Tag Archives: mathematics

On quadratic residue codes and hyperelliptic curves, revisited

Posted on by

I’ve recently been revisiting older mathematical papers with the help of AI, checking proofs, correcting errors, and looking for conclusions that were not apparent when the papers were written. Years ago I worked on symmetric objects in the theory of error-correcting codes and had the opportunity to teach coding theory at the Naval Academy. But I fell in love with a different topic, symmetric objects in graph theory, and stopped thinking about codes as an object of research. I’m learning to fall back in love with those old objects again.

For quadratic residue codes, the relationship with hyperelliptic curves gives a good example of how one topic can connect with a totally different one – the unity of mathematics. The connection comes from curves of the form

X_S:\quad y^2=\prod_{a\in S}(x-a),

where S is a nonempty subset of the finite field \mathbb{F}_p. Remarkably, the number of rational points on these curves is closely related to the weights of codewords in certain binary linear codes, called quasi-quadratic residue (QQR) codes. This means that a problem about constructing good codes can be translated into a problem about counting solutions to polynomial equations.

About 20 years ago I wrote a paper, “On quadratic residue codes and hyperelliptic curves” (published in Discrete Mathematics & Theoretical Computer Science 10 (2008), pdf here) to dive into that topic more. I had more questions than answers. Recall Goppa’s conjecture is that the binary asymptotic Gilbert-Varshamov bound is best possible for any family of binary codes. The asymptotic GV bound states that the rate R=dim(C)/length(C) of any binary linear block code C is greater than or equal to 1-H(\delta), where H(\delta)=\delta-\delta\log_2(\delta)-(1-\delta)\log_2(1-\delta). A refinement of the Bazzi-Mitter conjecture is given below and is still open.

Conjecture. There is a constant c < 2 such that |X_s(\mathbb{F}_p)| \leq cp for infinitely many primes p ≡ ±3 (mod 8).
(a) This is equivalent to: the quaternary quadratic residue codes of length p have minimum distance \geq \delta p for some fixed \delta > 0, for infinitely many such p.
(b) If it holds with c < 2(1 - H^{-1}(1/2)) (which is about 1.779), then Goppa’s conjecture is false.

Now, 20 years later, I’d like to know what AI thinks of the questions and conjectures in that paper. The result is the AI-aided updated and revised version (2026 pdf here), all my questions were answered by Claude but the conjectures remain open.

Conjectural permutation decoding of some AG codes, revisited

In a related paper, “Conjectural permutation decoding of some AG codes” (ACM SIGSAM Bull. 39(1) (2005)26–32, pdf here), I tried to exploit the large automorphism group of the hyperelliptic curve y^2=x^p-x over a finite field to construct good decoding methods for the associated Goppa code. The basic idea is that by permuting the coordinates of a received codeword message, with possible errors in transmission, to permute its error coordinates out of a chosen information set, making decoding simpler.

Open Question: How many errors can a given automorphism group guarantee to correct by permutation decoding, and how does the answer depend on the information set?

The original paper had two conjectures that addressed the above open question. (The conjectures are technical to state but see the original paper for details.) When I asked AI to referee this paper, ChatGPT and Claude showed both conjectures were false and provided specific counterexamples. They also corrected some sloppy writing on my part and the AI-aided revised paper is much better (pdf here). Again, it was a pleasure using one to referee the other while I was able to add or subtract material to improve readability and correctness. For more details, see the github README for that repo.

On the unitary dual of the metaplectic cover of SL(2), revisited

Posted on by

Over 25 years ago, I wrote a paper on the unitary dual of the n-fold metaplectic cover of SL(2,F), over a p-adic field (pdf of original), under some assumptions (such as p>2 and p does not divide n). It was mostly expository and never published, but I referenced it in other works. Although it boasted the overly pretentious title above, it does not give a full description of the unitary dual for arbitrary covering degree and wild residue characteristic.

I asked AI agents Argon-Gemini and Sol-ChatGPT to look it over and suggest corrections, each acting as referees. They found several errors, some serious, but they easily corrected them all. In addition, the paper was revised for readability, adding a literature review and glossary. The AI-aided corrected version is available (pdf here).

Here is an AI-generated summary of the corrected version:

This manuscript studies genuine representations of the n-fold metaplectic cover \widetilde{G} of G=\mathrm{SL}(2,F), where F is a nonarchimedean local field containing the nth roots of unity. The central complication is that the inverse image of the diagonal torus can be nonabelian. This affects its irreducible representations, orbital integrals, and principal-series characters.

