Choosing an MSc Mathematics project is an act of tradeoffs: ambition versus feasibility, novelty versus familiarity, and theory versus computation. 

This article groups ideas by discipline, gives practical advice for selecting and delivering a thesis, and provides concrete project examples that are realistic for a 6-12 month master’s program.

What to expect in a thesis

• literature review and precise problem statement
• either new proofs/rigorous analysis, or new models/simulations, or both
• clear deliverables: code, computations, theorems, plots, and a written thesis

Who is this guide for?

MSc students in mathematics who want practical, academically solid project ideas and a step-by-step playbook to finish.

How to choose the right MSc project?

Criteria for a strong project

• Interest: you will work on it for months; pick something that keeps you motivated.
• Feasibility: fits your timeframe and available resources.
• Supervisor fit: choose a topic your supervisor can advise on.
• Originality: small, achievable novelty is fine.
• Impact: potential for follow-up work, publications, or real-world use.
• Reproducibility: you should be able to share code, data, or proofs.

Quick Checklist (use before committing)

1. Do I have the required background courses?
2. Can I access needed software or data?
3. Is there a clear, focused research question?
4. Can the work produce at least one deliverable in 3 months?
5. Does my supervisor endorse the scope?

Risk Scoring (simple)

• Low: theory or computation with clear, bounded scope.
• Medium: requires some data or heavy computation, but manageable.
• High: depends on external datasets, new experimental setup, or very hard proofs.

MSC Mathematics Project Ideas

Discover inspiring MSc Mathematics project ideas that combine theory, computation, and real-world applications. From pure proofs to data-driven models, each topic challenges your mind and sharpens your research skills.

Pure Mathematics

Algebra and Number Theory

1. Fusion Systems of p-Groups

• Goal: Classify fusion patterns for p-groups of small order.
• Methods: Finite group theory, GAP computations.
• Difficulty: High.
• Deliverable: Classification charts, proofs, and example computations.

2. Growth and Amenability of Self-Similar Groups

• Goal: Analyze growth and amenability for automaton-defined groups.
• Methods: Combinatorial group theory, growth estimates.
• Difficulty: High.
• Deliverable: Growth plots, example proofs, summary report.

3. Diophantine Families with Parametric Bounds

• Goal: Derive integer solution bounds for parameterized equations.
• Methods: Elementary number theory, SageMath computation.
• Difficulty: Medium.
• Deliverable: Proofs, data verification, scripts.

4. p-adic Lifting in Exponential Equations

• Goal: Apply Hensel’s lemma to exponential Diophantine problems.
• Methods: p-adic analysis, lifting arguments.
• Difficulty: High.
• Deliverable: Proofs for sample equations, example calculations.

5. Class Groups of Quadratic Fields in Families

• Goal: Compute class numbers for a family and test heuristic patterns.
• Methods: Algebraic number theory, computational experiments.
• Difficulty: Medium.
• Deliverable: Data table, conjectural trends, proofs for small cases.

Representation Theory and Knot Theory

1. Character Tables of Finite Groups

• Goal: Compute character tables for non-abelian groups up to a fixed order.
• Methods: Character theory, GAP computations.
• Difficulty: Low.
• Deliverable: Tables, short analyses, pattern notes.

2. Modular Representations of Symmetric Groups

• Goal: Study decomposition behavior for small n and prime p.
• Methods: Modular representation theory, examples.
• Difficulty: Medium.
• Deliverable: Decomposition tables, explanatory notes.

3. Knot Invariants via Quantum Groups

• Goal: Compute and interpret Jones polynomials for simple knots.
• Methods: Quantum group representations, symbolic computation.
• Difficulty: Medium.
• Deliverable: Invariant tables, code for computations.

4. Braid Groups and Representations

• Goal: Explore linear representations of braid groups and connections to knots.
• Methods: Group representation theory, algebraic computation.
• Difficulty: High.
• Deliverable: Proof sketches, sample computations.

5. Diagrammatic Categorification

• Goal: Understand categorification of small quantum invariants.
• Methods: Diagrammatic algebra, literature synthesis.
• Difficulty: High.
• Deliverable: Survey report, worked examples.

