Hans Berliner – a life built around chess

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Lacking a suitable reference, here is a short bio of this fascinating researcher compiled by claude.

Early Life and Emigration from Germany

Hans Berliner was born in Berlin, Germany in 1929, into a family with a remarkable technological heritage. His great-uncle, Emile Berliner, was the inventor of the gramophone and held the patent on the disc recording format—a technology that would eventually prove superior to Thomas Edison’s cylindrical records and become the ancestor of all subsequent disc-based media formats, including hard drives and modern optical discs. The Berliner family had been involved in telephone technology at the turn of the century, with Emile owning the patent on the carbon receiver for the telephone and establishing a telephone company in Hanover, Germany.

Young Hans entered the German public school system just as Adolf Hitler was rising to power. The first hour of each school day was devoted to “religion” (meaning Christianity) and National Socialism—activities in which Berliner was not permitted to participate. He was also excluded from joining his friends in the Hitler Youth. “I was told that I was Jewish, and they didn’t want me,” Berliner later recalled. “That was quite a shock, and I guess that’s one of those things that sort of grows you up a little bit.”

Despite the oppressive political atmosphere, Germany in the 1930s remained a stimulating environment for a child interested in science. Berliner remembered it as “probably the best place in the world” for such interests. “The Germans were full of inventiveness and managed to produce things that were very, very good,” he recalled. He particularly remembered a metal wind-up toy car equipped with a working servomechanism that could sense when it was about to run off a ledge and automatically steer away—a sophisticated device for a child’s toy in 1935. German kindergarteners were reportedly “three years ahead” of their American counterparts in mathematics, and these formative years had a lasting positive effect on Berliner’s intellectual development.

The family’s fortunes changed dramatically in 1936 when two visitors from the United States came to stay with the Berliners. Seven-year-old Hans soon learned that the family would be leaving Germany. A nephew of Uncle Emile, Joseph Sanders, had arranged for approximately ten members of the extended family to emigrate to America. In 1937, the Berliner family arrived in the Washington, D.C. area, speaking very little English and with thick German accents.

Berliner’s father held a master’s degree in electrical engineering, while his mother had essentially no education beyond high school. Berliner would later describe his home environment as not particularly intellectual, with his parents “more just interested in surviving.” However, one of the most formative experiences of his life occurred before he was three years old, when his grandmother gave him a chalkboard with letters and numbers around the edges. He began making words and doing sums, and from that point on, learning came naturally. “I just wanted to know everything about everything,” he recalled.

Education and the Discovery of Chess

Berliner doggedly pursued his studies at Henry D. Cooke Elementary School in Washington, eventually graduating with the top grammar marks in his class. One of his fellow students was Carlos Fuentes, who would become a famous Mexican novelist and essayist. Fuentes vividly remembered Berliner as the “extremely brilliant boy” with “deep-set, bright eyes—a brilliant mathematical mind—and an air of displaced courtesy that infuriated the popular, regular, feisty, knickered, provincial, Depression-era sons-of-bitches.”

At age 11 and 12, Berliner read every book in the adult section of the library about astronomy, developing what he felt was phenomenal knowledge of the subject. He even reached the point where he recognized that certain theories about the formation of the solar system were patently wrong because they violated physical principles he understood. However, there was no outlet for such precocious knowledge—no school could handle a 12-year-old with advanced expertise, and no special programs existed for gifted children at that time.

Into this intellectual vacuum came chess. At age 13, Berliner was at a summer camp when rain forced everyone indoors. He noticed other children playing a game on a board that wasn’t checkers. He asked about the rules, learned them, and by the end of the day was already beating one of the other players. “Chess seemed a natural,” he recalled. “In a way, I sometimes thought maybe that was not the best path to go, but that was the one that presented itself.” Chess offered something that astronomy could not: a domain where a young person could immediately compete in the adult world.

Berliner progressed rapidly in chess, becoming a master by 1949 or 1950. He would later represent the United States at the 10th Chess Olympiad in Helsinki, Finland in 1952, where he met his future wife, a Finnish woman whom he married in 1954.

Academic and Professional Struggles

After high school, Berliner entered George Washington University to pursue a degree in physics with ambitions of becoming a top-notch physicist. However, he had no understanding of the academic path required—the need for a Ph.D. and years of study. He was one of the top physics students, but he grew bored with the program and began “majoring in bridge” rather than attending classes. Before long, his grades had plummeted, and the draft beckoned. He entered the U.S. Army in early 1951 and served with the German occupation forces. Throughout his tour of duty, Berliner continued to play chess, including one exhibition where he simultaneously played eight games against one of the top German teams—and won them all.

Upon his return to civilian life, Berliner was reluctant to return to college. It was Isador Turover, a fellow European immigrant and former top U.S. chess player who had become a successful lumber magnate in Washington, who set Berliner straight. Turover, whom Berliner had met through chess circles, told the younger man in no uncertain terms, “You will finish your degree.” He hired Berliner into his lumber company so he could earn money to pay his way through college, since there was no family money for education and Berliner had never secured a scholarship.

