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Hermann Weyl was one of the pioneers in the use of early algebraic/combinatorial topological methods in the problem of electrical currents on graphs and combinatorial complexes. The main article is "Repartición de Corriente en una red conductora", which appeared, written in Spanish, in 1923 in Revista Matemática Hispano-Americana:

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together with other less known papers.

Was there a particular cooperation around this topic between Hermann Weyl and the Spanish-speaking mathematical community? Did these articles just disappear from the mainstream of mathematical research?

I am aware of papers of Beno Eckmann from the 40's (Harmonische Funktionen und Randwertaufgaben auf einem Komplex).

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  • $\begingroup$ 'john mangual' answered the question already, in particular answered the first of the ?-postended sentences. The following mere 'data points' seem appropriate to add; they give some sort of weight to answering the second ?-postended sentence with 'Yes, they 'just' disappeared from the mainstream of mathematical research.' if the question is strictly construed as asking for this specific paper being cited often (needless to say: [...] $\endgroup$ – Peter Heinig Oct 3 '17 at 7:18
  • $\begingroup$ [...] the definition that Weyl is investigating in the 1923 paper is the standard 1-dimensional cycle group of a simplicial complex, which in model-dependent and 'metrical' considerations is better called the lattice of integral flows on the graph), and, as such, implicitly is cited countlessly often, ust not explicitly with reference to this Spanish paper; end.of.the.'needless to say bracket); now the data points: Weyl's 1923 paper does (0) not figure in [History of Topology. Elsever 1999], (1) [...] $\endgroup$ – Peter Heinig Oct 3 '17 at 7:38
  • $\begingroup$ [...] does not figure in [Jean Dieudonné: A History of Algebraic and Differential Topology, 1900 - 1960. Birkhäuser], (2) does not figure in Erhard Scholz: Hausdorffs Blick auf die entstehende algebraische Topologie, the latter two authors being professional historians of mathematics, (3) does not figure in [P. Hilton: A Brief, Subjective History of Homology and Homotopy Theory in This Century. Mathematics Magazine 1988], which is unsurprising in view of the title of that paper; however, notably, [...] $\endgroup$ – Peter Heinig Oct 3 '17 at 7:48
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    $\begingroup$ I find them hard to read as the font size is small. I also find in this case that it alters the character of the question. If the original poster approves, then keep the footnotes, but they seem more appropriate to comments or as part of an answer, not as additions to the question. I think it more appropriate if you copied the question text into an answer post and footnoted the copy as part of your response. Gerhard "Can Just Read The Comments" Paseman, 2017.10.03. $\endgroup$ – Gerhard Paseman Oct 3 '17 at 14:38
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    $\begingroup$ @GerhardPaseman: many thanks for the considered feedback. I agree, and I am sorry for the excessive editing of the OP. Also apologies to the opening poster; while using stronger words than 'distortion' wouldn't be fair, I would call my well-meant edits a 'distortion' of the OP. I'll proactively move the footnotes to my footnote-like 'answer' in this thread, while leaving the correction kindly suggested by F Zaldivar in place. This seems the right compromise. If the opening poster prefers any other change, either moving a foonote back into the OP, or taking out the picture, I'll do that too. $\endgroup$ – Peter Heinig Oct 4 '17 at 17:31
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The Spanish paper of Weyl is on John Baez' page. There's a brand new subject called Analysis Situs it's not even called "Topology" yet, and Weyl discusses what he could absorb from lectures at ETH Zurich in five years before, in 1918.

Here's some info from the Math Biography website on Weyl:

As a privatdozent at Göttingen, Weyl had been influenced by Edmund Husserl who held the chair of philosophy there from 1901 to 1916. Weyl married Helene Joseph, who had been a student of Husserl, in 1913; they had two sons. Helene, who came from a Jewish background, was a philosopher who was working as a translator of Spanish.

Not only did Weyl and his wife share an interest in philosophy, but they shared a real talent for languages. Language for Weyl held a special importance. He not only wrote beautifully in German, but later he wrote stunning English prose...

Looks like Hermann Weyl did not speak Spanish, but his wife did a translation.


This discussion of electrical networks and combinatorial topology is available in many places today in English. In Random Walks and Electric Networks Peter Doyle and J. Laurie Snell solve Markov chains with electrical networks.

These days we'd just say there is a planar graph $G = (V, E)$ and the currents are a functions $f: E \to \mathbb{R}$ satisfying a "conservation law" at each of the vertices $\sum_{v'\sim v} f(vv') = 0$. We could be even more abstract and say these are the elements of some chain complex: $V = C_0(X)$ and $E = C_1(X)$ and $f \in H^1(X)$.

