type of study you are conducting. Unlike experimental sciences, mathematics often relies more on logical proofs, theorems, simulations, and models rather than physical data. However, for applied and computational mathematics, numerical data, experimental values, and real-world datasets play an important role.

1. Data Collection in Mathematics

Mathematical data can come from multiple sources, depending on whether the research is theoretical, applied, or computational:

• Theoretical Research: No external data is required; existing theorems, axioms, and definitions are the “data.”
• Applied Mathematics: Data may come from physics, engineering, finance, economics, or biological systems. Example: collecting stock market numbers for mathematical modelling in finance.
• Computational Mathematics: Simulation results, algorithmic outputs, and numerical experiments form the raw data. For instance, solving partial differential equations using numerical methods and recording accuracy levels.
• Secondary Data: Many times, published mathematical tables, existing datasets, or government reports provide numerical input for testing new models.

2. Data Analysis in Mathematics

Once data is collected, the next step is mathematical analysis. This involves applying logical reasoning, formulas, and computational tools to extract results.

• Pure Mathematics: Proof-based analysis – testing whether a theorem holds true under specific assumptions, constructing counterexamples, or extending existing proofs.
• Applied Mathematics: Statistical methods, regression, optimisation, probability distributions, and matrix analysis are often applied to real-world data.
• Computational Mathematics: Numerical methods, programming (MATLAB, Python, R), and simulations are used to check stability, convergence, and accuracy of mathematical models.
• Graphical Analysis: Graphs, charts, and computational plots help in visualising mathematical results for better interpretation.

3. Example of Data Collection & Analysis

Suppose your PhD topic is on Mathematical Modelling of Infectious Diseases.

• Data Collection: Real-world data from hospitals, WHO reports, or government health databases.
• Analysis: Applying differential equations to model disease spread, solving equations using Runge–Kutta methods, and comparing predictions with actual cases.

4. Importance of Data Collection & Analysis

• Ensures your results are evidence-based and verifiable.
• Provides numerical support to theoretical findings.
• Makes your research applicable in real-world situations.
• Strengthens the credibility of your PhD thesis.

8) Mathematical Modelling & Proof Writing

Mathematical Modelling and Proof Writing are two pillars of PhD-level mathematics research. While modelling connects abstract mathematics to real-world problems, proof writing ensures that every statement in mathematics is logically valid and universally true. A strong thesis in Mathematics requires excellence in both areas.

1. What is Mathematical Modelling?

Mathematical modelling is the process of representing real-world systems (such as physics, economics, biology, or social sciences) using equations, algorithms, and structures. The goal is to describe, analyse, and predict outcomes using mathematical tools.

• Applied Models: Population growth models (Logistic, Lotka–Volterra), epidemic models (SIR, SEIR), financial risk models (Black–Scholes), and climate models.
• Purely Mathematical Models: Abstract representation of systems in graph theory, number theory, or algebra without direct physical connection.
• Computational Models: Numerical simulations using MATLAB, Python, or R to approximate solutions of complex equations.

Steps in Mathematical Modelling

1. Problem Identification: Define the system you want to study (e.g., spread of a disease).
2. Formulation: Translate the problem into equations (differential equations, matrices, probability models).
3. Solution: Solve equations analytically or numerically.
4. Validation: Compare results with real-world or experimental data.
5. Prediction: Use the model to forecast future outcomes or optimise decisions.

Example: In epidemic modelling, we use differential equations to model how a disease spreads over time. The solutions provide predictions about infection peaks and control measures.

2. Proof Writing in Mathematics

Proof writing is the language of mathematics. It is the process of showing that a mathematical statement follows logically from already established facts, definitions, and axioms. A PhD thesis in Mathematics is judged not just by results, but by the rigour and clarity of proofs.

Types of Proofs Common in PhD Maths

• Direct Proof: Step-by-step logical argument (e.g., proving divisibility properties).
• Indirect Proof: Proof by contrapositive (if not B, then not A).
• Proof by Contradiction: Assume the opposite, show it leads to impossibility.
• Induction: Common for sequences and series; prove base case, then general case.
• Constructive Proof: Provide a concrete example or method that satisfies the condition.
• Non-Constructive Proof: Prove existence without giving an explicit example.