Unable to return to physics due to his low grade point average, Berliner switched to psychology, naively believing that with a psychology degree he could “hang out a shingle as a psychologist and start counseling people.” Again fate intervened. A classmate who worked at the Naval Research Laboratory told Berliner, “We need people like you where I work.” This led to a position in what was then called “human engineering” or “engineering psychology”—a predecessor to modern interface design and human factors research.

Early Career and First Encounters with Computers

Starting at the Naval Research Laboratory in 1954, Berliner worked on human engineering problems, such as ensuring that Navy pilots didn’t pull the wrong lever and eject themselves when trying to lower the landing gear. During this period, he had his first brush with computers. A colleague at the lab was helping to build NRL’s own computer—one of very few computers in existence at that time—and suggested they might write a chess program together. They talked about it and even did a few preliminary things, but the project never got off the ground. Berliner had read the foundational articles by Claude Shannon and by Newell and Simon about computer chess in Scientific American, but he “didn’t have the foggiest notion how to implement it.”

From the Naval Research Lab, Berliner moved to Martin Company in Denver, then to General Electric’s Missile and Space Vehicle Division in Valley Forge, Pennsylvania—which he described as “one of the finest places I’ve ever operated,” with more talented people than perhaps even Carnegie Mellon—and finally to IBM in Bethesda, Maryland. Throughout these moves, his specialty remained human factors work, which he did not consider particularly fulfilling. “My pay was skyrocketing, but I had an awful lot of spare time,” he later recalled.

Rise to Chess Prominence

hroughout the 1950s, Berliner’s chess ranking continued to climb. He became a Senior Master with a rating in the 2420-2440 range, placing him among the top ten players in the United States. He played regularly in the U.S. Invitational Championship, and in the year Bobby Fischer won the title for the first time, Berliner finished fifth and drew with Fischer.

His reputation became especially strong in correspondence chess—games played through the mail—where his systematic approach and analytical mind could shine without the time pressure of tournament play. From 1965 to 1968, Berliner served as World Correspondence Chess Champion, winning his first championship with a remarkable record of 12 wins and 4 draws in 16 games, a margin of victory three times better than any previous champion. He remained the top-ranked U.S. correspondence chess player until 2005, long after he stopped competing.

Murray Campbell, who would later work on IBM’s Deep Blue, observed that Berliner wasn’t a “star” player in the mold of Bobby Fischer or Garry Kasparov, but achieved chess greatness “using a very systematic approach and a lot of hard work.” Berliner “had competitive fire” and an analytical mind that enabled him to beat players “who might very well have been more talented than him.”

Carnegie Mellon and Computer Chess Research

While at IBM in the late 1960s, Berliner began thinking seriously about computer chess. He had never written a program in his life, so he had to learn programming while developing his first chess program. Called “J. Biit” (an acronym for “Just Because It Is There”), it was written in PL/1 and played in the first U.S. Computer Chess Championship in 1970, finishing around the middle of the field.

In 1967, Berliner met future Nobel laureate Herbert Simon at a technical meeting. Simon, who in 1956 had famously predicted that within ten years a computer would become world chess champion, was still interested in chess-playing computers. He offered Berliner a job at Carnegie Mellon University, but Berliner turned down the position of researcher. “If I’m going to come there, you’ve got to put me on the student track,” he insisted. At nearly 40 years old, feeling that he wanted to do “something worthwhile” with his life, Berliner was accepted into CMU’s Computer Science Department as a doctoral student in 1969.

Berliner found CMU to be a “good” but “chaotic” environment, with the department “up on wobbly feet.” The founding department head, Alan Perlis, was “an amazing, wonderful person” with “a desire for progress and truth that was very, very commendable.” Newell allowed Berliner to begin his thesis work on computer chess even before completing the rigorous qualifying examinations.

For his doctoral dissertation, titled “Chess as Problem Solving,” Berliner developed a chess program called CAPS that incorporated what he called the “causality facility.” This mechanism dragged back descriptions of tactical events through the search tree, allowing the program to reason about why certain moves failed and what characteristics a move would need to prevent similar failures. The program could figure out how to block threats without trying every possibility, simply by reasoning about what was required.

However, even as Berliner refined CAPS, he became convinced that rule-based approaches had fundamental limitations. “I had a set of rules that were limited,” he explained. “They were the most important things—maybe 80 percent—but that’s nothing. The other 20 percent includes the things the top players know how to do. That’s why they’re the top players.” Newell and Simon kept encouraging him to create more rules, but Berliner recognized that something crucial was missing: the intuitive understanding that allowed grandmasters to evaluate positions at a glance.

The B* Algorithm and Backgammon

While searching for new research directions, Berliner learned backgammon from his father-in-law and decided to write a program for this simpler game. His initial attempts encountered a familiar problem: the program would reach a certain point and then bog down, “trying to optimize things that it should have forgotten about.” At some point, when a player was clearly winning, the strategy should change—the program needed to recognize when transitions were coming.

Berliner hit upon the idea of using fuzzy logic—still a novel concept in the 1970s—to assign different rules “weights” or “application factors” based on their relative importance at each stage of the game. The resulting program, called BKG, began winning games it would have previously lost. In July 1979, BKG became the first computer program to defeat a reigning world champion in any board game when it beat backgammon champion Luigi Villa.