Four main results

  1. Representations of the covering torus. Write \widetilde{A} for the covering torus, Z_A for its center, and d=n/\gcd(n,2). In the tame setting, each genuine character \lambda of Z_A determines a unique irreducible genuine representation \chi_\lambda of \widetilde{A}, of dimension d. Its character satisfies \rm{tr}\chi_\lambda(a)=d\lambda(a) for a\in Z_A, and \rm{tr}\chi_\lambda(a)=0 for a\notin Z_A. Thus the trace vanishes away from the center of the covering torus.
  2. Split orbital integrals. The corrected construction uses a subgroup that actually centralizes the element being integrated. For residue characteristic prime to 2n, the normalized split orbital-integral transform maps genuine compactly supported test functions onto the compactly supported, genuine, Weyl-invariant locally constant functions on Z_A.
  3. Principal-series character formula. For \Pi=\rm{Ind}_{\widetilde B}^{\widetilde G}(\chi\otimes\delta_B^{1/2}), the character on regular split elements is \Theta_\Pi(a)=\frac{\rm{tr}\chi(a)+\rm{tr}\chi^{\widetilde w}(a)}{|D(a)|^{1/2}}. The second term involves Weyl conjugation, not naive inversion. This distinction is essential for the character to transform correctly under the central roots of unity.
  4. A restricted character transfer. Suppose \gcd(p,2n)=1 and either n is odd or 4\mid n. A power map \nu(h(t))=(h(t^d),1) gives an ordinary torus character \eta_\lambda=\lambda\circ\nu. For test functions with matching normalized split orbital integrals, the corrected paper proves \rm{tr}\Pi(\chi_\lambda)(\bar f)=\rm{tr}\rm{Ind}_{B}^{G}(\eta_\lambda\delta_B^{1/2})(f). It also establishes the existence of a split-matching function f for each genuine test function \bar f on the cover.

What remains unresolved?

Despite its title, the revised paper does not claim a complete classification of the genuine unitary dual. It surveys known complementary-series and supercuspidal results, but does not establish a uniform complementary-series theorem in arbitrary degree. Nor does the transfer theorem cover n\equiv2\pmod4 (including the double cover), elliptic orbital integrals, or a full correspondence of all admissible representations.

The revision therefore identifies a precise, provable core of the original project while making clear where further work is needed.

“A note on gaps between zeros of L-functions,” revisited

Posted on by

Over 40 years ago I was fascinated by analytic number theory. I think it sprung from being gifted as a teen a copy of Ornstein Ore’s book, Number theory and it’s history. In any case, I was focused on learning as much about the Riemann zeta function \zeta(s) as I could. It was amazing to me then and now how much number-theoretic information a single function like that could encode. The non-trivial zeros \rho=\beta + i\gamma of \zeta(s) are supposedly all on the line \beta=1/2. Using arguments of A. Fujii (On the difference between r consecutive ordinates of the zeros of the Riemann zeta function, Proc. Japan Acad., 1975) and A. Ghosh (On the Riemann zeta function—mean value theorems and the distribution of |S(t)|, J. Number Theory, 1983), I wrote a preprint A note on gaps between zeros of L-functions (pdf here) to understand a conjecture of Hugh Montgomery. Years later I generalized the result from \zeta(s) to any Dirichlet series with several properties analogous those of \zeta(s) (such as an Euler product and so on), using the setup I had in the first book I wrote, Distribution theorems of L-functions (Pitman Research Notes,1986).

I don’t remember if I submitted it and it got rejected, or if I just posted it on my faculty webpage and forgot about it. In any case, it was never published. But, with the presence of AI as a math “paper referee”, I asked chatgpt and claude to be referees. They were asked to read it and suggest corrections, if any, to imprecise mathematical arguments, or revisions for readability. This process when through several revisions, with one or the other acting as the referee for various revisions. This is the AI-aided corrected version (pdf here).

.

Satake homomorphisms for the 2-fold cover of p-adic SL(2), revisited

Posted on by

In very vague, rough terms, the Langlands program gives a way of understanding certain types of maps between two specific types of groups that are related in a particular way. One way I like to think about it is that it gives mathematicians a template for conjectures of such “correspondences” which are likely to be true. Around 20-30 years ago, I was very interested specifically in the metaplectic covers of SL(2,F), where F is a p-adic field. I seems to have roped a number of victims into thinking and talking about such things with me. My colleagues Courtney Moen (who has retired from mathematics) and Jason Levy (who seems to have also left academia) are two examples.

