Norman L. Biggs

For the Welsh rugby player, see Norman Biggs.

Norman Linstead Biggs (born 2 January 1941) is a leading British mathematician focusing on discrete mathematics and in particular algebraic combinatorics.[1]

Education

Biggs was educated at Harrow County Grammar School and then studied mathematics at Selwyn College, Cambridge. In 1962, Biggs gained first-class honours in his third year of the University's undergraduate degree in mathematics.[2]

Career

He was a lecturer at University of Southampton, lecturer then reader at Royal Holloway, University of London, and Professor of Mathematics at the London School of Economics. He has been on the editorial board of a number of journals, including the Journal of Algebraic Combinatorics. He has been a member of the Council of the London Mathematical Society.

He has written 12 books and over 100 papers on mathematical topics, many of them in algebraic combinatorics and its applications. He became Emeritus Professor in 2006 and continue to teach History of Mathematics in Finance and Economics for undergraduates. He is also Vice-President of the British Society for the History of Mathematics.

Family

Biggs married Christine Mary Farmer in 1975 and has one daughter Clare Juliet born in 1980.

Interests and Hobbies

Biggs' interests include computational learning theory, the history of mathematics and historical metrology. Since 2006, he has been an Emeritus Professor at the London School of Economics.

Biggs hobbies consist of writing about the history of weights and scales. He currently holds the position of Chair of the International Society of Antique Scale Collectors (Europe), and a member of the British Numismatic Society.

Work

Mathematics

In 2002, Biggs wrote the second edition of Discrete Mathematics breaking down a wide range of topics into a clear and organised style. Biggs organised the book into four major sections; The Language of Mathematics, Techniques, Algorithms and Graphs, and Algebraic Methods. This book was an accumulation of Discrete Mathematics, first edition, textbook published in 1985 which dealt with calculations involving a finite number of steps rather than limiting processes. The second edition added nine new introductory chapters; Fundamental language of mathematicians, statements and proofs, the logical framework, sets and functions, and number system. This book stresses the significance of simple logical reasoning, shown by the exercises and examples given in the book. Each chapter contains modelled solutions, examples, exercises including hints and answers. Biggs, Norman L. (2002). Discrete Mathematics: Second edition. 

Algebraic Graph Theory

In 1974, Biggs published Algebraic Graph Theory which articulates properties of graphs in algebraic terms, then works out theorems regarding them. In the first section, he tackles the applications of linear algebra and matrix theory; algebraic constructions such as adjacency matrix and the incidence matrix and their applications are discussed in depth. Next, there is and wide-ranging description of the theory of chromatic polynomials. The last section discusses symmetry and regularity properties. Biggs makes important connections with other branches of algebraic combinatorics and group theory.[3]

Computational Learning Theory

In 1997, N. Biggs and M. Anthony wrote a book titled Computational Learning Theory: an Introduction. Both Biggs and Anthony focused on the necessary background material from logic, probability, and complex theory. This book is an introduction to computational learning.

History of Mathematics

Biggs contributed to thirteen journals and books developing topics such as the four-colour conjecture, the roots/history of combinatorics, calculus, Topology on the 19th century, and mathematicians.[4] In addition, Biggs examined the ideas of William Ludlam, Thomas Harriot, John Arbuthnot, and Leonhard Euler.[5]

Chip-Firing Game

Abstract

The chip-firing game has been around for less than 20 years. It has become an important part of the study of structural combinatorics. The set of configurations that are stable and recurrent for this game can be given the structure of an abelian group. In addition, the order of the group is equal to the tree number of the graph.[6][7]

Let graph G is a finite, simple, connected graph without loops such that G = (V, E) where V = {1, 2,..., n} and E = {e1, e2,..., em}. Let deg (vi) be the degree of vertex i where i = {v∈V(G)}. Each vertex of the graph will be assigned a nonnegative integer S(vi). Let S be the chip configurations of the game. S will show us how many chips are available at vi. A move will consist of selecting a vertex vj which has at least as many chips on it, as its degree; thus S(v) ≥ deg (v). One chip will be fired along each of its incident edges. Each time vj is fired, it loses its degree in chips. (i.e. If the deg(v) = 2 when v is fired, v loses 2 chips.) Each of vj neighbouring vertices will gain one chip per edge incident with vj. Let v(v, w) be the number of edges joining vj to w. Let the value of x(v) be the number of times v is fired in the sequence of moves.

After the sequence of firing s, the new chip configurations, s' is given by:

s'(v) = s(v) – x(v)deg(v) + ∑ x(v) v(v, w)

Example:

G:

Possible Firing Sequence:

Biggs's variant of the chip fire game is one where a vertex q is allowed to go into debt. In other words, it is allowed to be a negative integer unlike all the other vertices. This vertex q is called the bank. The bank does not fire. It just sits there collecting chips. Eventually, q will accumulate so many chips that no other vertex can fire which will make the configuration stable. Once the configuration is stable, vertex q can fire chips to neighbouring vertices to jump start the 'economy'. The bank can only fire if and only if the current configuration is stable.

Thus, in this alternate game, a configuration s is an integer-valued function satisfying:

s(q) = – ∑ s(v) ≤ 0 where s(v) ≥ 0 and not equal to q

We define a stable chip configuration to be:

0 ≤ s(v) < deg(v) (v ≠ q)

Publications

Summary of Biggs' published Books on Mathematics

Summary of Biggs' latest published Papers on Mathematics

2000

2001

2002

2004

2005

2007

2008

2009

2010

2011

For other published work on the history of mathematics, please see.[8]

See also

References

  1. ↑ Norman L. Biggs at DBLP Bibliography Server.
  2. ↑ "Norman Linstead Biggs". UK: London School of Economics. Retrieved 29 April 2013.
  3. ↑ "Algebraic Graph Theory". UK: Cambridge Mathematical Library. Retrieved 15 April 2014.
  4. ↑ "Personal Details". UK: London School of Economics. Retrieved 15 April 2014.
  5. ↑ "Thomas Harriot". UK: London School of Economics. Retrieved 15 April 2014.
  6. ↑ Biggs, Norman L. (25 June 1997). "Chip-Firing and the Critical Group of a Graph" (PDF). Journal of Algebraic Combinatorics: 25–45. Retrieved 10 May 2014.
  7. ↑ wikidot. "Chip-firing references". Retrieved 19 May 2014.
  8. ↑ "Contributions to Mathematics". UK: London School of Economics. Retrieved 15 April 2014.

External links

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