Mathematical modeling project: The student selects a real-world system or phenomenon that can be modeled using mathematical equations and analysis. They conduct research to understand the key factors involved, make simplifying assumptions if needed, and develop a system of equations to model the behavior over time. Common examples include modeling population growth, spread of diseases, traffic flow, weather patterns, financial markets, or physical systems. The student would validate the model by comparing its outputs to real data, do sensitivity analyses to study how the outputs change with different input parameters or assumptions, and discuss implications and limitations.

Advanced mathematical proof: The student develops an original proof of a significant open or unproven theorem in their area of mathematical focus. This requires thoroughly researching previous work, identifying gaps, and developing a logical multi-step argument to prove the statement is always true. Areas that could support such proof projects include advanced analysis, algebra, number theory, geometry, topology, or theoretical computer science. The written work must clearly explain all steps and assumptions in the proof.

Data analysis and machine learning project: For this applied mathematics project, the student selects a large, real-world dataset and applies techniques from fields like statistics, data science, machine learning or operations research to analyze patterns and relationships. Common tasks may include data cleaning, feature engineering, model building using techniques like regression, clustering, classification trees or neural networks, model selection, and interpretation of results. The modeling process, findings and limitations would be thoroughly discussed. Data could come from domains like biology, medicine, social sciences, business, engineering or physical sciences.

Graph theory application: The student explores applications of graph theory concepts to solve practical problems. This could involve representing a real network as a graph model, such as transportation, utility, computer or social networks. Analysis may include studies of connectivity, minimum spanning trees, max flow problems, shortest paths, centrality measures or community detection. The project would involve implementing graph algorithms in software and discussing how insights from the mathematical analysis can provide useful understanding or solutions for the target application domain.

Advanced statistical analysis: For data-driven projects, students could perform an in-depth statistical analysis of a real dataset to discover patterns and test hypotheses. This may involve techniques like regression, Bayesian modeling, nonparametric methods, time series analysis, multivariate analyses, graphical models, or advanced experimental design. The written work would include a literature review to contextualize the problem, clearly explaining the methodology, presenting and interpreting results, and discussing limitations and opportunities for future work. The findings would have practical implications.

History of mathematics research: For a more theoretical project, the student research’s the emergence and development of an important mathematical concept, theory or field throughout history. This could trace key contributors, ideas, milestones and evolution over multiple eras and civilizations. The write-up would synthesize information from primary and secondary sources to tell the story of how human understanding evolved. Examples could include number systems, geometry, calculus, group theory, probability/statistics, differential equations or more specialized topics like elliptic curves.

Graduate mathematics capstone projects provide an opportunity for students to conduct an in-depth investigation into an area of individual interest. By choosing topics that apply mathematical theory to solve practical problems or advance human knowledge, students can demonstrate mastery of high-level concepts while contributing new insights. Strong projects involve thorough research, rigorous analytical work, and clear communication of methods and findings. With proper scoping, planning and execution, any of the examples proposed here could serve as the foundation for impressive demonstration of a student’s mathematical skills and abilities at the graduate level.