Around 2004, with Jason, I wrote down a form of the correspondence that I thought the Langlands program template gave me as a conjectural “correspondence” between SL(2) over a p-adic field and the 2-fold cover. It’s very technical to write up, yes, but technically complicated and original are completely different things. I want to be clear, the notes/conversations had nothing original – we were writing down this template due to Robert Langlands in the case we wanted to investigate. I vaguely remember knowing that the conjectural guess was false, but it was just a template and I knew I’d have to sculpt it, like an artist, into a correct one. Sadly, my brain did not have the sharpness needed for the job, and I gave up.

That would have ended it forever if not for the facts that (a) I typed up notes of the template conjecture (or my version of how I remembered them) and, (b) now in 2026 20-something years later, AI is being used to prove the quasi-Riemann hypothesis. So, can it fix the false template guess I made long ago?

Thanks to both Opus-Claude and Sol-ChatGPT, the answer is yes. The AI-written paper The Satake Isomorphism, the Fundamental Lemma, and Transfer of Orbital Integrals for the Metaplectic Double Cover of SL(2, F) is the result. Basically, I tag-teamed one AI agent against another, one acting as the researcher and the other as the referee, and I would be adding or subtracting material at each stage to increase readability, preserve topicality, and maintain correctness. This went through 7 rounds of revision and the pdf above is the result.

The real work was done by AI, not Jason, not me. I think the reason it was so much fun working with that AI team in this way was solely because it taught me something new and interesting. I don’t like to give up on a problem. No one does. This helped me finish the job I gave up on in 2004.

Arithmetic of characters of generalized symmetric groups, revisited

Posted on by

Around 25-30 years ago, I was fascinated by the group structure of the Rubik’s cube group (whose size is roughly 4.3×10^19). It turns out (see my book on the Rubik’s cube for details) to be simply described using generalized symmetric groups. As a result, I wanted to know more about their representations and this lead to the question, what is the smallest field over which you can faithfully represent the generalized symmetric group (as matrices)? This lead to some lecture notes in 1998 and then a more refined preprint in 2002. After revisions, this was published in Archiv der Mathematik, vol 81 (2003)113–120.

What did AI think of this paper? While it found no serious errors (the main results and proofs were correct), Opus-Claude found an incorrect statement and that I’d mis-stated a result due to J.P. Serre. So I asked AI to write a more readable corrected version, adding a Glossary, Further directions paragraph, Literature review, and a Summary section. The corrected revision is here (11 page pdf). Hope you enjoy reading it as much as I did writing it!

Zeta functions and their associated operators, revisited

Posted on by

In 1981, as an undergraduate at Georgia Tech, I wrote a typewritten manuscript called “Zeta functions and their associated operators.” I’m not going to post it here. I think only Johan Belinfante (my professor at Georgia Tech) and John Benedetto (my PhD advisor at Univ Maryland) read it, and I hope to keep it that way! The notation is poor, the statements are poorly formulated, and the organization is worse.

But I remember what I was trying to say, and AI (mostly Opus-Claude, but also a lot of chatGPT) has the skills I lacked. This new paper, “A Grosswald–Schnitzer Invariance Theorem for Arithmetic Summation Operators,” is here: pdf.

I have to say, this co-working with AI was a very satisfying experience. I was able to take a paper I wrote as an undergraduate, which I still think has cool ideas, and put it into the form I like. Rereading that paper, I’m frustrated by my lack of skill as a writer and a disciplined thinker. It’s a huge relief to have something to show for all the thinking I did on it back in 1980-1981. The only thing I would not have dreamed of writing back then is (thanks to Opus-Claude) itemized in the “Future Directions” section at the end. Speculations of possible results that could be accomplished by going deeper with some of the ideas. I’ll come back to that later, but I’d also be happy if someone beats me to it.