The first inkling of the modern treatment could be by Yves Colin de Verdière:

Y. Colin de Verdière, Réseaux électriques planaires I, Comment. Math. Helv. 69 (1994), 351-374. Also available here.

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    $\begingroup$ Minor comments: the part of the answer above the 'line' is very relevant to the OP. The part below the 'line' are only tangentially relevant to the actual questions "Was there a particular cooperation around this topic between Hermann Weyl and the Spanish-speaking mathematical community? Did these articles just disappear from the mainstream of mathematical research?" in the OP. One can argue, though, that mentioning the book by Doyle--Snell is relevant in that it does not mention Weyl, so this is one data point supporting the implicit 'thesis' that "these articles just disappear[ed]". $\endgroup$ – Peter Heinig Oct 2 '17 at 20:15
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    $\begingroup$ The rendition of what "we" would "say" "These days" is wrong. (Though it seems best to simply leave it at that, since this way it is more informative for readers, and it is easily clarified: what you define are exactly (including the, unnecessary, choice of the transcendental notion $\mathbb{R}$ for the ring-of-coefficients) the 'zero-sum flows' in the sense of e.g. [S. Akbari, N. Ghareghani,G.B. Khosrovshahi,A. Mahmoody, On zero-sum 6-flows of graphs.Linear Algebra and its Applications 430 (2009) 3047–3052]. This is a notion very different from the one in Weyl's Spanish paper. $\endgroup$ – Peter Heinig Oct 3 '17 at 9:01
  • $\begingroup$ These 'zero-sum flows' are interesting, and I have no quarrel with them, but it is fair to say that they are currently irrelevant to topology. The strongest reason for this is that the class-function $\mathsf{UndirectedGraphs}\xrightarrow[]{\mathrm{ZeroSumFlows}}\mathsf{FreeAbelianGroups}$ is not invariant under homotopy-equivalences of objects of its domain: to see this, it suffices to consider the graph-theoretic 4-circuit $C^4$ (which is a 1-dimensional simpicial complex): evidently, its group of 'currents in your sense'(=zero-sum flows in the sense of Akbari et al) is [...] $\endgroup$ – Peter Heinig Oct 3 '17 at 9:06
  • $\begingroup$ [...] isomorphic to the free abelian group given by any standard model of $\mathbb{R}$ with the standard addition, aka $(\mathbb{R},+)$. But $C^4$ is homotopy-equivalent to the 3-circuit $K^3$ (just contract any edge), and the 3-circuit has trivial zero-sum group, which is obvious (for your conservation condition to hold, you'll have to have equal-magnitude-but-opposite-sign-values at one vertex, but then the conservation-conditions at the remaining two vertices cannot possibly be satisfied-except by the zero-(zero-sum-flow)). So a homotopy-equivalence has changed the isomorphism-type. $\endgroup$ – Peter Heinig Oct 3 '17 at 9:11
  • $\begingroup$ Again, personally I think one should simply leave it at that. The OP was asking a historical question, and was not asking for a discussion of simplicial homology (or deceptively similar-looking other definitions). $\endgroup$ – Peter Heinig Oct 3 '17 at 9:14
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It turns out that Beno Eckmann actually asked Hermann Weyl why he published this work in Spanish, as described here and here. The answer is remarkable and unexpected (and apparently does not involve his Spanish spouse).

I quote from the latter source (RK=Robert Kotiuga, BE=Beno Eckmann):

RK: In the hallway outside our offices was a high-tech espresso machine and every morning Beno would take a break to sit and enjoy an espresso outside our offices. The first day, I "coincidentally" joined him and he related wonderful anecdotes from 1950-1955, after Hermann Weyl retired from the IAS, resettled in Zürich, and frequented the department. The next day I resolved to ask Beno a question which I didn't think any living person could answer.

RK: Beno, there is something I really don't understand about Hermann Weyl.
BE: What is it?
RK: Well, in his collected works, there are are two papers about electrical circuit theory and topology dating from 1922/3. They are written in Spanish and published in an obscure Mexican mathematics journal. They are also the only papers he ever wrote in Spanish, the only papers published in a relatively obscure place, and just about the only expository papers he ever wrote on algebraic topology. It would seem that he didn't want his colleagues to read these papers.
BE: Exactly!
RK: What do you mean?
BE: Because topology was not respectable!

