Foreword

Since the 19th century, non-commutative structures have played an important

role in physics and mathematics. One can mention the work of Galois as an im-

portant early example in which non-commutativity greatly informed questions in

commutative mathematics. This role expanded considerably with the inception of

quantum mechanics in the early years of the 20th century, and since then it only has

grown. The late 20th century has seen the advent of non-commutative geometry

and a fertile ground of interaction between physics and mathematics.

This volume comprises the contributions of the XI Solomon Lefschetz Memo-

rial Lectures and the satellite conference: Topics in Deformation Quantization and

Noncommutative Structures held at the Centro de Investigacion y de Estudios Avan-

zados (CINVESTAV), Mexico City, Mexico, September 7-9 of 2005. This event was

organized in a close collaboration between the departments of Physics and Math-

ematics with its funding coming from both departments and from various grants

funded by the Consejo Nacional de Ciencia y Tecnologia (Conacyt).

The Lefschetz Memorial Lectures originated in honor of Prof. Solomon Lef-

schetz are held periodically in autumn when they spotlight a distinguished mathe-

matician. Prof. Lefschetz is well known for his contributions to mathematics but

perhaps it is less known that since 1945 he visited Mexico regularly, jump-starting

the Mexican school in topology. His contribution to mathematics in Mexico was,

as a result of these visits, of major importance. His contributions were recognized

when he was awarded the Order of the Aztec Eagle in 1964. The Mathematics

Department of the Cinvestav established the Solomon Lefschetz memorial lectures

in 1975 to honor him. A brief history of the lectures can be glimpsed by looking at

the impressive list of Lefschetz Lecturers throughout the years:

• (1975) Jack Hale

• (1976) Michael Atiyah

• (1977) Enrico Bombieri

• (1978) Shing S. Chern

• (1979) John Milnor

• (1982) David Mumford

• (1985) William Thurston

• (1989) Ya G. Sinai

• (1991) V. P. Maslov

• (1995) Sergio Albeverio

• (2005) Maxim Kontsevich

• (2006) Dennis Sullivan

• (2007) Graeme Segal

vii