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title: MA541 Final Exam  
subtitle: Fall 2025
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## What

- The final exam is in written take-home format.
- Graduate students will also present one of their problems/options 
- Undergraduate students are not required to present, but may do so if they wish.
- You may choose from 
    - **Option E** (text exercises), 
    - **Option O** (outline of text sections), or 
    - **Option R** (research in a related topic).
- See below for details. 

## When/Where

- Thursday, December 11th, 10:00 AM - JXJ 1315 


## Options

Choose **one** of the following options to complete for the final exam.  In each option you write up your solutions/notes/report and present during the final exam period.

### Option E - Exercises from the Text

Choose **two** exercises from the list below to complete.  Write up your solutions & present **one** of them during the final exam period.

- Exercises: Act IV 10
- Exercises: Act IV 11
- Exercises: Act IV 12
- Exercises: Act IV 13
- Exercises: Act IV 14

### Option O - Outline of Text Sections

Choose **one** of text sections below to summarize/discuss.  

- Discuss section 29.5.5 forward through section 29.8 in Needham's VDGAF.  
  - More detail expected here, compared to the sections below
  - Add a few examples not found in Needham
- Complete an outline of (as much of as possible) Act V of Needham's VDGAF.
- Complete an outline of Acts III-IV of Needham's VDGAF.


Write up (typeset) detailed notes that you can use to support a short lecture on the subject.  Present your outline during the final exam period.  

- Typesetting 
    - Use LaTeX (or an alternative typesetting system) to prepare your notes.
    - How do I use LaTeX ? One answer - [overleaf](https://www.overleaf.com/) is a good online LaTeX editor.
    - Thompson's "Get started with Overleaf" [tutorial](https://nmuwildcast.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=45cf8a65-521c-44a0-b606-b39a01597549)
    - Thompson's [Latex Template](article_template.tex) for papers with [bibliography example](article_template.bib)
    - Thompson's [Latex Template](exam_template.tex) for exams (for formatting ideas)

### Option R - Research in a Related Topic

Choose a topic related to the course material to research and present.  Prepare a short written report (5-10 pages) and present your findings during the final exam period.

- Same guidelines as Option O for formatting.
- Possible topics include (but are not limited to):
  - The role of curvature in General Relativity
  - Applications of differential geometry in modern physics
  - The history and development of Riemannian geometry
  - Computational methods in differential geometry
  - Connections between differential geometry & topology
  - Implement [Geomstats](https://geomstats.github.io/) and explore a specific manifold or application
- Be sure to include references and citations for any sources you use in your research. 