\begin{center} \renewcommand{\arraystretch}{1.3} \begin{tabular}{@{}>{\raggedright}p{0.44\textwidth}>{\raggedright\arraybackslash}p{0.48\textwidth}@{}} \toprule \textbf{Grosswald--Schnitzer Theorem} & \textbf{Operator analog}\\ \midrule perturbed primes $q_n$ & summation operator $E_q g(x)=\sum_\nu g(\nu x)$\\ the function $\zeta_q(s)$ & the operator $E_q$, via \eqref{eq:bridge}\\ factorization $\zeta_q=\varphi\,\zeta$ & factorization $E_q=\Phi\circ E_p$\\ $\varphi$ holomorphic and zero-free on $\Re s>0$ & $\Phi$ continuous and invertible on $\cS$\\ $\zeta_q$ and $\zeta$ have the same zeros & $\Ran E_q=\Ran E_p$\\ the zeros of $\zeta$ in $\Re s>0$ & the eigenvalues of $\Theta$ on $\cS/\Ran E_p$\\ a zero $\rho$ of multiplicity $m$ & a Jordan block of size $m$ for $\Theta$ at $\rho$\\ $1/\zeta_q=\sum_\nu\mu_q(\nu)\nu^{-s}$ & the M\"obius operator $W_q$, inverse of $E_q$ on $\Ran E_q$\\ the region $\Re s>0$ & the space $\cS$, built over $\Re s>0$\\ \bottomrule \end{tabular} \end{center}

Zeros of some self-reciprocal polynomials, revisited

Posted on by

Inspired by a blog post of Professor Alonso Castillo-Ramirez, I asked AI to look over an old survey paper I wrote in 2012. Long story short, ChatGPT-Sol (OpenAI) and Claude-Fable (Anthropic) found and corrected several errors. Embarrassing. (For the full mea culpa, see the abstract of the pdf of the corrected version.) AI even proved a conjecture I made in the paper and pointed out that another conjecture I made was a consequence of an Annals of Math paper published in 2020 . It’s embarrassing making a mistake, especially in a survey paper (apologies to the original authors!), but there is a rainbow at the end of the road.

On the teaching philosophy of Euphemia Haynes

Posted on by

I’m not an expert on her teaching philosophy but John Stigall is (the author of John Stigall, Euphemia Haynes’ philosophy of mathematics and mathematics education, Journal of Philosophy of Education, 2026). He pointed me to some of Euphemia’s work on the philosophy of teaching. The video below is inspired by, and based on, some of Dr. Haynes’ lectures, research and scholarly work. I could not have made it without AI (mostly Claude).

It’s titled “Euphemia Lofton Haynes – Mathematics as an open door”

Related posts:

The shell algorithm and a new book

Posted on by

A Peek Behind the Curtain

One feature of my forthcoming book with Caroline Melles, Exploring Graphs via Harmonic Morphisms, is its computational emphasis. Many of the examples and insights emerged not from pure contemplation, but from extensive experimentation using SageMath. This post gives a glimpse into the algorithmic machinery that made this exploration possible.

The fundamental problem I faced in generating examples was this: given two graphs Γ₁ and Γ₂, find all harmonic morphisms φ: Γ₂ → Γ₁. This approach is direct—a brute force algorithm that constructs all possible “color lists” (vertex labelings that encode potential morphisms) and tests each one for the harmonic property. Straightforward? Yes. Fast? Definitely not.

However, when the target graph Γ₁ is a path graph, one can do better. The “shell algorithm” exploits the geometry of distance in the source graph. Starting from a chosen vertex v₀, one builds outward through successive neighborhoods—vertices at distance 1, then 2, and so on—assigning colors constrained by distance from the starting point. At each stage, one checks necessary conditions before proceeding, pruning impossible branches early. The automorphism groups of both graphs then generate equivalent colorings, and after removing duplicates, one has a complete enumeration of the harmonic morphisms.

That’s what was done for various choices of small graphs, printing computational data to a file whose name used notation or terminology for the graphs \Gamma_1 and \Gamma_2. Months of computer searches resulted in a number of conjectures that were used to shape the material in the book.

In the works: a book “Exploring Graphs via Harmonic Morphisms”

Posted on by

Caroline Melles and I have been working for some years on a 2-volume book in graph theory which investigates harmonic morphisms. These are, roughly speaking, mappings from one graph to another that preserve locally harmonic functions on these graphs. Therefore, this topic fits into the general framework of harmonic analysis on graphs.

This post only concerns the first volume. The intent here is to mention some of the types of results we obtain. Of course, by no means is it intended to be a complete description.

The second volume will be summarized in a separate post.

