Joachim "Jim" Lambek (5 December 1922 – 23 June 2014) was a German-born Canadian
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
. He was Peter Redpath
Emeritus
''Emeritus'' (; female: ''emerita'') is an adjective used to designate a retired chair, professor, pastor, bishop, pope, director, president, prime minister, rabbi, emperor, or other person who has been "permitted to retain as an honorary title ...
Professor of Pure Mathematics at
McGill University
McGill University (french: link=no, Université McGill) is an English-language public research university located in Montreal, Quebec, Canada. Founded in 1821 by royal charter granted by King George IV,Frost, Stanley Brice. ''McGill Universit ...
, where he earned his
PhD degree in 1950 with
Hans Zassenhaus as advisor.
Biography
Lambek was born in
Leipzig
Leipzig ( , ; Upper Saxon: ) is the most populous city in the German state of Saxony. Leipzig's population of 605,407 inhabitants (1.1 million in the larger urban zone) as of 2021 places the city as Germany's eighth most populous, as wel ...
,
Germany
Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated betwe ...
, where he attended a
Gymnasium.
He came to England in 1938 as a
refugee
A refugee, conventionally speaking, is a displaced person who has crossed national borders and who cannot or is unwilling to return home due to well-founded fear of persecution. on the ''
Kindertransport
The ''Kindertransport'' (German for "children's transport") was an organised rescue effort of children (but not their parents) from Nazi-controlled territory that took place during the nine months prior to the outbreak of the Second World ...
''.
From there he was interned as an
enemy alien
In customary international law, an enemy alien is any native, citizen, denizen or subject of any foreign nation or government with which a domestic nation or government is in conflict and who is liable to be apprehended, restrained, secured and ...
and deported to a
prison work camp in
New Brunswick
New Brunswick (french: Nouveau-Brunswick, , locally ) is one of the thirteen provinces and territories of Canada. It is one of the three Maritime provinces and one of the four Atlantic provinces. It is the only province with both English and ...
,
Canada
Canada is a country in North America. Its ten provinces and three territories extend from the Atlantic Ocean to the Pacific Ocean and northward into the Arctic Ocean, covering over , making it the world's second-largest country by tot ...
. There, he began in his spare time a mathematical apprenticeship with Fritz Rothberger, also
interned, and wrote the
McGill Junior Matriculation in fall of 1941. In the spring of 1942, he was released and settled in
Montreal
Montreal ( ; officially Montréal, ) is the List of the largest municipalities in Canada by population, second-most populous city in Canada and List of towns in Quebec, most populous city in the Provinces and territories of Canada, Canadian ...
, where he entered studies at McGill University, graduating with an
honours
Honour (British English) or honor (American English; see spelling differences) is the idea of a bond between an individual and a society as a quality of a person that is both of social teaching and of personal ethos, that manifests itself as a ...
mathematics degree in 1945 and an
MSc a year later. In 1950, he completed his doctorate under
Hans Zassenhaus becoming McGill's first
PhD in mathematics.
Lambek became assistant professor at McGill; he was made a full professor in 1963. He spent his sabbatical year 1965–66 in at the
Institute for Mathematical Research
The Institute for Mathematical Research (''Forschungsinstitut für Mathematik'', FIM) is a mathematical research institution located at ETH Zurich and founded in 1964 by Beno Eckmann. Its main goals are to promote and facilitating the exchange be ...
at
ETH Zurich
(colloquially)
, former_name = eidgenössische polytechnische Schule
, image = ETHZ.JPG
, image_size =
, established =
, type = Public
, budget = CHF 1.896 billion (2021)
, rector = Günther Dissertori
, president = Joël Mesot
, ac ...
, where
Beno Eckmann
Beno Eckmann (31 March 1917 – 25 November 2008) was a Swiss mathematician who made contributions to algebraic topology, homological algebra, group theory, and differential geometry.
Life
Born in Bern, Eckmann received his master's degree fr ...
had gathered together a group of researchers interested in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
and
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, including
Bill Lawvere. There Lambek reoriented his research into category theory.