Analysis and Functional Analysis

1. Fixed Point Theorems in Banach Spaces

• Goal: Apply Schauder or Banach fixed point theorems to nonlinear ODEs.
• Methods: Rigorous analysis, small numerical demonstrations.
• Difficulty: Medium.
• Deliverable: Proof examples, applied illustrations.

2. Wavelet Construction and Applications

• Goal: Develop wavelet bases and apply them to simple signals.
• Methods: Harmonic analysis, MATLAB/Python coding.
• Difficulty: Medium.
• Deliverable: Code, plots, derivations.

3. Spectral Theorem in Hilbert Spaces

• Goal: Study and apply the spectral theorem to compact operators.
• Methods: Operator theory, proofs, small examples.
• Difficulty: High.
• Deliverable: Proof notes, operator examples.

4. Sobolev Spaces and Embeddings

• Goal: Verify Sobolev inequalities for selected domains.
• Methods: Functional analysis, proofs, computational checks.
• Difficulty: High.
• Deliverable: Analytical proofs, summary table.

5. Integral Equations and Compactness

• Goal: Explore Fredholm integral equations and compact operator theory.
• Methods: Integral transforms, functional analysis.
• Difficulty: Medium.
• Deliverable: Theoretical results, numerical tests.

Topology and Geometry

1. Homology of Configuration Spaces

• Goal: Compute homology for low-dimensional linkage spaces.
• Methods: Algebraic topology, chain computation.
• Difficulty: High.
• Deliverable: Chain complexes, computed examples.

2. Minimal Surfaces and Curvature Visualization

• Goal: Analyze minimal surfaces and visualize curvature.
• Methods: Differential geometry, symbolic computation.
• Difficulty: Medium.
• Deliverable: Plots, proofs, short report.

3. Fundamental Groups of Surfaces

• Goal: Compute presentations and examples for selected surfaces.
• Methods: Combinatorial topology, group presentations.
• Difficulty: Medium.
• Deliverable: Presentation tables, diagrams.

4. Morse Theory and Critical Points

• Goal: Apply Morse theory to study topology of manifolds.
• Methods: Differential topology, analysis.
• Difficulty: High.
• Deliverable: Proofs, illustrative plots.

5. Geodesics on Curved Surfaces

• Goal: Numerically approximate geodesics on given manifolds.
• Methods: Differential geometry, numerical methods.
• Difficulty: Medium.
• Deliverable: Simulation plots, derivations.

Combinatorics and Enumerative Problems

1. Asymptotics of Partition Functions

• Goal: Derive asymptotic formulas for restricted partitions.
• Methods: Generating functions, analytic combinatorics.
• Difficulty: High.
• Deliverable: Derivations, computed tables.

2. RSK Correspondence and Applications

• Goal: Illustrate the Robinson–Schensted–Knuth correspondence.
• Methods: Combinatorial proofs, Python scripts.
• Difficulty: Medium.
• Deliverable: Examples, visualizations.

3. Graph Coloring and Chromatic Polynomials

• Goal: Compute chromatic polynomials for graph families.
• Methods: Graph theory, recursion.
• Difficulty: Medium.
• Deliverable: Computed tables, proofs.

4. Extremal Set Theory

• Goal: Study bounds in set systems with intersection constraints.
• Methods: Combinatorial proofs.
• Difficulty: High.
• Deliverable: Proof summaries, results chart.

5. Generating Functions for Tilings

• Goal: Derive and analyze generating functions for simple tiling problems.
• Methods: Combinatorial analysis, recurrence relations.
• Difficulty: Medium.
• Deliverable: Proofs, computed examples.

Applied Mathematics

Differential Equations and Dynamical Systems

1. Stability in Nonlinear Predator–Prey Models

• Goal: Analyze local and global stability in modified Lotka–Volterra systems.
• Methods: Phase plane analysis, bifurcation theory, simulations.
• Difficulty: Medium.
• Deliverable: Analytical stability proofs, phase plots, report.

2. Reaction–Diffusion Pattern Formation

• Goal: Study Turing patterns in two-species reaction–diffusion systems.
• Methods: Linear stability, finite difference simulations.
• Difficulty: High.
• Deliverable: Simulations, pattern visualizations, derivations.

