Honors Project in Mathematics
The capstone of the mathematics program, in which students learn how mathematics is actually made by following a complete model project from first question to finished thesis: how an honors project differs from coursework and the expository, computational and research kinds of project, choosing and sizing a question with floor, target and stretch goals, finding a supervisor, writing a proposal and planning the year with a timeline and critical path, reading research papers in three passes and using the arXiv, searching the literature with MathSciNet, zbMATH Open, the Mathematics Subject Classification, the OEIS and citation trails, Pólya's heuristics, experiments and conjectures, with cautionary examples such as the circle-region problem and the Pólya conjecture, developing special cases into theorems, including a complete bijective proof that the number of ways to write n as a sum of consecutive positive integers equals the number of odd divisors of n, computation for exploration, verification and proof, exact arithmetic, certificates, reproducibility and the critical use of AI tools, proof assistants, Lean and mathlib, mathematical writing and the structure of a thesis, typesetting with LaTeX and BibTeX, talks, posters and the oral examination, and a capstone that assembles the model thesis with new chapters on sums of consecutive odd integers, the average number of representations and sums of consecutive squares, with runnable Python code throughout.
MATH-499 Year 4 3 credits 12 lessons · 4 h 46 min ▶ Video lessons coming soon
Offered by HSM University · Mathematics & Statistics
This is the American version of this course. See the Canadian version.
What you'll learn
- Explain how an honors project differs from coursework, distinguish expository, computational and research projects, and describe the roles of student, supervisor and examiners.
- Choose and size a project question using floor, target and stretch versions, approach a supervisor effectively, and plan the year with a proposal, a timeline and a critical path.
- Read mathematical literature efficiently in three passes, read proofs actively, and use the arXiv and reading logs to manage sources.
- Search the literature with MathSciNet, zbMATH Open, the Mathematics Subject Classification, the OEIS and citation trails, and verify every reference against its source.
- Generate and test conjectures with Pólya's heuristics and systematic experiments, and recognize misleading patterns such as the circle-region and Pólya-conjecture examples.
- Develop results from special cases to general theorems, and prove that the number of representations of n as a sum of consecutive positive integers equals the number of odd divisors of n.
- Use computation for exploration, verification and proof, with exact arithmetic, independent checks, certificates and reproducible code, and evaluate AI-generated mathematics critically.
- Describe how proof assistants such as Lean check proofs with a trusted kernel, read simple Lean proofs, and plan the formalization of a result.
- Write mathematics clearly: structure a thesis, state definitions and theorems precisely, write proofs at the right level of detail, and revise systematically.
- Typeset a thesis in LaTeX with theorem environments, cross-references and a BibTeX bibliography, and check documents for consistency automatically.
- Plan, rehearse and deliver a mathematics talk and poster, answer questions honestly, and prepare for an oral examination.
- Carry a project from question to finished thesis, separating classical, new and conjectural material, and verify every quoted result before submission.
Course content
12 lessons · 4 h 46 min total · final exam
- 1 What an honors project is: from coursework to original mathematics 24 min · quiz
- 2 Choosing a topic and a supervisor 25 min · quiz
- 3 Reading mathematics: papers, monographs and the arXiv 24 min · quiz
- 4 Searching the literature: MathSciNet, zbMATH and citation trails 24 min · quiz
- 5 Problem solving and conjecture: experiments, examples and heuristics 22 min · quiz
- 6 Developing results: from special cases to theorems 25 min · quiz
- 7 Computation in research: experimental mathematics and verification 25 min · quiz
- 8 Proof assistants and formal verification (Lean, mathlib) 26 min · quiz
- 9 Mathematical writing: structure, definitions and proofs 23 min · quiz
- 10 Typesetting with LaTeX and managing references 22 min · quiz
- 11 Presenting mathematics: talks, posters and the oral defense 22 min · quiz
- 12 Capstone: a model project from question to written thesis 24 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
The capstone of the mathematics program, in which students learn how mathematics is actually made by following a complete model project from first question to finished thesis: how an honors project differs from coursework and the expository, computational and research kinds of project, choosing and sizing a question with floor, target and stretch goals, finding a supervisor, writing a proposal and planning the year with a timeline and critical path, reading research papers in three passes and using the arXiv, searching the literature with MathSciNet, zbMATH Open, the Mathematics Subject Classification, the OEIS and citation trails, Pólya's heuristics, experiments and conjectures, with cautionary examples such as the circle-region problem and the Pólya conjecture, developing special cases into theorems, including a complete bijective proof that the number of ways to write n as a sum of consecutive positive integers equals the number of odd divisors of n, computation for exploration, verification and proof, exact arithmetic, certificates, reproducibility and the critical use of AI tools, proof assistants, Lean and mathlib, mathematical writing and the structure of a thesis, typesetting with LaTeX and BibTeX, talks, posters and the oral examination, and a capstone that assembles the model thesis with new chapters on sums of consecutive odd integers, the average number of representations and sums of consecutive squares, with runnable Python code throughout.
Part of the Science stream, typically taken in year 4.