Ask HN: Research topics at the intersection of topology and computer science

[syadegari]

I am interested in research topics at the intersection of topology/geometry and computer science. My previous line of research has been focused on numerical methods with application to mechanical engineering problems (Using methods such as Finite Element and Finite Difference). After working for two and half years in industry and being exposed to new computational problems (often with their roots in computer science), I’ve decided to pursue a master degree that combines my interests in geometry/topology and computer science. Some of the key/important topics of interests that I’ve come across so far have been:

1- A short article with three examples of computational homology https://www.math.upenn.edu/~ghrist/preprints/nieuwarchief.pdf

2- https://www.youtube.com/watch?v=8XJes6XFjxM

3- Computable Topology, the study of “topological and algebraic structure of computation”.

4- Topological data analysis (https://www.youtube.com/watch?v=x3Hl85OBuc0), application of geometrical methods in learning algorithms and in natural language processing/parsing (for the last point though I was not able to find a lot of resources).

I appreciate comments or suggestions regarding the following questions:

1- Research Ideas at the intersection of math (geometry and topology) and computer science. This could be a new topic that I haven’t mentioned here or a refinement/variation of the above ideas.

2- Graduate schools with similar research interests, particularly in the Netherlands and the western part of Germany.

3- Given the interdisciplinary nature of the topics I’ve mentioned, would you recommend a master study in a math department or a computer science/informatics department?

As for me, I have obtained my PhD in Computational Mechanics and I am familiar with some topics in applied math (self study during and after my graduate studies) such as Algebra, Linear Algebra, Analysis, Functional Analysis, Variational Methods and Point-Set Topology.

[officialchicken]

I liked a recent paper by Baltimore Raven's center John Urschel et. al. on "Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians" - that was mentioned on ESPN of all places (of course not the title, it was just "math" to them). Maybe "shortest-path" isn't your geometric interest, but there is definitely an overlap with this material, topology, and requires Comp Sci.

arXiv:1412.0565v1

Edit: add link http://arxiv.org/abs/1412.0565

[syadegari]

Thanks for mentioning the paper, this comes close to one of my topics of interest, spectral graph theory. Although quite an interesting subject (I can see myself easily being drawn to the subject), I wonder if there has been any attempts to understand or study the topological structure at the heart of algorithms that are used for learning or natural language processing/parsing. I am quite curious and eager about these methods so if you happen to know of a paper or reference please let me know.

[GFK_of_xmaspast]

You found that Ghrist article but you didn't find this: https://www.math.upenn.edu/~ghrist/notes.html ?

Also I haven't read it but: http://www.maa.org/press/maa-reviews/topological-signal-proc...

There's also the entire field of 'computational geometry'.