3. Chaos in Discrete Dynamical Systems

• Goal: Investigate onset of chaos in logistic and Hénon maps.
• Methods: Numerical iteration, Lyapunov exponents, bifurcation diagrams.
• Difficulty: Medium.
• Deliverable: Plots, code, written analysis.

4. Nonlinear Oscillators and Limit Cycles

• Goal: Explore periodic solutions using Poincaré–Bendixson theorem.
• Methods: Analytical study, numerical confirmation.
• Difficulty: Medium.
• Deliverable: Proofs, diagrams, small code demo.

5. Delay Differential Equations in Biology

• Goal: Model biological feedback with delay effects.
• Methods: Stability analysis, MATLAB simulations.
• Difficulty: High.
• Deliverable: Time series plots, stability conditions, report.

Fluid Dynamics and Geophysical Models

1. Navier–Stokes Flow in Simple Geometries

• Goal: Simulate steady laminar flow between plates or in pipes.
• Methods: Finite difference/element methods, MATLAB or FEniCS.
• Difficulty: High.
• Deliverable: Solver, plots, analysis of velocity fields.

2. Vortex Shedding Behind Cylinders

• Goal: Model flow separation and vortex streets.
• Methods: CFD simulation, stability analysis.
• Difficulty: High.
• Deliverable: Flow animations, vorticity plots, brief report.

3. Shallow Water Equations for Ocean Currents

• Goal: Implement a 1D or 2D shallow-water model.
• Methods: Finite difference scheme, boundary condition analysis.
• Difficulty: High.
• Deliverable: Simulation, plots, method validation.

4. Boundary Layer Flow Analysis

• Goal: Derive and simulate Blasius-type boundary layer profiles.
• Methods: Similarity solutions, numerical integration.
• Difficulty: Medium.
• Deliverable: Plots, derivation notes, code.

5. Heat Transfer with Convection–Diffusion

• Goal: Study combined conduction and convection in 2D.
• Methods: PDE modeling, finite element simulation.
• Difficulty: Medium.
• Deliverable: Simulation results, convergence analysis.

Optimization and Operations Research

1. Linear Programming and Simplex Visualization

• Goal: Implement and visualize the simplex algorithm.
• Methods: Linear algebra, MATLAB/Python coding.
• Difficulty: Medium.
• Deliverable: Working code, step-by-step visual output.

2. Integer Programming for Scheduling Problems

• Goal: Solve a resource allocation or timetabling problem.
• Methods: Mixed-integer optimization, solver tools.
• Difficulty: High.
• Deliverable: Optimization model, solution report.

3. Network Flow Optimization

• Goal: Model and optimize logistics or supply chains.
• Methods: Graph algorithms, max-flow/min-cost analysis.
• Difficulty: Medium.
• Deliverable: Network diagrams, algorithm implementation.

4. Portfolio Optimization under Constraints

• Goal: Formulate and solve constrained quadratic optimization.
• Methods: Convex optimization, Python (cvxpy).
• Difficulty: Medium.
• Deliverable: Optimization code, result summary.

5. Multi-objective Optimization Methods

• Goal: Compare Pareto-front algorithms for two criteria.
• Methods: Evolutionary or scalarization techniques.
• Difficulty: Medium.
• Deliverable: Graphs, algorithm report, visualizations.

Stochastic Processes and Control

1. Random Walks and Hitting Times

• Goal: Analyze expected hitting times in simple random walks.
• Methods: Probability theory, Markov chains.
• Difficulty: Medium.
• Deliverable: Analytical solutions, simulation verification.

2. Stochastic Differential Equations (SDEs)

• Goal: Compare numerical methods for SDE simulation.
• Methods: Euler–Maruyama, Milstein schemes.
• Difficulty: High.
• Deliverable: Code, convergence plots, error table.

3. Queueing Models and Waiting Times

• Goal: Analyze M/M/1 or M/M/c queues and simulate results.
• Methods: Stochastic modeling, Monte Carlo.
• Difficulty: Medium.
• Deliverable: Analytical results, simulation plots.

4. Markov Decision Processes (MDPs)

• Goal: Model a control problem using finite-state MDPs.
• Methods: Dynamic programming, reinforcement learning basics.
• Difficulty: High.
• Deliverable: Policy computation, performance report.