Graphs in our book are unweighted and, unless stated otherwise, have no loops or multiple edges. The basic idea is this: in chapter 2 we classify harmonic morphisms using a criteria expressed as a matrix identity. For various graph-theoretical constructions (such as edge deletion or join or a graph product or …) that can be performed on a given graph Gamma, we pick a graph morphism associated to the construction (such as sending a vertex in the constructed graph to the given graph). That morphism is associated to a matrix (which we called the vertex map matrix in chapter 3 of our earlier book, Adventures in Graph Theory). When this matrix satisfies the above-mentioned matrix criteria then the associated morphism is harmonic.

Chapter 1 is on Graph Morphisms.

This chapter is devoted to background on graph morphisms and some of the methods we use to study them.

  1. Roughly speaking, a morphism is a mapping between graphs that preserves incidence structure. After defining horizontal and vertical edges, vertical multiplicities, local horizontal multiplicities, it recalls well-known graph families like cycle graphs, path graphs, and complete graphs.
  2. There are a few very useful degree identities. First, there is a fundamental formula relating vertex degrees to multiplicities under morphisms. There is also a formula for the degree
    of the morphism in terms of vertical multiplicities and local horizontal multiplicities.
  3. A topic threading through the book is that of matrix-theoretic methods. This first chapter introduces vertex map matrices and edge map matrices that encode morphisms. After establishing key matrix identities and products, reviews adjacency matrices and their spectra, with detailed analysis of cycle graph eigenvalues using Chebyshev polynomials and complex roots of unity.
  4. It recalls signed and unsigned incidence matrices, with and without edge orientations, and establishes the fundamental Graph Homomorphism Identity relating incidence matrices to morphism matrices,
  5. introduces Laplacian matrices as differences of degree and adjacency matrices, connecting to the incidence matrix framework.
  6. Introduced graph blowup morphisms via a blowup construction where vertices are replaced by independent sets, creating natural homomorphisms with specific structural properties.
  7. Some functorial properties of graph morphisms are established, such as how morphisms behave under graph constructions like subdivisions, smoothing, deletions, and substitutions.
  8. The chapter ends with exercises and a chapter summary.

Chapter 2 on Harmonic Morphisms

    This chapter is devoted to the basics of harmonic morphisms.
  1. Introduces the core definition: a graph morphism is harmonic if local horizontal multiplicities are constant across edges incident to each vertex’s image.
  2. Cycle space and cocycle space – Develops the algebraic framework using homology and cohomology of graphs. Covers Urakawa’s theorem on pullbacks of harmonic 1-forms and Baker-Norin results on divisors and Jacobians.
  3. Matrix-theoretic methods – Establishes the fundamental matrix characterization: a morphism is harmonic iff there exists a diagonal multiplicity matrix satisfying specific adjacency matrix identities. Proves equivalence with an analogous Laplacian matrix identity and an analogous incidence matrix criteria.
  4. The Riemann-Hurwitz formula – Presents the graph-theoretic analogue relating genera of graphs via harmonic morphisms, with matrix proof and applications to regular graphs.
  5. Some functorial consequences – Demonstrates how harmonic morphisms interact with graph constructions like subdivision, edge substitution, leaf addition, and deletion. Shows these
    operations preserve harmonicity under appropriate conditions.
  6. The chapter ends with exercises and a chapter summary.

All harmonic morphisms from this graph to C4 are covers.

  1. Fundamental Problem: Given a graph Gamma1, for which graphs Gamma2 is there a non-trivial harmonic morphism phi from Gamma2 to Gamma1?
  1. Follow-up question: Can the number of such phi be counted?

Chapter 3 on Counting Problems

This chapter looks at various families, such as the path graphs. What is especially remarkable is that, as we will see, the problem of counting harmonic morphisms often boils down to solving certain recurrance relations, some of which arose (in a completely different context of course) in
the work of medieval mathematicians, both in Europe and in India.

  1. Regarding harmonic morphisms between path graphs, we show how to construct and count the harmonic morphisms from longer path graphs to shorter ones.
  2. Regarding harmonic morphisms between cycle graphs, we show how to construct and count the harmonic morphisms from larger cycle graphs (when they exist) to smaller ones. It turns out all such harmonic morphisms are necessarily covers.
  3. Regarding harmonic morphisms between complete graphs, we show how to construct and count the harmonic morphisms from larger complete graphs (when they exist) to smaller ones.
  4. Harmonic morphisms to P2 (arising from the Baker-Norin Theorem) can be counted.
  5. Harmonic morphisms to P3 (the path graph with only 3 vertices) can be counted in special cases.
    There are lots of open questions, such as which trees have a harmonic morphism to P3.
  6. The chapter ends with exercises and a chapter summary.