One of the highlights of Berliner’s research was the B* (“B-star”) algorithm, designed to emulate what he called the “jumping around” process in human thought. Rather than using traditional best-first or depth-first searches with arbitrary termination criteria, B* assigned an “optimistic” and a “pessimistic” score to each node. The algorithm continued searching a branch as long as the pessimistic value of the best node was no worse than the optimistic value of its sibling nodes. B* found paths that were sufficient for a task rather than theoretically “perfect” ones—emulating how a human chess master stops searching when finding a move that seems clearly best.

As Andy Palay, one of Berliner’s students, observed, B* represented “a much more directed search toward what appear to be the most promising paths” compared to simple best-first searches.

HiTech: A Chess-Playing Machine

In 1983, Berliner embarked on what he would later describe as one of the best periods of his professional life. His student Carl Ebeling proposed building a chess-playing machine using the then-novel technology of very-large-scale integrated (VLSI) circuits. Ebeling custom-designed a processor to generate chess moves, and the resulting machine, named HiTech, used 64 of these processors—one for each square of the chessboard—operating in parallel.

The enthusiasm for HiTech was extraordinary. “Everyone wanted to know what the latest developments were, and if they could help,” Berliner recalled. Volunteers from inside and outside CMU’s Computer Science Department took turns wire-wrapping connections in a third-floor lab in Wean Hall. With a hardware budget of only about five thousand dollars, the team scavenged parts and bought 32K memory chips at twenty or thirty dollars each.

A working prototype was completed in 1984. HiTech could consider 175,000 positions per second—compared to the one or two moves per second a top human player might examine. The machine also incorporated pattern recognizers that Ebeling designed to identify specific positional features, such as trapped bishops, pawn structures, and king safety issues. These pattern recognizers “sat like gargoyles on the edge of the thing watching the positions go by,” detecting problems and opportunities that pure calculation might miss.

“From the very beginning, I could see that it had the potential—maybe not every single time—it had the potential to play better than any device that existed,” Berliner recalled. The team worked hundred-hour weeks, with Monday morning meetings to discuss progress and problems. “Every week it got better. Every week the machine played better than it did the week before; now that’s incredible, that’s truly amazing.”

In October 1985, HiTech won Pittsburgh’s Gateway Open chess competition, earning the rank of “master.” That same year it won the ACM tournament for chess programs. The machine went on to win the Pennsylvania State Championship three consecutive years—something Berliner himself had never achieved as a human player. By 1987, HiTech was ranked 190th in the United States and was the only computer among the top 1,000 players, with a rating around 2440—Senior Master level.

One particularly memorable achievement was a four-game match against grandmaster Arnold Denker, which HiTech won 3.5 to 0.5. “It outplayed him in every game, in every single game. It completely beat him,” Berliner recalled.

Rivalry and the Road to Deep Blue

While Berliner’s team was developing HiTech, fellow CMU graduate students Feng-hsiung Hsu and Thomas Anantharaman began work on another chess-playing computer called ChipTest. Like HiTech, it relied on VLSI technology, but it was significantly faster—by 1987, ChipTest was searching 500,000 moves per second.

Daniel Sleator, a CMU professor who had founded the Internet Chess Club, noted that having two competing chess systems developed simultaneously at CMU “reflects a number of important things about the culture in the Computer Science Department. For one thing, there is a tremendous amount of respect for the work of graduate students. The faculty gives them the benefit of the doubt, and in many cases, including this one, it pays off.”

However, the relationship between the teams deteriorated over time. “There was some tension that never got resolved, and there were some hard feelings in terms of the competition between the two groups,” Campbell acknowledged. ChipTest evolved into Deep Thought, which won the World Computer Chess Championship in 1989. IBM subsequently hired Hsu, Campbell, and Anantharaman, and the machine that became Deep Blue eventually defeated world champion Garry Kasparov in 1997.

Legacy and Perspective

In retrospect, Berliner was characteristically blunt about the field he helped pioneer. Computer chess was “a research dead-end” as far as artificial intelligence was concerned, he concluded. “The whole AI thesis was wrong. AI researchers thought more knowledge would do everything.” Instead, more powerful processors and machine-learning techniques powered by statistical analysis, rather than human-devised rules, proved capable of cracking data-intensive problems in speech, image analysis, and data retrieval.

However, those who studied Berliner’s work disagreed with his self-assessment. “He covered a lot of ground, and he achieved excellence in all of those areas,” said Campbell. Jonathan Schaeffer, a professor at the University of Alberta who led the team that created the unbeatable Chinook checkers program, noted that Berliner “has a legacy of excellent papers that contain insights, algorithms and new ideas that aren’t as common today as they should be. People continue to reference his work when they realize there are other ways to do things, and then they point at Hans.”

Schaeffer observed that “most scientists aren’t willing to take the kinds of risks that Hans would take. But that’s why his papers are still around, while the papers of his contemporaries are long gone and forgotten.”