5. Stochastic Gradient Descent as a Continuous Process

• Goal: Study SGD dynamics via stochastic differential approximation.
• Methods: Probability theory, differential equations.
• Difficulty: High.
• Deliverable: Mathematical derivation, simulation demo.

Computational and Statistical Mathematics

Numerical Analysis

1. Finite Difference Schemes for PDEs

• Goal: Derive and test schemes for the heat or wave equation.
• Methods: Taylor expansion, stability analysis.
• Difficulty: Medium.
• Deliverable: Code, convergence plots, summary report.

2. Convergence of Iterative Solvers

• Goal: Analyze multigrid or conjugate gradient convergence.
• Methods: Linear algebra, numerical benchmarking.
• Difficulty: Medium.
• Deliverable: Data table, convergence graphs.

3. Spectral Methods for Differential Equations

• Goal: Implement spectral collocation for a simple PDE.
• Methods: Fourier/Chebyshev basis, numerical integration.
• Difficulty: High.
• Deliverable: Plots, solver code, validation.

4. Finite Element Method (FEM) Basics

• Goal: Build a small FEM solver for Poisson’s equation.
• Methods: Variational formulation, FEniCS/Python.
• Difficulty: High.
• Deliverable: FEM implementation, visualization, analysis.

5. Numerical Methods for SDEs 

• Goal: Compare strong and weak convergence rates for SDE solvers.
• Methods: Stochastic calculus, simulation.
• Difficulty: High.
• Deliverable: Error analysis, plots, report.

Statistics and Machine Learning Theory

1. Foundations of Bayesian Inference

• Goal: Prove basic convergence properties of Bayesian estimators.
• Methods: Probability theory, asymptotic analysis.
• Difficulty: Medium.
• Deliverable: Theoretical proofs, small simulations.

2. Kernel Methods and RKHS Theory

• Goal: Derive theoretical properties of kernel regression or SVMs.
• Methods: Functional analysis, proofs, Python experiments.
• Difficulty: High.
• Deliverable: Proofs, illustrative code.

3. Regularization and Bias–Variance Tradeoff

• Goal: Explore L1/L2 regularization in regression models.
• Methods: Optimization, numerical simulation.
• Difficulty: Medium.
• Deliverable: Plots, derivations, experiment results.

4. Statistical Learning Bounds

• Goal: Study PAC bounds for simple classifiers.
• Methods: Probability inequalities, theoretical derivation.
• Difficulty: High.
• Deliverable: Proof notes, example applications.

5. Bootstrap Methods in Inference

• Goal: Evaluate the accuracy of bootstrap confidence intervals.
• Methods: Simulation, resampling.
• Difficulty: Medium.
• Deliverable: Code, data analysis, error estimates.

Cryptography and Coding Theory

1. RSA and Integer Factorization

• Goal: Explore mathematical security and small-scale attacks.
• Methods: Number theory, modular arithmetic.
• Difficulty: Medium.
• Deliverable: RSA demo, analysis report.

2. Elliptic Curve Cryptography (ECC)

• Goal: Implement ECC operations and study security aspects.
• Methods: Algebraic geometry, Python coding.
• Difficulty: High.
• Deliverable: Working code, key exchange demo.

3. Reed–Solomon Codes

• Goal: Construct and decode RS codes using polynomial algebra.
• Methods: Finite fields, error correction algorithms.
• Difficulty: Medium.
• Deliverable: Encoder/decoder code, example results.

4. LDPC Codes and Iterative Decoding

• Goal: Analyze low-density parity-check code performance.
• Methods: Probability, linear algebra, simulation.
• Difficulty: High.
• Deliverable: Simulation plots, decoding results.

5. Lattice-based Cryptography (Post-Quantum)

• Goal: Implement small lattice cryptosystems and analyze hardness.
• Methods: Linear algebra, number theory.
• Difficulty: High.
• Deliverable: Prototype code, parameter analysis.

6. Interdisciplinary Directions

Mathematical Biology

1. Epidemic Modeling and Fitting

• Goal: Fit SIR/SEIR models to real epidemic data.
• Methods: ODE modeling, parameter estimation.
• Difficulty: Medium.
• Deliverable: Model code, parameter tables, report.

2. Tumor Growth Models

• Goal: Model tumor growth using logistic or diffusion-based systems.
• Methods: PDE/ODE theory, numerical fitting.
• Difficulty: High.
• Deliverable: Simulations, plots, discussion.

