This two-day Short Course on Combinatorics: Past, Present, and Future is organized by Robin Wilson, The Open University, and will take place on Friday and Saturday, January 4 and 5.

The object of this short course is to learn about the development of a wide range of combinatorial topics, from earliest times up to the present day and beyond. The topics presented will include early combinatorics from China and the Islamic and Hebrew traditions, European combinatorics during the Renaissance, the legacy of Leonhard Euler, and combinatorial topics in the modern era.

EARLY COMBINATORICS (up to the 17th century)

Andrea Breard (Université des Sciences et Technologies de Lille, France): China
Combinatorial practices in China go back to high Antiquity, when divinatory techniques relied on configurations of broken and unbroken lines. The Book of Change (Yijing), compiled under the Zhou dynasty, has transmitted these practices until today, and has been a widely commented and read source. But combinatorial practices in China are not limited to divination and magic squares: a large number of early sources also describe games like Go, chess and games with cards, dominoes and dice that show a combinatorial interest from a more mathematical point of view. The earliest source that systematically discusses permutations and combinations is an 18th-century manuscript. Although mathematics had by then been introduced from Europe, the manuscript is clearly based on traditional mathematical concepts and algorithmic modes. In this talk I shall show how early combinatorial practices provide a framework for later mathematical developments in imperial China.

Victor Katz (University of the District of Columbia, US): Combinatorics in the Islamic and Hebrew traditions
Among the earliest records we have of combinatorial questions being asked is a Hebrew mystical work which asks us to calculate the number of words that can be created out of the letters of the Hebrew alphabet. This is a way of 'quantifying' God's work, because God created things by naming them. A similar question was posed in some early Islamic treatises, using, of course, the Arabic alphabet. In both traditions, however, mathematicians generalized these questions and went on to figure out the rules for calculating permutations and combinations, and by the fourteenth century, they were able to give formal proofs of the results.

EUROPE

Eberhard Knobloch (Technical University of Berlin, Germany): European combinatorics, 1200-1700
Combinatorial studies are to be found in philosophical, religious, musical, and game-theoretical contexts: Ramon Lull was especially important in this respect. Pacioli, Cardano and Tartaglia deduced the first results, and Jesuits such as Clavius and Lullists such as Mersenne, Kircher, Izquierdo, and Caramuel played a crucial role, and only in the 17th century did mathematicians such as Schooten and Wallis deal with combinatorics in a purely mathematical context. Pascal's treatise on the arithmetical triangle might be called the first modern treatise on combinatorics. Leibniz's contributions to symmetric functions, partitions, determinants remained unpublished until very recently. James Bernoulli's Ars conjectandi comprehended an exhaustive treatment of early modern combinatorics.

EULER's LEGACY

Robin Wilson (Open University, UK): Early graph theory
In this talk, covering the 18th and 19th centuries, I shall trace the origins of graph theory the problem of the Königsberg bridges, via the polyhedron formula, Kirchhoff's electrical networks, Hamilton's Icosian game, and Cayley's work on trees, to the early attempts to solve map-coloring problems.

George Andrews (Pennsylvania State University, US): Euler's 'De Partitio Numerorum'
Chapter 16 of Euler's Introductio in Analysin Infinitorum is titled De Partitio Numerorum. I shall examine how Euler began the theory of partitions, note some surprising connections with subsequent developments, and reveal how Euler's bizarre presentation of an ancient chestnut, that every integer is uniquely the sum or difference of distinct powers of 3 (e.g., 80 = 34 - 30), leads to new insights in the theory of partitions.

Lars Andersen (University of Aarhus, Denmark): Latin squares
A Latin square of order n is an n × n array with entries from a set of n symbols arranged in such a way that each symbol occurs exactly once in each row and exactly once in each column. From this simplistic starting point, the theory of Latin squares has developed to become an interesting discipline in its own right, as well as a very important tool in design theory in general. I shall describe this development, including the invention by Leonhard Euler in the 18th century of the influential concept of orthogonal Latin squares, originally with the purpose of constructing 'magic' squares, a method that was actually anticipated by amulet makers in medieval times.

Robin Wilson (Open University, UK): Triple systems, schoolgirls and designs
In this talk I shall outline the early history of block designs, (so-called) Steiner triple systems, and the problem of the fifteen young ladies, with particular reference to the work of Thomas Kirkman.

COMBINATORICS COMES OF AGE

Lowell Beineke (Indiana University - Purdue University at Fort Wayne, US): 20th-century graph theory
Around the middle of the 20th century graph theory began to burgeon, and in the second half of the century, it had become one of the most active areas of mathematics. From its roots in such topics as map coloring and network connections, I shall follow some of the branches that have recently borne the most fruit.

Herb Wilf (University of Pennsylvania, Philadelphia, US) and Lily Yen (Capilano College, US): Sister Celine as I knew her
Sister Mary Celine Fasenmyer is responsible for the seminal ideas in the now-burgeoning field of computerized proofs of identities. I met her, following a suggestion of Doron Zeilberger, when she was well into her 80s, and, together with some of my graduate students, interviewed her on videotape. In my talk I shall show parts of that video and will recount some stories about her life, her appearance at a conference in Florida, and her work.

Bjarne Toft (Southern Denmark University, Odense, Denmark): The game of Hex: history, results and problems
The game of hex was invented in 1942 by the Danish scientist, designer and poet Piet Hein, and independently in 1948 by John Nash. The history of the game, its mathematics and its unsolved problems have been intensely studied in recent years by (among others) Thomas Maarup, Cameron Brown, Ryan Hayward and Jack van Rijswijck. In this talk I shall present a general survey of the game and its history, and of the life and achievements of Piet Hein.

TOWARDS THE FUTURE

Ron Graham (University of California, San Diego, US) and Doron Zeilberger (Rutgers University): Combinatorics: The Future and Beyond
In this talk, we shall describe some evolving trends in the general subject of combinatorics, discuss some challenging open problems, and speculate on what we might expect to see in the future.

Registration

Advance registration fees are US$125/member; US$175/nonmember; and $50/student, unemployed, emeritus. On-site registration fees are US$140/member; US$190/nonmember; and US$60/student, unemployed, emeritus.

The deadline for pre-registration has passed. Please register at the meeting. The Registration Desk for the MAA Short Course will be located on the upper level of the SDCC outside rooms 5A/5B. This desk will be open on Friday, January 4, from 8:00 am to Noon.