Berliner’s willingness to question conventional wisdom led to controversy, including his accusation that Russian computer scientist and chess grandmaster Mikhail Botvinnik had committed fraud by manipulating published results. Schaeffer and others reviewed Berliner’s evidence and concluded that Botvinnik had indeed massaged his results. Similarly, Berliner’s 1999 book The System: A World Champion’s Approach to Chess attracted sharp criticism from some reviewers, but the attacks left him unbowed.

His former students remembered not just his demanding standards but his generosity. “Working with Hans was a lot of fun,” recalled Palay. “There was a great deal of graciousness, both on a personal level and a professional level. He 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.

The shell algorithm and a new book

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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.

The Elevator Problem

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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!

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

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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.

The mathematician and the Pope

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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 their knowledge about Haynes’ life and work and philosophy, but if there are mistakes, and I’m sure there are, they are my responsibility alone.

This is a non-technical (I promise!) introduction to the life and work of Euphemia Haynes. She’s a fascinating character known not just for her pure mathematics PhD thesis earned at the age of 53, but for her extraordinary devotion to improving mathematics education for everyone, as well as her service to many charities, especially those related to the Catholic Church.

At the time, the prestigious Papal Decoration of Honor medal, the Pro Ecclesia et Pontifice, was the top award for non-clergy (in particular, all women) bestowed by the Pope. That papal recognition was given to Haynes by Pope John XXIII, when she was almost 70. In fact, hers was the only Pro Ecclesia et Pontifice medal bestowed by the Pope to anyone during his entire tenure.

This is an introduction to her life’s journey.

Upbringing

Born Martha Euphemia Lofton in Washington D.C. on September 11, 1890, Euphemia preferred using her middle name.

Her father, Dr William Lofton, was a dentist while her mother Lavinia was very active in her church and later became an elementary school teacher in the D.C. school district. According to saved correspondence, the family lived on 17th Street and attended, until Euphemia was in her 20s, St Augustine’s which was a few blocks away. Lavinia and Euphemia and Joseph were part of the church choir for many years. Indeed, Lavinia was the organist for the junior choir since Euphemia was a baby.

This familial foundation within the Catholic community likely instilled in Euphemia her values of service, justice, and community engagement.

Education

Lavinia Lofton started teaching in the DC public schools in the fall of 1901 (when Euphemia was 11, as a kindergarten assistant. She was permanently appointed teacher in the DC school system a few years later in the summer of 1903.

Inspired no doubt by her mother, Euphemia’s educational journey was marked by consistent excellence. In fact, she distinguished herself early, graduating as valedictorian from M Street High School in 1907.

Euphemia Lofton began her own teaching career in the DC elementary school system in the fall of 1909. She taught there until the summer of 1912. Then she left for Smith College, where she earned an undergraduate mathematics major (and psychology minor) in 1914. During this time, the letters from her fiance Harold Haynes discuss his plans to visit her, as well as keeping her up to date on various family and business matters in DC. Upon graduation, she immediately returned to DC and started teaching in the high school system in the fall of 1914.

She taught  in the Miner Normal School and various local high schools, such as Dunbar, until 1930. At that point she began teaching at the Miner Teachers College (later assimilated with others to become the University of the District of Columbia). As an early indication of her extraordinary administrative talents, Euphemia quickly founded and established the mathematics department at Miner as well.

Marriage

In 1917 she married Harold Appo Haynes, a teacher like herself. The couple had no children. He was a childhood friend and, based on saved letters between he and Euphemia, a source of constant encouragement and strength for her. Harold had a EE degree from the University of Pennsylvania in 1910, and later earned a masters in education from the University of Chicago in 1930, and a doctorate in education from New York University in 1946.

Besides Euphemia’s promotion to teach at the college level, another significant event occurred in 1930. In that year, Euphemia obtained a Master’s degree in education from the University of Chicago (with her husband Harold).

Master’s thesis

Euphemia’s thesis was a significant piece of scholarship. In it, she discussed test validity and student assessment methods. With the goal to trace the evolution of testing in elementary and secondary school mathematics from 1900 to 1930, she focused on the main mathematics subjects arithmetic, algebra, and plane geometry. She surveyed published education literature, analyzed actual test instruments in those subjects, and summarized reports by educators and administrators on their own test development.

In the early 1900s, teachers gave a large group of students the same problem and compared how they did, without a grading key or separation into different skill metrics. In her thesis she notes that educators since those early 1900s started moving away from subjective grading to data-driven evaluation. Tests need to focus objectively on specific abilities. Tests moved from general surveys to fine-grained skill analysis. Indeed, by the 1910s–1920s, grade school tests measured discrete skills within core mathematics subjects. Testing evolved as a tool for diagnosis, curriculum evaluation, and teacher development. She also notes the progression toward standardized tests and its use for diagnosing student challenges.

In fact, after retiring from teaching she because president of the DC Board of Education (the first woman to hold that office). By that point she had grown to be an extremely gifted administrator with a single-minded focus on service to both education and her church. The timing of her presidency, following the 1954 Brown v. Board decision and amidst ongoing civil rights litigation,positioned her to directly implement and enforce desegregation and equity policies. Indeed, it was during her term as president that the track system was eliminated. (For further details, see the discussion of the 1967 Hobson v Hansen District Court case in [KSZ14]).