3. Population Genetics Models

• Goal: Study gene frequency dynamics under drift and selection.
• Methods: Stochastic modeling, diffusion approximation.
• Difficulty: High.
• Deliverable: Analytical results, simulation figures.

4. Neural Field Equations

• Goal: Model activity propagation in simplified neural fields.
• Methods: Integral equations, simulation.
• Difficulty: High.
• Deliverable: Plots, stability analysis.

5. Predator–Prey Pattern Formation

• Goal: Simulate spatial predator–prey systems with diffusion.
• Methods: Reaction–diffusion modeling, finite differences.
• Difficulty: Medium.
• Deliverable: Simulation output, analysis report.

Financial Mathematics

1. Option Pricing via Black–Scholes

• Goal: Derive and solve the Black–Scholes PDE.
• Methods: Stochastic calculus, finite difference methods.
• Difficulty: Medium.
• Deliverable: Code, plots, short report.

2. Portfolio Optimization and Risk

• Goal: Formulate and solve a mean–variance optimization model.
• Methods: Convex optimization, Python (cvxpy).
• Difficulty: Medium.
• Deliverable: Solver code, efficient frontier plots.

3. Monte Carlo Simulation for Derivatives

• Goal: Price options using Monte Carlo techniques.
• Methods: Stochastic simulation, variance reduction.
• Difficulty: Medium.
• Deliverable: Simulation code, convergence plots.

4. Stochastic Volatility Models

• Goal: Study and simulate Heston-type models.
• Methods: SDE simulation, calibration.
• Difficulty: High.
• Deliverable: Simulations, parameter report.

5. Interest Rate Modeling

• Goal: Derive and analyze Vasicek or CIR short-rate models.
• Methods: Stochastic calculus, parameter fitting.
• Difficulty: High.
• Deliverable: Model equations, calibration charts.

Game Theory and Economics

1. Evolutionary Game Dynamics

• Goal: Study replicator equations and their equilibria.
• Methods: Differential equations, stability analysis.
• Difficulty: Medium.
• Deliverable: Phase portraits, proof notes.

2. Auction Theory Models

• Goal: Analyze equilibria in first- and second-price auctions.
• Methods: Game theory, mathematical proofs.
• Difficulty: Medium.
• Deliverable: Analytical derivations, graphs.

3. Repeated Games and Cooperation

• Goal: Explore conditions for cooperation in iterated games.
• Methods: Dynamic programming, simulation.
• Difficulty: Medium.
• Deliverable: Simulations, equilibrium discussion.

4. Network Games and Centrality

• Goal: Study Nash equilibria on network structures.
• Methods: Graph theory, linear algebra.
• Difficulty: High.
• Deliverable: Proofs, numerical examples.

Mechanism Design Basics

• Goal: Explore incentive-compatible mechanisms.
• Methods: Optimization, equilibrium theory.
• Difficulty: High.
• Deliverable: Analytical report, example mechanisms.

Mathematics Education and Outreach

1. Visualizing Core Math Concepts

• Goal: Create interactive visualizations for undergraduate topics.
• Methods: GeoGebra, Python, or web tools.
• Difficulty: Low.
• Deliverable: Interactive demo, tutorial writeup.

2. Effect of Problem-Based Learning on Proof Skills

• Goal: Measure learning impact of problem-based teaching.
• Methods: Experimental design, data analysis.
• Difficulty: Medium.
• Deliverable: Study report, statistical results.

3. Gamified Learning in Algebra

• Goal: Design a simple game to teach algebraic concepts.
• Methods: Educational modeling, software design.
• Difficulty: Medium.
• Deliverable: Prototype app, evaluation summary.

4. Mathematical Storytelling Techniques

• Goal: Study effect of narrative framing on concept retention.
• Methods: Survey-based experiment, data statistics.
• Difficulty: Low.
• Deliverable: Survey report, insights.

5. Concept Mapping for Calculus

• Goal: Evaluate effectiveness of concept maps in learning calculus.
• Methods: Cognitive mapping, survey design.
• Difficulty: Medium.
• Deliverable: Concept maps, data analysis report.

