Ian R. Porteous
Encyclopedia
Ian Robertson Porteous was a Scottish mathematician at the University of Liverpool
and an educator on Merseyside. He is best known for three books on geometry
and modern algebra. In Liverpool
he and Peter Giblin
are known for their registered charity Mathematical Education on Merseyside which promotes enthusiasm for mathematics
through sponsorship of an annual competition.
(later a noted theologian and Old Testament expert), from Crossgates, Fife
and May Hadwen Robertson of Kirkcaldy, Fife. He attended George Watson's College
in Edinburgh, and the University of Edinburgh
, obtaining his first mathematical degree in 1952. After a time in national service
, he took up study at Trinity College, Cambridge
. Porteous wrote his thesis Algebraic Geometry under W.V.D. Hodge and Michael Atiyah
at University of Cambridge
in 1961.
in New York, Porteous was influenced by Serge Lang
. He continued to do research on manifold
s in differential geometry.
In 1971 his article "The normal singularities of a submanifold" was published in Journal of Differential Geometry
5:543–64. It was concerned with the smooth embeddings of an m-manifold in Rn.
In 1969 Porteous published Topological Geometry with Van Nostrand Reinhold and Company. It was reviewed in Mathematical Reviews
by J. Eells, who interpreted it as a three-term textbook for a sequence in abstract algebra
, geometric algebra
, and differential calculus in Euclidean and Banach spaces and on manifolds. Eells says "Surely this book is the product of substantial thought and care, both from the standpoints of consistent mathematical presentation and of student's pedagogical requirements." In 1981 a second edition was published with Cambridge University Press
.
s, 13:The classical group
s, 15:Clifford algebra
s, 16:Spin groups, 17:Conjugation
, 20:Topological spaces, 21:Manifold
s, 22:Lie group
s. In the preface Porteous acknowledges the contribution of his master's degree
student Tony Hampson and anticipatory work by Terry Wall. See references to a link where misprints may be found.
The textbook Geometric Differentiation (1994) is a modern, elementary study of differential geometry. The subtitle, "for the intelligence of curves and surfaces" indicates its extent in the differential geometry of curves
and differential geometry of surfaces
.
The review by D.R.J. Chillingworth says it is "aimed at advanced undergraduates or beginning graduate students in mathematics..." Chillingworth notes "a peculiar feature of the book is its use of compact notation for differentiation using numerical subscripts that allow tidy presentation of calculations." For instance, Porteous gives Faa di Bruno's formula
. Furthermore, the reviewer notes that this mathematics has "connections to optics, kinematics and architecture as well as (more recently) geology, tomography, computer vision and face-recognition."
These applications follow from the theories of contact
, umbilical point
s, ridge
s, germ
s, and cusp
s. Porteous has suggestions for readers wanting to know more about singularity theory
. The underlying theme is the study of critical points of appropriate distance-squared functions. A second edition was published in 2001, where the author was able to report on related work by Vladimir Arnold
on spherical curves. In fact, Porteous had translated Arnold's paper from the Russian
.
can be seen through the work of his charity "Mathematical Education on Merseyside" (see references). As recounted in the book Challenging Mathematics, in 1978 Giblin and Porteous began to organize a Challenge competition for first and second formers in secondary school. By 1989 they were drawing 3500 participants each year. Held over two weekends in the Spring Term, students considered six questions each round. Marking was arranged through the mathematics department of Liverpool University, and prizes were awarded at "an evening of mathematical recreation". Broad participation was encouraged by making half the problems widely accessible. Solutions to the problems appear in their book.
Beyond mathematics, he enjoyed hill-walking and sang in his church choir
. He served as a Liberal
councillor on Liverpool City Council
from 1974 to 1978.
He passed away suddenly of a suspected heart attack on 30 January 2011.
University of Liverpool
The University of Liverpool is a teaching and research university in the city of Liverpool, England. It is a member of the Russell Group of large research-intensive universities and the N8 Group for research collaboration. Founded in 1881 , it is also one of the six original "red brick" civic...
and an educator on Merseyside. He is best known for three books on geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
and modern algebra. In Liverpool
Liverpool
Liverpool is a city and metropolitan borough of Merseyside, England, along the eastern side of the Mersey Estuary. It was founded as a borough in 1207 and was granted city status in 1880...
he and Peter Giblin
Peter Giblin
Peter John Giblin is an English mathematician whose primary research involves singularity theory and its application to geometry, computer vision, and computer graphics. Giblin is an emeritus professor of mathematics at the University of Liverpool where he has served on staff for more than 40 years...
are known for their registered charity Mathematical Education on Merseyside which promotes enthusiasm for mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
through sponsorship of an annual competition.
