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EDIT: I was told by my supervisor to implement the algorithm first and then look back over the question because "biologists' papers do not always contain the information that is necessary to reproduce their calculations". That is to say that I will come back to this question at some point when I've had an epiphany.

I've just started my mathematics thesis on Topological Data Analysis last week and I was given a question to resolve for next week:

"What are the 3-dimensional subspaces to which the projections are made in figure A? Are they really subspaces or are they quotient spaces?"

I am actually not sure how to proceed to resolve this question. I've skimmed over the paper and I didn't spot anything helpful, except for the figure description:

"leads to nine distinct clusters in a two-dimensional state space inferred by generalized regression modelling"

  1. Why do they talk about a two-dimensional state space. The question mentions a three-dimensional space. j.c.: "The three lines you call axes actually arise from the branching analysis performed by the software Monocle 2. Relevant documentation"
  2. Using some method we project an n-dimensional space to a 2 (or 3)-dimensional space. What does he mean by "subspaces/quotient spaces to which the projections are made" How would I check this?
  3. Do I need to find out how the branching analysis or Generalized Regression Modelling works? By G.R.M. do they mean this: Generalized linear model?

It would be great if someone could assist me with this problem, or maybe make the question more understandable.

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