Sample Project Template (for each idea)

Use this template when you write your project proposal:

• Title
• Goal: one sentence.
• Background: needed prerequisites.
• Methods: theory, computation, experiments.
• Deliverables: code, theorems, plots, thesis chapters.
• Difficulty: low / medium / high.
• Timeline: milestones.

Example deliverable set for a computational project

• Well-documented code repository with README.
• Reproducible scripts to regenerate key figures.
• Thesis chapters: introduction, methods, results, conclusion.
• Short conference-style summary (2-4 pages).

Practical Workflow And Timeline (10-16 weeks example)

This is a condensed plan you can expand for a semester-long or year-long thesis.

• Week 1-2: finalize topic and supervisor, set precise question.
• Week 3-4: focused literature review and finalize methods.
• Week 5-7: set up tools and run first experiments or first proofs.
• Week 8-10: extend experiments/proofs, start writing methodology and results.
• Week 11-12: polish results, run additional checks, reproducibility tests.
• Week 13-14: write introduction, literature review, and conclusions.
• Week 15-16: proofreading, formatting, create presentation slides, rehearse viva.

Adjust for a 6-12 month project: repeat iteration cycles, add midterm milestones and progress reports.

Backup Plans

• If data are unavailable: use synthetic data and prove robustness.
• If code runs slow: move to HPC/Colab or reduce problem size and argue scaling.

Literature Review Tips

Efficient search strategy

• Start with surveys and textbooks.
• Use citation mining: find a 2-3 core papers and follow forward and backward citations.
• Use arXiv for preprints, Google Scholar for impact and related work.

Summarize each paper in one paragraph: aim, method, results, gap. Build a concept map to identify where your project sits.

Tools, Data, And Reproducibility

Recommended software

• Computation: Python (NumPy, SciPy, Matplotlib), MATLAB, Julia, Firedrake, SageMath.
• Statistics: R, Stan, PyMC.
• Proof and notes: LaTeX.
• Version control: Git + GitHub.
• Reproducibility: provide requirements.txt or environment.yml; use notebooks for demos.

Data sources

• UCI, Kaggle, government open data portals, or synthetic datasets you construct and describe.

Publishing code and data

• Host code on GitHub and archive with Zenodo for DOI. Include a clear README and license.

Common Hurdles And Fixes

• Scope creep – fix: narrow to one precise question and a single extension.
• Missing data – fix: use synthetic data or focus on theory
• Non-reproducible code – fix: containerize or provide environment specs.
• Weak literature review – fix: make a 2×2 map of methods versus results to identify a clear gap.

Evaluation Criteria Examiners Look for

Typical rubric (example percentages)

• Originality and technical depth: 35%
• Correctness and rigor: 25%
• Methodology and experiments: 20%
• Presentation, writing, and reproducibility: 15%
• Viva performance and defense: 5%

Focus on clarity and reproducibility to score well across categories.

Presentation And Viva Tips

A 10-12 slide viva structure

1. Title and objective
2. Motivation and background
3. Problem statement and contributions
4. Methodology – theory or experiments
5. Key results – theorem statements or plots
6. Validation and reproducibility
7. Discussion and limitations
8. Conclusion and future work
9. References and acknowledgements
10. Backup slides with extra figures or proofs

Anticipate common questions with one-line answers

• Why this problem?
• What is original here?
• What would you do next?
• What failed and why?
• What are assumptions and limitations?

Present negative results positively – explain what they reveal and how they shape next steps.

Resources And Further Reading

Short list to get started

• Textbooks in your chosen field (classic graduate texts).
• arXiv for recent preprints.
• University thesis repositories for sample structures.
• Software docs: NumPy, SciPy, Matplotlib, Stan, Firedrake.
• Data sources: UCI, Kaggle, government portals.

Frequently Asked Questions

Final Encouragement And Next Steps

Actionable next steps

1. Pick 2-3 ideas from the shortlist that excite you.
2. Check prerequisites and do a rapid literature scan.
3. Talk to potential supervisors and get feedback on scope.
4. Use the week-by-week plan above to draft a project timeline.
5. Set up a reproducible coding environment from day one.

A thesis is a craft project that rewards focus, clear communication, and reproducible work. Pick something that makes you curious and manageable within your timeframe, and you will find the months more rewarding and productive.