Volunteer service

Here we mention just a few of the many volunteer efforts Euphemia Haynes generously devoted her time to.

A common thread throughout her life was her commitment to social justice and her leadership in various Catholic organizations.

She co-founded the Catholic Interracial Council of the District of Columbia, an organization she helped establish to promote racial harmony and justice within the Church and broader community. These Catholic Interracial Councils, which united under a national umbrella in 1959-1960, were founded with the explicit aim of bridging racial divides between White and Black populations. The D.C. Council, for instance, played a vital role in coordinating Catholic participation in significant civil rights events.

Haynes also held a significant leadership position as the first vice president of the Archdiocesan Council of Catholic Women, which focused on education, social justice, and assisting immigrants. She later served as president of the local chapter of that Council.

After retiring from teaching in 1959, she didn’t stop working but served others through organizations such as  

  • the AAUW (American Association of University Women),
  • the Committee of International Social Welfare,
  • the Executive Committee of the National Social Welfare Assembly,
  • the D.C. Health and Welfare Council,
  • the USO,
  • the Urban League,
  • the NAACP, and
  • the League of Women Voters,

to name a few. Her unwavering commitment to social justice was fueled by the Catholic moral imperative to combat racism.

Her involvement with the Catholic Interracial Council and her broader civil rights work exemplify how Catholic social teaching provided a moral framework for her and other social activists during the Civil Rights era. The Church’s evolving stance on racial justice provided a powerful moral authority and a network (such as the Catholic Interracial Council and the Archdiocesan Council of Catholic Women) through which individuals like Haynes could actively pursue desegregation and equality. Her life demonstrates the practical application of religious principles to pressing social issues, showcasing how faith communities were critical sites for civil rights organizing and advocacy.

PhD thesis

Haynes earned her PhD from Catholic University of America (CUA) in 1943. The thesis topic itself is quite technical, to say nothing of the methods and proofs in the thesis itself. Just to understand the setup for the problem she solved in her thesis requires, at a minimum, knowing differential calculus.

In essence, Haynes’ thesis delves into the core of enumerative geometry, a field concerned with counting geometric objects satisfying specific conditions. For example, since the time of the ancient Greeks, mathematicians have known of the Problem of Apollonius: what is the number of circles tangent to three circles in general position? (It turns out the answer is 8.) Euphemia’s problem was in a similar spirit but was of course much harder and, as pointed out, even quite technical to state. None-the-less, her work is rooted in “synthetic” methods from the 1800s emphasizing geometric constructions and reasoning without explicit reliance on algebraic calculations. One of her innovations was to make extensive use of those algebraic calculations that the synthetic method was designed to avoid! In some sense, she solved her thesis problem by first reformulating it in a more complicated mathematical framework, then she worked out the solution.

It may be worth noting that after the 1940s, the methods she used were virtually abandoned for a direct algebraic approach, using machinery borrowed from the relatively recent methods of commutative algebra. For a few more technical details are her thesis see my earlier post “Remarks on …“, also available on this blog.

By the way, she was the first Black woman to ever earn a PhD in mathematics anywhere in the United States. She never requested recognition for this achievement. Indeed, it was over 50 years later when historians of mathematics recognized it was actually her and not someone else!

Unassuming but determined, Euphemia immediately put her understanding of advanced mathematics to work, inspiring not only students in the classroom, but also teachers. In a 1945 address to a meeting of DC mathematics teachers, Euphemia spoke about the unifying nature of what she called symbolic logic. Just as the physicist studies the natural world using rocks, plants, and other physical materials, the mathematician works within the world of logic. She explained that, instead of stones or chemicals, the ”tools” of the mathematician are facts, ideas, relationships, and implications. The abstract objects of logic are the raw materials of advanced mathematics, shaping the universe in which mathematicians explore and create.

Boiled down to its essence, her message to the mathematics teachers in the audience was: Through your service, you are teaching your students to better understand the world around them.

She achieved many academic awards in her life for her service. Another kind of award was bestowed on her from the Catholic Church.

The Pro Ecclesia et Pontifice medal

Just shy of 70 years old, in 1959 Euphemia Lofton Haynes was awarded the Papal Medal, Pro Ecclesia et Pontifice, by Pope John XXIII. This recognized her “outstanding valour and bravery on behalf of the Church and Society,” as well as her extraordinary services to the Church while maintaining fidelity to God and to the Pope. This Papal Decoration of Honor was a powerful affirmation of Euphemia Haynes’s entire life — her academic, professional, and civil rights endeavors. They were expressions of her deep faith and unwavering service to humanity. It’s an appreciation from the Catholic Church for her professional accomplishments in education, including founding departments and teaching for decades, her direct, hands-on service to the community, especially to Catholic students and teachers.

Her life lesson

Euphemia Haynes showed us that mathematics is not just about numbers. For her, it’s also about perseverance, leadership, and service. She broke barriers in higher education, stood firm in her beliefs, and used her talents to uplift others. Her life is a reminder that your passion — whether in math or something else — can connect to something larger than yourself. You don’t have to choose between mathematical expertise and helping others. As Euphemia Haynes showed us, you can do both.