Family and early life
Porteous was born on 9 October 1930. He was one of six children of Reverend Norman Walker PorteousNorman Walker Porteous
Norman Walker Porteous was a noted theologian and writer on Old Testament issues, and the last surviving officer of the First World War....
(later a noted theologian and Old Testament expert), from Crossgates, Fife
Crossgates, Fife
Crossgates is a small village in Fife, Scotland.It is located close to the junction of the M90 and A92, about two miles east of Dunfermline and a similar distance south west of Cowdenbeath...
and May Hadwen Robertson of Kirkcaldy, Fife. He attended George Watson's College
George Watson's College
George Watson's College, known informally as Watson's, is a co-educational independent day school in Scotland, situated on Colinton Road, in the Merchiston area of Edinburgh. It was first established as a hospital school in 1741, became a day school in 1871 and was merged with its sister school...
in Edinburgh, and the University of Edinburgh
University of Edinburgh
The University of Edinburgh, founded in 1583, is a public research university located in Edinburgh, the capital of Scotland, and a UNESCO World Heritage Site. The university is deeply embedded in the fabric of the city, with many of the buildings in the historic Old Town belonging to the university...
, obtaining his first mathematical degree in 1952. After a time in national service
National service
National service is a common name for mandatory government service programmes . The term became common British usage during and for some years following the Second World War. Many young people spent one or more years in such programmes...
, he took up study at Trinity College, Cambridge
Trinity College, Cambridge
Trinity College is a constituent college of the University of Cambridge. Trinity has more members than any other college in Cambridge or Oxford, with around 700 undergraduates, 430 graduates, and over 170 Fellows...
. Porteous wrote his thesis Algebraic Geometry under W.V.D. Hodge and Michael Atiyah
Michael Atiyah
Sir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...
at University of Cambridge
University of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...
in 1961.
Early career
Porteous began teaching at the University of Liverpool as a lecturer in 1959, becoming senior lecturer in 1972. During a year (1961–62) at Columbia UniversityColumbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
in New York, Porteous was influenced by Serge Lang
Serge Lang
Serge Lang was a French-born American mathematician. He was known for his work in number theory and for his mathematics textbooks, including the influential Algebra...
. He continued to do research on manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s in differential geometry.
In 1971 his article "The normal singularities of a submanifold" was published in Journal of Differential Geometry
Journal of Differential Geometry
The Journal of Differential Geometry is a peer-reviewed mathematical journal published by the Lehigh University. The journal publishes nine issues per year and is distributed by International Press. Surveys in Differential Geometry] It also publishes an annual supplement in book form called...
5:543–64. It was concerned with the smooth embeddings of an m-manifold in Rn.
In 1969 Porteous published Topological Geometry with Van Nostrand Reinhold and Company. It was reviewed in Mathematical Reviews
Mathematical Reviews
Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses of many articles in mathematics, statistics and theoretical computer science.- Reviews :...
by J. Eells, who interpreted it as a three-term textbook for a sequence in abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, geometric algebra
Geometric algebra
Geometric algebra , together with the associated Geometric calculus, provides a comprehensive alternative approach to the algebraic representation of classical, computational and relativistic geometry. GA now finds application in all of physics, in graphics and in robotics...
, and differential calculus in Euclidean and Banach spaces and on manifolds. Eells says "Surely this book is the product of substantial thought and care, both from the standpoints of consistent mathematical presentation and of student's pedagogical requirements." In 1981 a second edition was published with Cambridge University Press
Cambridge University Press
Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by Henry VIII in 1534, it is the world's oldest publishing house, and the second largest university press in the world...
.
Later career and works
In 1995 Ian Porteous published Clifford Algebras and the Classical Groups which was reviewed by Peter R. Law. In praise, Law says "Porteous' presentation of the subject matter sets a standard by which others may be judged." The book has 24 chapters including 8:quaternionQuaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
s, 13:The classical group
Classical group
In mathematics, the classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces. Their finite analogues are the classical groups of Lie type...
s, 15:Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
s, 16:Spin groups, 17:Conjugation
Conjugate element (field theory)
In mathematics, in particular field theory, the conjugate elements of an algebraic element α, over a field K, are the roots of the minimal polynomialof α over K.-Example:The cube roots of the number one are:...
, 20:Topological spaces, 21:Manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s, 22:Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s. In the preface Porteous acknowledges the contribution of his master's degree
Master's degree
A master's is an academic degree granted to individuals who have undergone study demonstrating a mastery or high-order overview of a specific field of study or area of professional practice...
student Tony Hampson and anticipatory work by Terry Wall. See references to a link where misprints may be found.