Best Practices in Proof Writing

• Define all terms clearly before starting the proof.
• Break down arguments into short logical steps.
• Use proper mathematical symbols and notation consistently.
• Highlight key theorems or lemmas used as a base.
• End with a conclusion sentence (e.g., “Hence, proved” or “Therefore, the statement holds true”).

3. Linking Modelling with Proofs

A unique aspect of mathematical research is when models and proofs work together:

• First, a model is constructed to represent a system.
• Then, theorems and proofs validate the assumptions and properties of that model.
• Finally, the results are generalised into new theorems or applied in practical domains.

Example: In graph theory, you may model a network as a graph, then use proofs to establish properties like connectivity, optimisation, or colouring, which are later applied in computer networks or scheduling problems.

9) Chapter-wise Thesis Writing in Mathematics

1. Introduction
2. Literature Review
3. Methodology
4. Results & Proofs
5. Discussion
6. Conclusion & Future Work

10) Mathematics Research Paper Writing

Writing a research paper in Mathematics is one of the most essential parts of a PhD journey. Unlike the thesis, which is long and detailed, a research paper is usually short (4,000–8,000 words) and focuses on a specific idea, result, or theorem. It allows you to communicate your findings to the academic world, publish in reputed journals, and contribute to the wider field of mathematics.

1. Purpose of Mathematics Research Papers

• Communicate New Results: Share your original theorems, proofs, or models with the research community.
• Get Peer Recognition: Establish credibility and gain feedback from other mathematicians.
• Career Growth: Journal publications are essential for academic jobs, postdocs, and fellowships.
• Advance the Field: Add to existing knowledge and inspire further research in your area.

2. Structure of a Mathematics Research Paper

1. Title: Concise, specific, and reflective of the theorem, concept, or model studied. Example: “On the Stability of Nonlinear Dynamical Systems Using Lyapunov Functions.”
2. Abstract: A 200–300 word summary including the problem, method, and main result.
3. Introduction: Explains the background, research gap, and significance of the study.
4. Preliminaries & Definitions: State all notations, assumptions, and basic lemmas needed.
5. Main Results & Proofs: Present theorems step-by-step with clear, logical proofs.
6. Applications / Examples: Show how the results apply in a real-world or theoretical context.
7. Discussion: Compare with earlier results, highlight strengths and possible limitations.
8. Conclusion: Short summary of contributions and suggestions for future research.
9. References: All books, articles, and sources cited in a consistent format (APA, AMS, Harvard).

3. Best Practices for Writing Mathematics Papers

• Clarity First: Write in short, logical steps so that even non-specialists can follow.
• Notation Consistency: Use standard symbols and avoid unnecessary complexity.
• Proof Rigor: Every step in a proof must follow logically; avoid skipping justifications.
• Illustrative Examples: Provide small examples or counterexamples to make abstract results clearer.
• Figures and Diagrams: Use graphs, plots, or models wherever possible (especially in applied mathematics).
• Formatting: Follow the journal’s author guidelines carefully (LaTeX is standard for mathematics papers).

4. Common Mistakes to Avoid

• Writing too broadly instead of focusing on one main result.
• Poorly written or incomplete proofs.
• Lack of references to earlier work (ignoring literature makes papers weak).
• Excessive technical jargon without explanation.
• Submitting without checking plagiarism/similarity index (should be below 10%).

5. Journal Publication in Mathematics

• UGC-CARE and Scopus Journals: Required for Indian PhD regulations.
• International Journals: Such as *Journal of Algebra*, *Annals of Mathematics*, *SIAM Journal on Applied Mathematics*.
• Submission Process: Prepare paper in LaTeX, check similarity report, submit with cover letter, wait for peer review.
• Peer Review: Be ready to revise proofs, add more examples, or strengthen the discussion as per reviewer comments.

Example: A research scholar working on Graph Theory may publish a paper on new colouring algorithms with proofs and show applications in network optimisation. Another working in Number Theory might prove a new property of prime numbers and publish it in a theoretical mathematics journal.

11) PhD Mathematics Thesis Summary & PPT (Viva-Voce)

After completing the thesis, every PhD scholar must prepare a Thesis Summary and a PPT presentation for the final Viva-Voce examination. This stage is crucial because it is the oral defense of your research in front of examiners and experts. A well-prepared summary and PPT make your work clear, structured, and impactful.

