Introduction to Computational Topology

COSC 49.09, Fall 2020

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Projects

Project proposal

The project proposal is due by October 30 (Friday). (The deadline is fixed as we want to give people a chance to read each others' proposals and form groups.) Each student has to submit their project proposal individually. The proposal should be about one or two pages long, with

• a self-contained description of the project goal,
• a short summary of the relevant work, and
• a few words about your plan, including materials you want to read, half-baked ideas and attempts, and/or resources you might need.

I'll collect all the submitted proposals and compile them into one single document. After reading all the proposals, feel free to reach out to other students and form groups to work on the project together. At most three members are allowed in each group. It does not have to be the same as your homework group; and there is no restriction on forming groups between graduate students and undergrads.

Update (Nov 1): Now all the project proposals can be found and downloaded on Canvas.

Final presentation and report

At the end of the term (probably during finals week), we will schedule Zoom meetings for the final presentation. Each group will have 20-30 minutes to present their work. The presentation will be graded towards your final project score, so make sure your talk is polished!

Each group should also submit a final report, summarizing all the work you have done towards the project goal. The report should be submitted after the presentation (to incorporate the feedbacks from the presentation), but no later than December 7 (Monday). The length of the report should be around 10-15 pages (this is a soft limit). You are welcome to include all the failed and partial attempts, conjectures backed up by experimental evidence, future plans, and so on. The report will be graded based on its originality, correctness, creativity, and presentation. Here's a rule of thumb: your project report should be presentable to anyone outside this class with sufficient background knowledge, including your future employers and colleagues. It is understood that you won't be able to resolve an open question completely or finish the painting the way you want it to; after all, one or two months are really short if we are talking about real research. But since you already put in the effort, make sure it is attracting and something that you are proud to stand behind.

Possible topics

The project can be in (potentially more than) one of the following categories:

• Theory research: Trying to tackle some questions raised in the literature or puzzles that nobody has thought about, or developing new tools with provable properties in any subareas of computational topology.
• Experiments: Implementing an existing algorithm, or applying prebuilt libraries on data sets that you personally care about with proper analysis and interpretations.
• Surveys: Summarizing everything that is known on a specific topic, explaining the various approaches to the problem, together with a comprehensive list of citations.
• Creative work: Make a drawing of sculpture or choreograph a dance to convey some topological ideas; publish an scicomm article on your webblog; having a trending twitter feed on topology; do a magic trick that relies on some topological fact; etc, etc.

It is important to choose a project topic that you personally care about; it is not fruitful to spend your time on things just for the sake of satisfying course requirements. Ideally, find a project that you might want to continue to work on even after the term is over. The ultimate goal is to have your result published (at a suitable venue for the project type); this can happen years after the term is over. In such case, I will be more than happy to write a strong and unreserved recommendation letter for your future employment.

You are strongly encouraged to contact me and schedule a meeting to talk about your ideas for the project! Part of my job is to provide guidance and point you to resources that might be helpful. Remember, professors are also human being, although sometimes we don't look like one. I won't bite you; I promise.

Here is a sample list of papers published in the last five years that I find interesting; you can use them as inspiration to find your own research topic. (More to come; stay tune!)

• You Can Find Geodesic Paths in Triangle Meshes by Just Flipping Edges, Nicholas Sharp and Keenan Crane, 2020.
• Origamizer: A practical algorithm for folding any polyhedron, Erik D. Demaine and Tomohiro Tachi, 2017.
• Automatic Machine Knitting of 3D Meshes, Vidya Narayanan, Lea Albaugh, Jessica Hodgins, Stelian Coros, and Jim McCann, 2018.
• Dynamical and Topological Tools for (Modern) Music Analysis, Mattia Giuseppe Bergomi, 2015.
• Road Network Reconstruction from Satellite Images with Machine Learning Supported by Topological Methods, Tamal K. Dey, Jiayuan Wang, and Yusu Wang, 2019.
• On the Structure of Unique Shortest Paths in Graphs, Greg Bodwin, 2018.
• Min-Cost Flow in Unit-Capacity Planar Graphs, Adam Karczmarz and Piotr Sankowski, 2019.
• Minimum-cost integer circulations in given homology classes, Sarah Morell, Ina Seidel, and Stefan Weltge, 2020.
• Fréchet-stable signatures using persistence homology, Donald R. Sheehy, 2018.
• On the structure of Schnyder woods on orientable surfaces, Kolja Knauer, Daniel Gonçalves, and Benjamin Leveque, 2019.
• Even maps, the Colin de Verdière number and representations of graphs, Vojtech Kaluza and Martin Tancer, 2020.
• When and Why the Topological Coverage Criterion Works, Nicholas J. Cavanna, Kirk P. Gardner and Donald R. Sheehy, 2017.