References

[Ha30] E. L. Haynes, The Historical Development of Tests in Elementary and Secondary Mathematics, Masters Thesis, University of Chicago, 1930. pdf: click here

[Ha43] ——, Determination of Sets of Independent Conditions Characterizing Certain Special Cases of Symmetric Correspondences, Doctoral Thesis (advisor Aubrey Landry), The Catholic University of America, Washington DC, 1943. pdf: click here

[Ha45] —-, Mathematics – symbolic logics (typewritten and hand-written notes for a talk on the nature of advanced mathematics), address to Teachers of Mathematics in Jr. and Sr. High Schools (1945), Washington DC. (Available from the collected works of Euphemia Haynes at Catholic University of America.)

[KSZ14] S. Kelly, C. Shimmers, K. Zoroufy, Euphemia Lofton Haynes: Bringing education closer to the “goal of Perfection’‘, available online at the url arxiv.org/abs/1703.00944.

Mathematics PhD students of Aubrey Edward Landry 

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Based on information primarily sourced from the Mathematics Genealogy Project and university records, here are the Ph.D. students in Mathematics who graduated from The Catholic University of America between 1910 and 1950 under the advisement of Professor Aubrey Edward Landry:

1. Sister Mary Gervase Kelley (1917)*
Thesis Title: On the Cardioids Fulfilling Certain Assigned Conditions
2. Joseph Nelson Rice (1917)*
Thesis Title: On the In-and-Circumscribed Triangles of the Plane Rational Quartic Curve
3. Louis Antoine De Cleene (1927)*
Thesis Title: On Triangles Circumscribed about a Conic and Inscribed in a Cubic Curve
4. Frank Engelbert Smith (1928)*
Thesis Title: The Triangles In and-Circumscribed to the Triangular-Symmetric Rational Quartic
5. James Norman Eastham (1931)*
Thesis Title: The Triangles In-and-circumscribed to the Tacnodal Rational Quartic Curve with Residual Crunode
6. Sister Marie Cecilia Mangold (1929)*
Thesis Title: The Loci Described by the Vertices of Singly Infinite Systems of Triangles Circumscribed about a Fixed Conic
7. Sister Leonarda Burke (1931)*
Thesis Title: On a case of the triangles in-and-circumscribed to a rational quartic curve with a line of symmetry
8. Sister Mary de Lellis Gough (1931)
Thesis Title: On the Condition for the Existence of Triangles In-and-Circumscribed to Certain Types of Rational Quartic Curve and Having a Common Side
9. Sister Charles Mary Morrison (1931)*
Thesis Title: The Triangles In-and-Circumscribed to the Biflecnodal Rational Quartic
10. Sister Mary Felice Vaudreuil (1931)*
Thesis Title: Two Correspondences Determined by the Tangents to a Rational Cuspidal Quartic with a Line of Symmetry
11. Sister Mary Domitilla Thuener (1932)*
Thesis Title: On the Number and Reality of the Self-Symmetric Quadrilaterals In-and-Circumscribed to the Triangular-Symmetric Rational Quartic
12. Sister Mary Nicholas Arnoldy (1932)*
Thesis Title: The Reality of the Double Tangents of the Rational Symmetric Quartic Curve
13. Sister Mary Helen Sullivan (1934)*
Thesis Title: The Number and Reality of the Non-Self-Symmetric Quadrilaterals In-and-Circumscribed to the Rational Unicuspidal Quartic with a Line of Symmetry
14. Sister Mary Laetitia Hill (1935)*
Thesis Title: The Number and Reality of Quadrilaterals In-and-Circumscribed to a Rational Unicuspidal Quartic with Real Tangents from the Cusp
15. Sister Mary Henrietta Reilly (1936)*
Thesis Title: Self-Symmetric Quadrilaterals In-and-Circumscribed to the Plane Rational Quartic Curve with a Line of Symmetry
16. Sister Mary Charlotte Fowler (1937)*
Thesis Title: The discriminant of the sextic of double point parameters of the plane rational quartic curve
17. Sister Catherine Francis Galvin (1938)*
Thesis Title: Two Geometrical Representations of the Symmetric Correspondence C(N,N) with Their Interrelations
18. Euphemia Lofton Haynes (1943)*
Thesis Title: Determination of Sets of Independent Conditions Characterizing Certain Special Cases of Symmetric Correspondences
19. Brother Cyprian John McNamara (1948)*
Thesis Title: The Reality of the Double Tangents of the Rational Bicircular Symmetric Quartic

Note, most of these are nuns. Other than Brother McNamara, Landry advised de Cleene (1927) who had received his Holy Orders 8 years earlier.

Landry had three male “civilian” students (Rice, 1917, Smith 1928, Eastham, 1931) and one female “civilian” student (Haynes, 1943). Rice was later hired full time at Catholic University. Eastham was soon hired to teach at Nazareth College. Haynes, 16 years later, was awarded the highest award the Pope (Pope John XXIII) could give to a woman: the Pro Ecclesia et Pontifice medal.

