This book is a tribute to Uno Kaljulaid, an eminent mathematician who was born in 1941 in Estonia and died there in 1999. Several of his teachers, Yuri Manin, Boris Plotkin and most especially Jaak Hion, influenced his mathematical work a lot. Kaljulaid studied at Tartu University in 1959–1963, got his diploma in algebraic geometry in 1966 under Professor Manin of Moscow University, did postdoctoral studies at Tartu University in 1968–1972, and defended brilliantly his Tartu/Minsk Candidate thesis on representation theory in 1979, thesis which was supervised by Professor Plotkin. Kaljulaid taught at Tartu University from 1972. In addition, he was involved in the Computer Science Institute there from 1993, and was simultaneously part time senior research fellow at the Institute of Cybernetics in Tallinn.
The book contains, in addition to a biographical sketch of Kaljulaid, his bibliography, some fifty four works published mainly in Estonian journals. Many of these works, appear in this book translated from Russian into English by one of the editors, Jaak Peetre. The book contains six chapters:
Chapter I “Representations of semigroups and algebras” Kaljulaid did pioneering work in the theory of the variety of semigroup representations and the variety of linear automata. This chapter contains in particular an English translation of Kaljulaid’s Candidate thesis.
Chapter II “Automata theory” During the last decade of his life, Kaljulaid’s main area of interest became automata theory. He became particularly interested in automata and their decomposition, more specifically in the algebraic construction of the wreath product. Category theoretic representation of automata were extensively used by Kaljulaid. He drafted a rather ambitious course on automata theory including his own results. The main part of this course was given as a series of colloquium presentations at Lund University in 1994 (Part A: Automata, Languages and Rationality) and at Stockholm University in 1996 and 1997 (Part B: On the Languages of and Rationality of Formal Series using Order and Topology). Professor Peetre wrote part of these lectures from Kaljulaid’s slides and handwritten notes, some of which remained unfinished by Kaljulaid and were never presented.
Chapter III “Majorization” A grant was made by the Crafoord Foundation that allowed Kaljulaid to travel to Lund several times. The purpose of these trips was to study, with Professor Peetre, generalized majorization. This chapter contains, in particular, a fragment of a larger unfinished paper on this subject.
Chapter IV “Combinatorics” Kaljulaid was led to combinatorics through teaching. This chapter contains in particular an unpublished manuscript on Fibonacci numbers of graphs c. 1991. It also contains the draft of a letter he wrote to Professor Tambour related to some cooperative work.
Chapter V “History of mathematics” This chapter represents material found by Kaljulaid in archives in Tartu. Kaljulaid was interested in the history of mathematics, particularly in the life and mathematical work of the algebraist, Theodor Molien, who won a gold medal from the Paris Academy of Sciences for his innovative work in algebra. This chapter also contains notes on four other 19th century Tartu mathematicians.
Chapter VI “Popularization of mathematics” Kaljulaid wrote several popular scientific papers on his mathematical interests. For instance, Section 2 is an éloge to his teacher, Professor Manin, who won the Lenin prize for work in Diophantine geometry; Section 6 is an essay devoted to Galois theory; and Section 9 describes applications of formal languages in computer science and biology.
The book is supplied by an index of names as well as a subject index. Quotes appear throughout the book. Comments from Professors Manin and Plotkin also appear in the book.
This book, a glimpse at Uno Kaljulaid’s work, will be of interest to mathematicians, and will hopefully attract several people into mathematics.