Preface

IPM, the Institute for Research in Fundamental Sciences in Tehran, Iran, com-

memorated the 20th anniversary of its founding by a major international conference

IPM 20 - Combinatorics 2009 on May 15-21, 2009. The conference was dedicated

to Reza Khosrovshahi, one of the founders of IPM and the director of its School of

Mathematics from 1996 to 2007, on the occasion of his 70th birthday. This volume

contains a collection of some of the papers presented at the conference.

Mathematics has a long history in the Iranian plateau. Between the ninth and

the ﬁfteenth centuries, Iranian mathematicians played a central role in the develop-

ment of mathematics in the Islamic world. Mathematicians such as Khawrizmi,

Mahani, Nayrizi, Buzjani, Quhi, Karaji, Biruni, Khayyam, Tusi, and Kashani

hailed—as their names suggest—from all corners of the Iranian world, and over

a seven hundred period transformed the mathematical heritage inherited from In-

dia, Persia, and most notably Greek Alexandria. Algebra—whose name comes

from Khawrizmi’s ﬁrst book on the subject— was developed and eventually be-

came a discipline distinct from Geometry with its own problems and methods. The

concept of number was enlarged to include Euclid’s magnitudes, and, by the time

of Kashani in the ﬁfteenth century, real numbers and their decimal expansions

were used easily and productively. With applications to astronomy and geography

in mind, trigonometry and spherical geometry were systematized and developed.

Binomial coeﬃcients and their properties (including the so-called “Pascal’s trian-

gle”) were developed and used extensively. In this period, applied mathematics

also blossomed. Mathematicians working in collaboration with artisans, architects,

and astronomers developed many practical algorithms (in fact, the word algorithm

comes from the name of Khawrizmi) and approximation procedures.

Nevertheless, after the ﬁfteenth century, there was a marked decline in origi-

nal mathematical activity in Iran. While the study of Euclid and the classics of

Islamic mathematics continued unabated, mathematics lost the center stage. Fast

forwarding to the most recent period, Iranian mathematics started its reawaken-

ing in the years following World War II when mathematicians centered around

University of Tehran, most notably Dr. Gholamhossein Mosahab, trained a gener-

ation of mathematics educators. In turn, in the 1960s and 70s, a dedicated group

of legendary high school level math teachers disseminated a love for mathematics

among the youth. This was the period when mathematical problem solving and

mathematical books for non-experts became popular and when a number of high

school mathematics teachers developed a national reputation. The next big step in

the development of mathematics in Iran came in 1989 with the founding of IPM

(Institute for Studies in Theoretical Physics and Mathematics since renamed the

Institute for Research in Fundamental Sciences). This was complemented by the

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