An asterisk next to a name means that I’ve obtain the PhD by inter-library loan (usually from CUA). If there are any additions or missing students, please let me know!

The Beautiful Math of Tangent Lines: Unveiling the Lemniscate’s Secrets

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An experimental post

I asked Gemini to create a blog post based on this plot (below) and the Python code I wrote (based on ideas explained to me by Edray Goins) that generated it. (The prompt I fed to Gemini was modified from a prompt found online for generating a blog post about food.) The response, basically unedited, is below. The topic is inspired by the PhD thesis of Euphemia Haynes (a student of Aubrey Landry, Catholic University of America, 1943). See this earlier post for more information on it.

Mathematics can be beautiful, and sometimes that beauty is hidden in plain sight. Take, for instance, this striking image:

a graceful lemniscate curve adorned with a web of tangent lines, all seemingly converging on a single point. This isn’t some artistic rendering; it’s the output of a fascinating algorithm, written in Python and SageMath, plot_lemniscate_quartic_tangents.

This algorithm doesn’t just draw a pretty picture; it solves a sophisticated geometric problem: finding all the tangent lines to a lemniscate quartic that also pass through a specific point on the curve. Let’s dissect the magic:

  1. The Lemniscate: The algorithm begins by defining the lemniscate quartic curve. This curve, shaped like a sideways figure-eight, has a unique and elegant equation: (x² + y²)² = a(x² – y²), where ‘a’ is a constant.
  2. The Target Point: We choose a specific point, P0, on the lemniscate. This is the point where all our tangent lines will magically meet.
  3. The Tangent Hunt: This is where the algorithm gets clever. It doesn’t directly solve for the tangent lines. Instead, it cleverly iterates through numerous points (P) on the lemniscate. For each point, it calculates two things:
    • The slope of the secant line connecting P0 and P.
    • The slope of the tangent line to the lemniscate at point P.
  4. The Convergence: If the slopes from step 3 are approximately equal (within a tolerance), it means the secant line through P0 and P is also a tangent line to the curve at P. This line is added to our collection of tangent lines.
  5. The Reveal: Finally, the algorithm plots the lemniscate along with all the tangent lines it discovered. The result is the captivating image we see—a testament to the hidden geometric relationships within this seemingly simple curve.

The algorithm isn’t just about finding tangents; it reveals a beautiful correspondence between points on the lemniscate and its tangent lines. It’s a visual exploration of the intricate geometry underlying algebraic curves, demonstrating how computation can unveil hidden mathematical beauty. So next time you see a seemingly simple curve, remember the rich and sometimes surprising geometry hiding within.

Statistics of fielding errors in baseball: BAL2019

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“An error is a statistic charged against a fielder whose action has
assisted the team on offense, as set forth in this Rule 9.12.”
– MLB Rule book, 2023

These questions seem natural:
1) Are the number of fielding errors correlated with a team’s winning percentage? If so, how were are they correlated?
2) Are errors uniformly distributed? That is, given the frequency of the 24 game states (as described in the game states post), do errors occur in the same rough frequency?

In the case of the 2nd question, the states with bases empty (0, 1, or 2 outs) are the most commonly occurring states, followed by the states with a runner on 1st (only). For a randomly selected game from the 81 home games of a given MLB team in a given season, are the proportions

“plays with bases empty (batter up, pitcher about to throw first pitch)”: “plays with runner on 1st (only)”

and

“plays with bases empty that have an error”: “plays with runner on 1st (only) that have an error”

roughly equal? (Short answer: I doubt it.)

To these and related questions, we use Retrosheet to compile data. The plays below use the Retrosheet event file notation, as explained in the Retrosheet documentation.

For example, what we examine these questions in the case where the team is the Baltimore Orioles and the (regular) season is 2019? The O’s had a low winning percentage that year: 33.3% in 2019 (the year Brandon Hyde was hired as the Orioles’ manager).
The Orioles finished the 2019 season with 54 wins and 108 losses. The team was also below the league average in offensive categories runs scored, batting average, on-base percentage, and slugging percentage. In the 81 home games, 111 errors were committed, 59 by a visiting team and 52 by the Orioles. The Os played 12 games with at least 3 errors (by both sides), and 6 games with 2 errors. A significant proportion of the errors were in those 12 games: about 15% of the games had over 1/3rd the total number of errors.

Here are three particularly interesting plays from that season of Orioles home games. Back-to-back errors that occur for consecutive batters, such as Alberto–Mancini below, are relatively uncommon. But so are put outs with an error modifier that didn’t get negated by that error, as in the 3rd play below.