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    $\begingroup$ Just two minor corrections, to the answer and to the OP question: The journal where Weyl published this paper is "Revista Matemática Hispano-Americana" (as can be read after the title of the paper in the link provided by the OP), a precursor of the "Revista Matemática Iberoamericana". Both journals are from mathematical societies of Spain, and the second one is now published by the EMS. $\endgroup$ – F Zaldivar Oct 3 '17 at 0:46
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    $\begingroup$ This is not a convincing explanation, since Weyl discussed Analysis Situs in 1919 in Raum Zeit Materie. $\endgroup$ – Matt F. Oct 3 '17 at 13:55
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    $\begingroup$ ... but if this is the explanation Weyl gave to Eckmann, it was at least convincing to himself.... $\endgroup$ – Carlo Beenakker Oct 3 '17 at 15:25
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    $\begingroup$ Just to clarify, there is no need to go through Kotiuga, Beno Eckmann reported himself on this conversation with Weyl in the first of the two sources I linked to, on page 9: I asked him the above question [Why did you publish your two 1923/1924 papers on Algebraic Topology in Spanish] in 1954 [..]. Hermann Weyl answered that he simply did not want to draw attention to those two publications, the colleagues should not read them! $\endgroup$ – Carlo Beenakker Oct 4 '17 at 6:10
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This is a relevant comment on an answer given in this thread, which the comment boxes cannot conveniently accommodate.

The summary of the 'current' given after "These days" in an answer in this thread seems technically wrong (to me): if by 'planar graphs' is meant (which is quite usual) 'planar undirected simple finite graph-theoretic graph', then this is not how 'currents' are most usually formalized these days: your definition of the 'conservation condition as ' $\forall v\ \sum_{v'\sim v} f(vv')=0$ ', with $f$ having been defined as a function on a set of unordered 2-sets is precisely the central definition of the field of zero-sum flows, which is a, admittedly deceptively similar-looking-to-the-casual-observer, interesting niche-subject mostly practiced in Central Asia; but this is not what Weyl was investigating.

The free-abelian-group-that-is-modelled-by-said-zero-sum-flows is also (as yet) rather irrelevant to topology, as it still 'awaits its categorification' (this might be a gap in my knowledge, but to my knowledge the class function $\mathsf{Graphs}\xrightarrow[]{\mathrm{ZeroSumFlowGroup}}\mathsf{FreeAbelianGroups}$ is is-not/has-not-emphasized-to-be a functor for any noteworthy category of graphs (though I don't doubt it can, rather easily, I am simply trying to substantiate that this definition is rather irrelevant to the OP proper). The main point is that with the definition in the answer, the edges are not oriented, so a natural interpretation of a direction in which the quantity flows is lacking, the possibility of having negative values on an edge non-withstanding. Again: what you defined are 0-sum-flows. To add context and a reference, let me mention that a recommendable, light-on-graph-theory-focused-on-the-linear-algebra relevant article is S. Akbari, M. Kano, S. Zare: A generalization of 0-sum flows in graphs. Linear Algebra and its Applications 438 (2013) 3629–3634, from whose abstract I take the following excerpt:

about which one should note that (0) calling an arbitrary function from the (unordered-edge)-set to the group-of-coefficients is a bit unusual, (1) the heavily-yellowed equation is essentially exactly your definition of the 'currents'. This is quite a different theory from classical simplicial homology/currents-à-la-Weyl's-Spanish-paper, and rather irrelevant to the OP's question. Please consider the crucial part of Weyl's 1923 paper:

whose first sentence I now translate, for convenience of readers:

To address the problem of the distribution of current, we assume that each segment is given a sense of traversal [i.e.: Weyl unambiguously does first orient the lines, only later defines a function thereon; this is exactly good old simplicial homology, not zero-sum-flows]

It will be necessary for me to interrupt now, but let me summarize that

  • currently the mathematical part of the answer in this thread which starts with 'The Spanish paper of Weyl' is at least a little misleading, since the crucial step in the formalization, usual since the 1920s, to first orient the line segment, and only then define a function on the oriented segments, is not sufficiently emphasized. I am not sure what the best correction is. Maybe this extended footnote is more informative than a slick corrected version. Finally, to my mind the most usual treatment of simplicial homology is of course via exterior algebra: the boundary operator in question is, needless to say, the usual derivation $\partial_2\colon\bigwedge^2\bigoplus_{v\in V}\mathbb{Z}v\longrightarrow\bigoplus_{v\in V}\mathbb{Z}v$, where $\mathbb{Z}v$ denotes the free $\mathbb{Z}$-module (=abelian group) on the set $\{v\}$.

I am aware that Baez does it, and does it correctly, with quivers, and hence has 'orientations built-in'.

I am also aware that perhaps you simply intended your $G=(V,E)$ to denote a quiver; my little-point-to-be-made is only that this distinction will be lost on the majority of readers, and perhaps also be lost on the OP. The orientation-issue is underemphasized currently. (Whether it should be emphasized I am not sure, as the OP is asking quite a different thing from being served with a summary of simplicial homology. The OP is focusing on historical questions.)

Will not be able to respond until tomorrow.

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