1. Thesis Summary in Mathematics

The Thesis Summary is a short version of the entire research, usually between 25–40 pages, depending on university guidelines. It should highlight only the most important points of your research. A good summary includes:

• Introduction: Topic background, problem statement, and importance of the research.
• Objectives: What your research aimed to achieve.
• Review of Literature: The research gap in earlier mathematical studies.
• Methodology: The mathematical approach, modelling, and tools used (e.g., proof techniques, simulations).
• Key Theorems/Results: A clear explanation of the main theorems, proofs, or models developed.
• Applications: Real-world or theoretical significance of the findings.
• Conclusion: Summary of contributions and possible directions for future research.

Example: In a thesis on Nonlinear Dynamical Systems, the summary may include definitions, key theorems with short proofs, stability analysis results, and a discussion of applications in physics or biology.

2. PPT (PowerPoint) for Viva

The PPT is a visual presentation (15–25 slides) designed to present your research clearly and quickly. It should be simple, professional, and visually engaging. The main structure of the PPT is:

• Slide 1: Title, Name, University, Supervisor.
• Slide 2–3: Introduction & research problem.
• Slide 4: Objectives and research questions.
• Slide 5–6: Literature review highlights (only 3–4 key points).
• Slide 7–8: Research methodology (mathematical tools, models, proofs).
• Slide 9–14: Main theorems, models, or results (with equations, graphs, charts).
• Slide 15–16: Applications of results.
• Slide 17–18: Conclusion and future scope.
• Slide 19: References (short list).
• Slide 20: Acknowledgements & Thank You.

3. Tips for Effective Thesis Summary & PPT

• Keep explanations short and simple – avoid overloading slides with text.
• Use equations, graphs, or diagrams instead of only text (especially in applied mathematics).
• Practice presenting your PPT in 10–15 minutes.
• Prepare answers for likely examiner questions (e.g., limitations, future work).
• Check formatting – neat, professional slides give a strong impression.

Example: A viva presentation in Graph Theory might include definitions, a new algorithm for graph coloring, proofs with diagrams, and applications in computer networks.

12) Plagiarism Check in Mathematics Thesis

UGC guidelines: similarity should be below 10%... use paraphrasing, proper citations.

12) Indicative Timeline (PhD Mathematics)

A PhD in Mathematics usually takes 3–5 years, depending on the university regulations, research scope, and complexity of the chosen topic. To ensure smooth progress, it is important to follow a stage-wise plan. Below is an indicative timeline that most mathematics scholars can adapt.

Year 1 – Foundation Stage

• Months 1–3: Selection of topic, supervisor discussions, finalizing research area.
• Months 4–6: Preparation of Research Proposal and presentation before the Research Committee.
• Months 7–12: Writing and submission of Synopsis; conducting extensive literature review on related theories, theorems, and models.

Year 2 – Literature & Methodology

• Months 13–18: Deep study of existing mathematical frameworks, problem identification.
• Months 19–24: Developing Research Methodology — proof strategies, computational methods, or mathematical modelling.
• Begin drafting initial chapters: Introduction, Literature Review, Methodology.

Year 3 – Data Collection & Analysis

• Months 25–30: Data collection (theorems, problem sets, computational simulations).
• Months 31–36: Analysis of mathematical models, testing hypotheses, proof verification.
• Drafting of results and analysis chapters.

Year 4 – Writing & Refinement

• Months 37–42: Complete Results & Discussion chapters.
• Months 43–46: Thesis writing in full format (chapter-wise).
• Months 47–48: Plagiarism check (<10%), corrections, and formatting.

Year 5 – Final Stage (if required)

• Months 49–52: Preparation of Thesis Summary and PPT (Viva-Voce).
• Months 53–60: Thesis submission, examiner evaluation, viva-voce defense.

Quick Timeline Chart

14) Common Mistakes and How to Avoid

• Too broad topics
• Weak objectives
• Poor proof structure
• No citations
• High plagiarism

15) Get Started

Share your current stage — proposal, modelling, analysis, or writing. We’ll guide you step-by-step.