Lambek retired in 1992 but continued his involvement at
McGill's mathematics department. In 2000 a
festschrift
In academia, a ''Festschrift'' (; plural, ''Festschriften'' ) is a book honoring a respected person, especially an academic, and presented during their lifetime. It generally takes the form of an edited volume, containing contributions from the h ...
celebrating Lambek's contributions to mathematical structures in
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
was published. On the occasion of Lambek's 90th birthday, a collection ''Categories and Types in Logic, Language, and Physics'' was produced in tribute to him.
Scholarly work
Lambek's PhD thesis investigated
vector fields using the
biquaternion
In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions co ...
algebra over
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
, as well as
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
immersion in a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
. The second component was published by the
Canadian Journal of Mathematics
The ''Canadian Journal of Mathematics'' (french: Journal canadien de mathématiques) is a bimonthly mathematics journal published by the Canadian Mathematical Society.
It was established in 1949 by H. S. M. Coxeter and G. de B. Robinson. The cur ...
. He later returned to
biquaternion
In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions co ...
s when in 1995 he contributed "If Hamilton had prevailed: Quaternions in Physics", which exhibited the
Riemann–Silberstein bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalar (mathematics), scalars and Euclidean vector, vectors. If a scalar is considered a degree-zero quantity, and a vector is a d ...
to express the free-space electromagnetic equations.
Lambek supervised 17 doctoral students, and has 75 doctoral descendants as of 2020. He has over 100 publications listed in the
Mathematical Reviews
''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science.
The AMS also pu ...
, including 6 books. His earlier work was mostly in
module theory
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mod ...
, especially torsion theories, non-commutative localization, and
injective module
In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module ''Q'' that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if ''Q'' is a submodule of ...
s. One of his earliest papers, , proved the
Lambek–Moser theorem
The Lambek–Moser theorem is a mathematical description of partitions of the natural numbers into two Complement (set theory), complementary sets. For instance, it applies to the partition of numbers into even number, even and odd number, odd, or ...
about integer sequences. In 1963 he published an important result, now known as Lambek's theorem, on
character module In mathematics, especially in the area of abstract algebra, every module has an associated character module. Using the associated character module it is possible to investigate the properties of the original module. One of the main results discovere ...
s characterizing flatness of a module.
His more recent work is in
pregroups and
formal language
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.
The alphabet of a formal language consists of symb ...
s; his earliest works in this field were probably and . He is noted, among other things, for the
Lambek calculus
Categorial grammar is a family of formalisms in natural language syntax that share the central assumption that syntactic constituents combine as functions and arguments. Categorial grammar posits a close relationship between the syntax and sema ...
, an effort to capture mathematical aspects of natural language syntax in
logical form
In logic, logical form of a statement is a precisely-specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguo ...
, and a work that has been very influential in
computational linguistics
Computational linguistics is an Interdisciplinarity, interdisciplinary field concerned with the computational modelling of natural language, as well as the study of appropriate computational approaches to linguistic questions. In general, comput ...
, as well as for developing the connections between
typed lambda calculus
A typed lambda calculus is a typed formalism that uses the lambda-symbol (\lambda) to denote anonymous function abstraction. In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a ...
and
cartesian closed categories
In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in ma ...
(see
Curry–Howard–Lambek correspondence). His last works were on
pregroup grammar Pregroup grammar (PG) is a grammar formalism intimately related to categorial grammars. Much like categorial grammar (CG), PG is a kind of type logical grammar. Unlike CG, however, PG does not have a distinguished function type. Rather, PG uses in ...
.
Selected works
Books
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Articles
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* Reprinted in
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References
External links
Faculty profile of Joachim Lambekat McGill University
Lambek festival(80th anniversary)
{{DEFAULTSORT:Lambek, Joachim
1922 births
2014 deaths
20th-century Canadian mathematicians
21st-century Canadian mathematicians
21st-century German mathematicians
Algebraists
Canadian logicians
Category theorists
Kindertransport refugees
German emigrants to Canada