Ian Robertson Porteous (9 October 1930 – 30 January 2011) was a Scottish mathematician at the

University of Liverpool The University of Liverpool (abbreviated UOL) is a Public university, public research university in Liverpool, England. Founded in 1881 as University College Liverpool, Victoria University (United Kingdom), Victoria University, it received Ro ...

and an educator on Merseyside. He is best known for three books on

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

and modern algebra. In

Liverpool Liverpool is a port City status in the United Kingdom, city and metropolitan borough in Merseyside, England. It is situated on the eastern side of the River Mersey, Mersey Estuary, near the Irish Sea, north-west of London. With a population ...

he and

Peter Giblin Peter John Giblin (10 July 1943) 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 Univer ...

are known for their

registered charity A charitable organization or charity is an organization whose primary objectives are philanthropy and social well-being (e.g. educational, Religion, religious or other activities serving the public interest or common good). The legal definitio ...

''Mathematical Education on Merseyside'' which promotes enthusiasm for mathematics 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 Porteous Norman Walker Porteous (9 September 1898 in Haddington, East Lothian, Scotland – 3 September 2003 in Edinburgh, Scotland) was a noted theologian and writer on Old Testament issues, and the last surviving military officer of the First World ...

(later a theologian and Old Testament academic), from Crossgates, Fife and May Hadwen Robertson of

Kirkcaldy, Fife Kirkcaldy ( ; ; ) is a town and former royal burgh in Fife, on the east coast of Scotland. It is about north of Edinburgh and south-southwest of Dundee. The town had a recorded population of 49,460 in 2011, making it Fife's second-largest s ...

. He attended

George Watson's College George Watson's College is a co-educational Private schools in the United Kingdom, private day school in Scotland, situated on Colinton Road, in the Merchiston area of Edinburgh. It was first established as a Scottish education in the eighteenth ...

in Edinburgh, and the

University of Edinburgh The University of Edinburgh (, ; abbreviated as ''Edin.'' in Post-nominal letters, post-nominals) is a Public university, public research university based in Edinburgh, Scotland. Founded by the City of Edinburgh Council, town council under th ...

, obtaining his first mathematical degree in 1952. After a time in

national service National service is a system of compulsory or voluntary government service, usually military service. Conscription is mandatory national service. The term ''national service'' comes from the United Kingdom's National Service (Armed Forces) Act ...

, he took up study at

Trinity College, Cambridge Trinity College is a Colleges of the University of Cambridge, constituent college of the University of Cambridge. Founded in 1546 by King Henry VIII, Trinity is one of the largest Cambridge colleges, with the largest financial endowment of any ...

. Porteous wrote his thesis ''Algebraic Geometry'' under W.V.D. Hodge and

Michael Atiyah Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded the ...

University of Cambridge The University of Cambridge is a Public university, public collegiate university, collegiate research university in Cambridge, England. Founded in 1209, the University of Cambridge is the List of oldest universities in continuous operation, wo ...

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 University Columbia University in the City of New York, commonly referred to as Columbia University, is a Private university, private Ivy League research university in New York City. Established in 1754 as King's College on the grounds of Trinity Churc ...

in New York, Porteous was influenced by

Serge Lang Serge Lang (; May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the i ...

. He continued to do research on

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

s in

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

. In 1971 his article "The normal singularities of a submanifold" was published in

Journal of Differential Geometry The ''Journal of Differential Geometry'' is a peer-reviewed scientific journal of mathematics published by International Press on behalf of Lehigh University in 3 volumes of 3 issues each per year. The journal publishes an annual supplement in book ...

5:543–64. It was concerned with the smooth embeddings of an ''m''-manifold in R^''n''. In 1969 Porteous published ''Topological Geometry'' with Van Nostrand Reinhold and Company. It was reviewed in

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 ...

by J. Eells, who interpreted it as a three-term textbook for a sequence in

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

geometric algebra In mathematics, a geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric pr ...