  1. During the game TBA (Tampa Bay Rays) vs BAL (Baltimore Orioles) on 2019-08-22, the state before was (0,0,0,0,1,0,0,1). The batter came up to the play and a simple type of error occurred: play,1,1,albeh001,01,CX,E6/TH/G. The state after was (0,1,0,0,1,0,0,1).
    Summary: In the bottom of the 1st inning with bases empty, BAL batter Hanser Alberto hit a ground ball to short. A throwing error by the shortstop allowed the batter to reach 1st.
  2. Trey Mancini is next in the batting line-up. The state before is, as we know, (0, 1, 0, 0, 1, 0, 0, 1), where the runner on 1st is Alberto. Mancini’s play is: play,1,1,manct001,00,X,D7/G+.1-H(UR);B-H(TH)(E6/TH)(NR). The state after is (0, 0, 0, 0, 1, 0, 2, 1).
    Summary: In the bottom of the 1st inning with a runner on 1st, BAL batter Trey Mancini hit a hard ground ball double to left field. The runner on 1st base scored (1 RBI). A throwing error by the shortstop allowed the batter to stretch the double into a home run (no RBI).
    [Video showed more details of this unusual play: the left fielder attempted to throw out Alberto at home, but the throw was to the shortstop, not the 3rd baseman who could have held Mancini at 2nd. This is not an error. However, the shortstop’s errant throw home, hoping to stop Alberto form scoring, went into the camera well, so by MLB Rule 5.06(b)(4)(A), the Umpire awarded Mancini an extra base.]
  3. During the game SDN (San Diego Padres) vs BAL on 2019-06-25, the state before was (2, 1, 0, 0, 3, 5, 1, 1). The batter came up to the play and a simple type of error occurred: play,3,1,smitd007,00,X,D9/G+.1-H;BX3(95)(E9). The state after was (0, 0, 0, 0, 4, 5, 2, 0).
    Summary: In the bottom of the 3rd inning, BAL batter Dwight Smith hit a ground-rule double to right field. The runner on 1st base scored (1 RBI). The batter stretched the hit into a triple, due to an error by the right fielder. Here, the error (E9) does not negate the tag (BX3) in this case but the explanation is complicated: The runner reached 3rd but then (mistakenly?) stepped off and was tagged out. It was ruled that the error by the center fielder allowing the batter to reach 3rd was unrelated to the put out at 3rd. The batter was tagged out at 3rd base by the third baseman after safely reaching the bag. This out was the 3rd out, ending the half-inning.

There are 108 others described in the full report “Game states via Retrosheet: Errors in BAL 2019 home games”. It’s available as a (approx 40-50 page) pdf for anyone who wants it. Just email me at wdjoyner@gmail.com.


Remarks on the 1943 PhD  thesis of E. Haynes

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Euphemia Haynes was a trailblazer as the first African-American woman to earn a doctorate in mathematics. The Catholic University of America holds a copy of her thesis, entitled “The Determination of Sets of Independent Conditions Characterizing Certain Special Cases of Symmetric Correspondences”, in their archives. 

A short biography: Euphemia Lofton daughter of Dr William Lofton, a dentist, and Lavinia Day Lofton, a kindergarten teacher. She was the valedictorian of M Street High School in 1907 and then graduated from (what is now known as) University of the District of Columbia with distinction and a degree in education in 1909. She was a  mathematics major at Smith College, which she graduated from in 1914. She married Harold Appo Haynes, a teacher in 1917 and earned her master’s degree in education from the University of Chicago in 1930. In 1943 she was awarded her PhD from The Catholic University of America, advised by Professor Aubrey E. Landry. An excellent, more detailed biography can be found at [KSZ] (see also http://www.math.buffalo.edu/mad/PEEPS/haynes.euphemia.lofton.html).

More of her biography is given in the post The Mathematician and the Pope, also available on this blog.

In this blog post, we merely try to explain her title. What is a “symmetric correspondence”?

We refer to Dolgachev’s notes [Do14], section 5.5: A correspondence of degree d between nonsingular curves X and Y is a non-constant morphism T:X\to Y^{(d)} to the d-th symmetric product Y^{(d)} of $Y$. Its graph is denoted \Gamma_T\subset X\times Y. The projection \Gamma_T\to X is a finite map of degree $d$, while the projection \Gamma_T\to Y is a finite map of degree e, say. It defines a correspondence Y\to X^{(e)} denoted by T^{-1}, called the inverse correspondence. Its graph \Gamma_{T^{-1}}\subset Y\times X is the image of \Gamma_T under the swap X\times Y\to Y\times X. If d is the degree of T and e is the degree of T^{-1}, we say that $T$ is a correspondence of type (d, e). This correspondence is symmetric if T = T^{-1}. 

Very roughly speaking, in her thesis, Haynes looks at various special cases of curves and in these cases she derives (technically defined) conditions that characterize the types of symmetric correspondences that arise in those cases.

While a scan of her thesis is archived at Catholic University, I have typed up her thesis in latex. For a digital copy, just email me (wdjoyner@gmail.com).

References

[Do14] I. Dolgachev, Classical Algebraic Geometry: a modern view, Cambridge Univ. Press, 2012.

(at https://mathweb.ucsd.edu/~eizadi/207A-14/CAG.pdf)

[KSZ17] Susan Kelly, Carly Shinners, Katherine Zoroufy, “Euphemia Lofton Haynes: Bringing Education Closer to the “Goal of Perfection“, preprint, 2017 (available at https://arxiv.org/abs/1703.00944). A version of this paper was also published in the Notices of the American Mathematics Society.