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Please excuse my naive question, but what kind of rôle does the visualization of (especially high-dimensional) data play in mathematical research?

I know, that it plays an important rôle in the analysis and interpretation of data and utilizes advanced mathematics, but, apart from being a servant to mathematical research, are there also examples, where mathematical research is solely dedicated to data-visualisation or, where data-visualisation brought up questions, that initiated important mathematical research?

A prominent example of mathematical research that has been initiated by data visualisation is Guthries observation, that apparently four colors suffice to color every map; hat observation had very fruitful consequences for mathematics.


Questions:

  • what are further examples of problems that originated in data-visualization and became the subject of mathematical research?

  • are there examples of mathematical research, that is dedicated to improving the visualization of data, i.e. to make features of interest, that are "burried" in the data, more "prominent"?

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Manfred Weis
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  • If somebody has an idea to add one or several broader tag it would be useful (being confused by the claim that the 4-color conjecture was based on "data visualization", I have little idea of the scope of the question) – YCor Jan 13 '18 at 17:09
  • The 4 color conjecture came from the task to make countries with common border visually distinguishable; the naive solution would be to assign a different color to each country. If also economical considerations come into play, then one might strive for using as few colors as possible and a natural question is then for the minimal number of colors necessary. As it turned out, that question could not be answered with then existing mathematical theorems and dedicated research was necessary. – Manfred Weis Jan 13 '18 at 18:33
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    I understand that it comes from visualization. I'm rather puzzled by the use of the word "data". Data means something already encoded in some way. If you're asking whether visualization plays a role mathematical research the answer is obviously yes and if you're asking how, this sounds way too broad. – YCor Jan 13 '18 at 18:58
  • @YCor: I concur with your last comment. I also struggled with the scope of the question, but could not resist the temptation to post a constructive response, however imperfect. – Victor Protsak Jan 13 '18 at 19:27
  • @YCor: let me explain, why I consider drawing a political map as visualizing data: a political territory is defined via its boundaries, that are documented in field survey, which yields the original data, that can be interpreted and visualized in various way, depending what kind of information one is interested in. Even if coloring a map doesn't resemble mining or filtering of data, it is still the problem of emphasizing that adjacent territories are e.g. controlled by different political authorities. – Manfred Weis Jan 13 '18 at 20:34
  • Yes, but then any more or less information "can" be made data. So, to avoid discussing further on the meaning of data, let me ask in another way: if the question were termed as: "what are further examples of problems that originated in visualization and became the subject of mathematical research?", what kind of answers would not qualify to your question? – YCor Jan 13 '18 at 20:57
  • @YCor: I would not accept problems that were first observed in the visualization of data (in the broadest sense), that didn't or do initiate mathematical research. Cartography is a good example: there are different requirements for mapping the approximately spherical shape of the earth to the Euclidean plane, but to my knowledge the theory about mappings between surfaces was not inspired by cartography, but rather developed "autonomously" as part of differential geometry. [to be continued] – Manfred Weis Jan 13 '18 at 21:30
  • [continued] what I am looking for, are problems, that were encountered in data-visualisation and for which it is documented, that they were brought to (or caught) the attention of professional mathematicians, which then noticed, that the problem could not be solved with contemporary mathematics, necessitating some research efforts. Merely determing the values of parameters or solving systems of equations wouldn't count as research in that respect. – Manfred Weis Jan 13 '18 at 21:39
  • This sounds very strange. You really believe that geometry of surfaces wasn't initially inspired by visualization? – YCor Jan 13 '18 at 21:48
  • @YCor: that isn't quite what I said; I said that I don't think that cartography initiated investigating the geometry of surfaces; it might well be, that Gauss drew some motivon from his work in field survey and it is certainly true, that painting was major driving force in investigating the (mathematical) laws of perspectivic mappings of all kinds of surfaces. – Manfred Weis Jan 13 '18 at 22:24
  • OK, but if so it also does not qualify to my alternative formulation (without the word data). So I maintain it sounds far too broad for me. – YCor Jan 13 '18 at 22:42

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This is a rather open-ended question, so let me respond by offering some observations in the same spirit, focusing primarily on posets, graphs and simplicial complexes.

In a very general way, graphs and digraphs are used throughout mathematics to represent relations and other structures with metric, topological and enumerative-combinatorial properties, such as the diameter, connectedness and the number of subgraphs with prescribed properties reflecting some of the deeper or less obvious features of the represented structure. (These relations may originate from a mathematical model or be innate to a mathematical theory.) Advanced examples along these lines are provided by the order complexes of posets and Stanley-Reisner complexes of square-free ideals, where the complex both visualizes the underlying structure and brings forth its important features. In a different direction, diagrams in category theory and quivers in algebra, including advanced constructions such as the Auslander-Reiten quiver, are types of visualizations.

Here is a specific illustration already firmly within mathematical setting. The notion of Hasse diagram of a partially ordered set provides an important visualization of this structure that has been used to study its properties. For example, it is easy to see that the diamond lattice $M_3$ and the pentagon lattice $N_5$ are not distributive. In fact, a lattice is distributive if and only if it does not contain a sublattice isomorphic to $M_3$ or $N_5$. On a more rudimentary level, the fundamental concepts such as maximal and minimal elements, chains and maximal chains of a poset can be thought of as visualizations. Some research directions in group and module theory deal with the structure of groups/modules based on their properties of their subgroup/submodule lattice and one can argue that, for example, uniserial modules (where all submodules form a single chain) first became objects of interest because of this visualization aspect. Later, the success of this theory motivated investigating more complicated cases.

Finally, going beyond combinatorial and algebgraic structures, configuration spaces and moduli spaces of all kinds fit into a visualization paradigm: they "visualize" the totality of objects or "states of a system" and can be used to study generic or typical properties, statistics and often dynamics (when the configuration or moduli space arises from geometry).

Victor Protsak
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    I did not touch upon topological data analysis, which is a thing in itself and provides perhaps the most direct instance of visualization of data. – Victor Protsak Jan 13 '18 at 19:18