Chapter 4 on Harmonic Quotient Morphisms

    This chapter studies quotient graphs arising from group actions and from vertex partitions.
  1. Quotient graphs from group actions. Harmonic actions and transitive actions are studied separately.
  2. Quotient graphs from paritions. Orbit partitions and equitable partitions are studied.
  3. As a nice application of harmonic morphisms with particularly nice structural properties, we consider multicovers and blowup graphs.
  4. The last section provides explicit formulas for the eigenvalue spectra of harmonic blowups of bipartite graphs, connecting the eigenvalues of the source and target graphs through the blowup parameters. The main result is the Godsil-McKay Theorem.
  5. The chapter ends with exercises and a chapter summary.

Chapter 5 on Graph Morphisms and Graph Products

    This chapter studies graph morphisms associated to tensor products of graphs and lexicographical products of graphs.

    Roughly speaking, a graph product of Gamma1 with Gamma2 is a graph Gamma3 = (V3, E3), where V3 = V1 x V2 is the Cartesian product and there is a rule for the edges E3 based on some conditions on the vertices. The graph products considered in this book are the disjunctive, Cartesian, tensor, lexicographic, and the strong products.

    The most basic questions one wants answered are these:
    is the projection pr1 : Gamma1 x Gamma2 to Gamma1 harmonic, and
    is the projection pr2 : Gamma1 x Gamma2 to Gamma2 harmonic?
    If they do turn out to be harmonic morphisms, we also want to know the vertical and horizontal multiplicities as well. If they do not turn out to be harmonic morphisms, we also want (if possible) to establish conditions on the graphs under which the projections are harmonic.

    However, we want to not only consider products of graphs but also products of morphisms.
    In this case, the most basic question one wants answered is this:
    Given harmonic morphisms phi : Gamma2 to Gamma1 and phi’ : Gamma2′ to Gamma1′, is the
    product phi x phi’ harmonic?

  1. For example, we show that projection morphisms from tensor products are always harmonic with explicit horizontal multiplicity formulas.
  2. Moreover, we prove that the tensor product of harmonic morphisms (without vertical edges) yields a harmonic morphism with horizontal multiplicity matrix given by the Kronecker product of the original multiplicity matrices.
  3. If Gamma x Gamma’ is a lexicographical product then the projection onto the first factor, pr1, is a harmonic morphism. However, the projection onto the second factor is not in general.
  4. We establish a connection between the balanced blowup graph and a lexicographical product. One corollary of this connection is that the blowdown graph agrees with the first projection of the product, so is a harmonic morphism.

Chapter 6 on More Products and Constructions

  1. This chapter studies graph morphisms associated to Cartesian/strong/disjunctive products of graphs as well as joins and NEPS graphs.
  2. For example, we show that projection morphisms from Cartesian products or from strong products are always harmonic with explicit horizontal multiplicity formulas.
  3. Roughly speaking, one of the results states:
    Given two m-quasi-multicovers phi from Gamma2 to Gamma1 and phi’ from Gamma2′ to Gamma1′, the Cartesian product phi x phi’ is also an m-quasi-multicover (hence harmonic).
  4. Another result, roughly speaking, states:
    Given two harmonic morphisms phi from Gamma2 to Gamma1 and phi’ from Gamma2′ to Gamma1′, the
    strong product phi x phi’ is also harmonic.
  5. Can one classify the graphs for which the disjunctive product projections pr1 or pr2 are graph morphisms?
  6. For example, we show that if phi from Gamma2 to Gamma1 and phi’ from Gamma2′ to Gamma1′ are graph morphisms, then the associated product map from Gamma2 x Gamma2′ to Gamma1 x Gamma1′ (where x is the disjunctive product) is, in general, not a graph morphism.
  7. Given two harmonic morphisms phi from Gamma2 to Gamma1 and phi’ from Gamma2′ to Gamma1′, the join morphism phi wedge phi’ is harmonic if and only if a certain technical condition is true.
  8. A theorem due to Urakawa states that projection morphisms from a NEPS graph to one
    of its factors are always harmonic. Moreover, we give explicit horizontal multiplicity formulas.
  9. The chapter ends with exercises and a chapter summary.

Computations are supported throughout using SageMath and Mathematica. The plan is the publish the volume with Birkhauser. We thank the editors there, especially John Benedetto, for their encouragement and guidance.