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was very much concerned with making sure that he was treating me well not just as his student but as a person ebeling noted that berliner led by example more than anything else there was a constant attention to detail and he was always thinking looking out for the next idea that might work berliner retired from carnegie mellon in 1998 his advice to students learn all the substantive knowledge that you can in the final analysis all knowledge hangs together and the more you know the easier it will be to make good decisions in the future learn something that has value something quantitative hopefully have something you can do that someone else will want to pay you for a product if you don t have that it s going to be tough for you hans berliner passed away on january 13 2017 at the age of 87 leaving behind a remarkable legacy as a world chess champion a pioneer in computer game playing and a scientist who was never afraid to challenge conventional wisdom in pursuit of truth main references interview with gardner hendrie on 2005 03 07 oral history of hans berliner chm reference number x3131 2005 computer history museum 2005 j togyer the iconoclast in the link the magazine of the carnegie mellon university school of computer science may 7 2012 posted in chess research software tagged carnegie mellon university chess games hitechm deep blue programming research leave a comment the shell algorithm and a new book posted on 2026 01 06 by wdjoyner 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 γ and γ find all harmonic morphisms φ γ γ 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 γ 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 v 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 and months of computer searches resulted in a number of conjectures that were used to shape the material in the book posted in harmonic morphisms math research sage software teaching mathematics tagged combinatorics math mathematics research sage leave a comment the elevator problem posted on 2025 11 02 by wdjoyner you are on the bottom floor floor 0 lets call it of an apartment building with no basement there are n elevators which we index 1 2 n assume the elevators are on floors f1 f2 where fk 0 is the floor elevator k is currently on 1 k n assume you only like one of the elevators elevator e the way the elevator logic works is this when you press the elevator button one closest to you one on floor number min fk k 0 is told to go to 0 if there is a tie then of those on the same lowest floor the elevator with the smallest index is told to go to 0 move if you press the button and some other elevator than elevator e arrives you can tell it to go to any floor you wish taboo you can press the elevator button if and only if no elevator is moving down goal you want to use elevator e for some fixed e 1 2 n problem is there a finite sequence of moves that allows you to ride in elevator e my plan is to post the answer sometime later but have fun with it posted in math tagged combinatorics math 1 comment in the works a book exploring graphs via harmonic morphisms posted on 2025 10 07 by wdjoyner 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 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 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 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 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 introduces laplacian matrices as differences of degree and adjacency matrices connecting to the incidence matrix framework introduced graph blowup morphisms via a blowup construction where vertices are replaced by independent sets creating natural homomorphisms with specific structural properties some functorial properties of graph morphisms are established such as how morphisms behave under graph constructions like subdivisions smoothing deletions and substitutions the chapter ends with exercises and a chapter summary chapter 2 on harmonic morphisms this chapter is devoted to the basics of harmonic morphisms introduces the core definition a graph morphism is harmonic if local horizontal multiplicities are constant across edges incident to each vertex s image 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 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 the riemann hurwitz formula presents the graph theoretic analogue relating genera of graphs via harmonic morphisms with matrix proof and applications to regular graphs 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 the chapter ends with exercises and a chapter summary all harmonic morphisms from this graph to c4 are covers fundamental problem given a graph gamma1 for which graphs gamma2 is there a non trivial harmonic morphism phi from gamma2 to gamma1 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 regarding harmonic morphisms between path graphs we show how to construct and count the harmonic morphisms from longer path graphs to shorter ones 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 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 harmonic morphisms to p2 arising from the baker norin theorem can be counted 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 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 quotient graphs from group actions harmonic actions and transitive actions are studied separately quotient graphs from paritions orbit partitions and equitable partitions are studied as a nice application of harmonic morphisms with particularly nice structural properties we consider multicovers and blowup graphs 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 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 for example we show that projection morphisms from tensor products are always harmonic with explicit horizontal multiplicity formulas 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 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 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 this chapter studies graph morphisms associated to cartesian strong disjunctive products of graphs as well as joins and neps graphs for example we show that projection morphisms from cartesian products or from strong products are always harmonic with explicit horizontal multiplicity formulas 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 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 can one classify the graphs for which the disjunctive product projections pr1 or pr2 are graph morphisms 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 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 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 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 posted in graph theory group theory harmonic morphisms math research sage tagged combinatorics math mathematics research sage science writing 1 comment the mathematician and the pope posted on 2025 08 21 by wdjoyner acknowledgement this could not have been written without the helpful conversations and correspondences with these brilliant scholars edray goins pomoma john stigall howard nathan alexander howard and susan kelly univ wisconsin retired also i thank the librarians at the catholic university of america for their help i m deeply indebted to them for sharing...