The textbook Geometric Differentiation (1994) is a modern, elementary study of differential geometry. The subtitle, "for the intelligence of curves and surfaces" indicates its extent in the differential geometry of curves
Differential geometry of curves
Differential geometry of curves is the branch of geometry that dealswith smooth curves in the plane and in the Euclidean space by methods of differential and integral calculus....
and differential geometry of surfaces
Differential geometry of surfaces
In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric....
.
The review by D.R.J. Chillingworth says it is "aimed at advanced undergraduates or beginning graduate students in mathematics..." Chillingworth notes "a peculiar feature of the book is its use of compact notation for differentiation using numerical subscripts that allow tidy presentation of calculations." For instance, Porteous gives Faa di Bruno's formula
Faà di Bruno's formula
Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives, named after , though he was not the first to state or prove the formula...
. Furthermore, the reviewer notes that this mathematics has "connections to optics, kinematics and architecture as well as (more recently) geology, tomography, computer vision and face-recognition."
These applications follow from the theories of contact
Contact (mathematics)
In mathematics, contact of order k of functions is an equivalence relation, corresponding to having the same value at a point P and also the same derivatives there, up to order k. The equivalence classes are generally called jets...
, umbilical point
Umbilical point
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points that are locally spherical. At such points both principal curvatures are equal, and every tangent vector is a principal direction....
s, ridge
Ridge (differential geometry)
For a smooth surface in three dimensions a ridge point occurs when a line of curvature has a local maximum or minimum of principal curvature. The set of ridge points form curves on the surface called ridges....
s, germ
Germ (mathematics)
In mathematics, the notion of a germ of an object in/on a topological space captures the local properties of the object. In particular, the objects in question are mostly functions and subsets...
s, and cusp
Cusp (singularity)
In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....
s. Porteous has suggestions for readers wanting to know more about singularity theory
Singularity theory
-The notion of singularity:In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor...
. The underlying theme is the study of critical points of appropriate distance-squared functions. A second edition was published in 2001, where the author was able to report on related work by Vladimir Arnold
Vladimir Arnold
Vladimir Igorevich Arnold was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory,...
on spherical curves. In fact, Porteous had translated Arnold's paper from the Russian
Russian language
Russian is a Slavic language used primarily in Russia, Belarus, Uzbekistan, Kazakhstan, Tajikistan and Kyrgyzstan. It is an unofficial but widely spoken language in Ukraine, Moldova, Latvia, Turkmenistan and Estonia and, to a lesser extent, the other countries that were once constituent republics...
.
Death and legacy
Porteous' commitment to mathematics educationMathematics education
In contemporary education, mathematics education is the practice of teaching and learning mathematics, along with the associated scholarly research....
can be seen through the work of his charity "Mathematical Education on Merseyside" (see references). As recounted in the book Challenging Mathematics, in 1978 Giblin and Porteous began to organize a Challenge competition for first and second formers in secondary school. By 1989 they were drawing 3500 participants each year. Held over two weekends in the Spring Term, students considered six questions each round. Marking was arranged through the mathematics department of Liverpool University, and prizes were awarded at "an evening of mathematical recreation". Broad participation was encouraged by making half the problems widely accessible. Solutions to the problems appear in their book.
Beyond mathematics, he enjoyed hill-walking and sang in his church choir
Choir
A choir, chorale or chorus is a musical ensemble of singers. Choral music, in turn, is the music written specifically for such an ensemble to perform.A body of singers who perform together as a group is called a choir or chorus...
. He served as a Liberal
Liberal Party (UK)
The Liberal Party was one of the two major political parties of the United Kingdom during the 19th and early 20th centuries. It was a third party of negligible importance throughout the latter half of the 20th Century, before merging with the Social Democratic Party in 1988 to form the present day...
councillor on Liverpool City Council
Liverpool City Council
Liverpool City Council is the governing body for the city of Liverpool in Merseyside, England. It consists of 90 councillors, three for each of the city's 30 wards. The council is currently controlled by the Labour Party and is led by Joe Anderson.-Domain:...
from 1974 to 1978.
He passed away suddenly of a suspected heart attack on 30 January 2011.
External Links
- Vladimir Arnold (1995) "The geometry of spherical curves and the algebra of quaternions", translated by Ian Porteous, Russian Mathematical Surveys 50:1–68.
- D.R.J. Chillingworth Review of Geometric Differentiation
- Peter R. Law Review of Clifford Algebras and the Classical Groups.
- Ian Porteous (2005) Mathematical Education on Merseyside.
- Ian Porteous corrections to Clifford Algebras and the Classical Groups