, 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 was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

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:

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...

s, 13:The classical groups, 15:

Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of a distinguished subspace. As -algebras, they generalize the real number ...

s, 16:

Spin group In mathematics the spin group, denoted Spin(''n''), page 15 is a Lie group whose underlying manifold is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathbb_2 \to \o ...

s, 17:

Conjugation Conjugation or conjugate may refer to: Linguistics *Grammatical conjugation, the modification of a verb from its basic form *Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics *Complex conjugation, the change o ...

, 20:

Topological spaces In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...

, 21:

Manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

s, 22:

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

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

. The subtitle, "for the intelligence of curves and surfaces" indicates its extent in the

differential geometry of curves Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the ...

and

differential geometry of surfaces In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth manifold, smooth Surface (topology), surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensiv ...

. 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 The Federal Aviation Administration (FAA) is a U.S. federal government agency within the U.S. Department of Transportation that regulates civil aviation in the United States and surrounding international waters. Its powers include air traffi ...

. 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 In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are e ...

ridge A ridge is a long, narrow, elevated geomorphologic landform, structural feature, or a combination of both separated from the surrounding terrain by steep sides. The sides of a ridge slope away from a narrow top, the crest or ridgecrest, wi ...

germ Germ or germs may refer to: Science * Germ (microorganism), an informal word for a pathogen * Germ cell, cell that gives rise to the gametes of an organism that reproduces sexually * Germ layer, a primary layer of cells that forms during embry ...

s, and

cusp A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth. Cusp or CUSP may also refer to: Mathematics * Cusp (singularity), a singular point of a curve * Cusp catastrophe, a branch of bifu ...

s. Porteous has suggestions for readers wanting to know more about

singularity theory In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...

. 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 Igorevich Arnold (or Arnol'd; , ; 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. He is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, and contributed to s ...

on spherical curves. In fact, Porteous had translated Arnold's paper from the Russian.

Death and legacy

Porteous' commitment to

mathematics education In contemporary education, mathematics education—known in Europe as the didactics or pedagogy of mathematics—is the practice of teaching, learning, and carrying out Scholarly method, scholarly research into the transfer of mathematical know ...

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 organise a Challenge competition for first and second formers in secondary school. By 1989 they were drawing 3,500 participants each year. The competition was held over two weekends in the Spring Term. Students considered six questions in 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, Porteous enjoyed hill-walking and sang in his church

choir A choir ( ), also known as a chorale or chorus (from Latin ''chorus'', meaning 'a dance in a circle') is a musical ensemble of singers. Choral music, in turn, is the music written specifically for such an ensemble to perform or in other words ...

. He served as a Liberal councillor on

Liverpool City Council Liverpool City Council is the Local government in England, local authority for the City status in the United Kingdom, city of Liverpool in Merseyside, England. Liverpool has had a local authority since 1207, which has been reformed on numerous ...

from 1974 to 1978. He died suddenly of a suspected

heart attack A myocardial infarction (MI), commonly known as a heart attack, occurs when Ischemia, blood flow decreases or stops in one of the coronary arteries of the heart, causing infarction (tissue death) to the heart muscle. The most common symptom ...

on 30 January 2011.

Selected publications

* *Corrections to ''Clifford Algebras and the Classical Groups''
* * * Vladimir Arnold (1995) "The geometry of spherical curves and the algebra of quaternions", translated by Ian Porteous, ''Russian Mathematical Surveys'' 50:1–68.

References

External links

* * Hodge Institute (2011
Ian Porteous
* Peter Giblin (2012
In Memoriam, Ian R. Porteous 9 October 1930 – 30 January 2011
''Journal of Singularities'', Volume 6. {{DEFAULTSORT:Porteous, Ian R. 1930 births 2011 deaths People from Fife Academics of the University of Liverpool Alumni of the University of Edinburgh Alumni of Trinity College, Cambridge Differential geometers Liberal Party (UK) councillors in Liverpool People educated at George Watson's College Scottish mathematicians British textbook writers