Peter McMullen

Not to be confused with Peter McMullin.

Peter McMullen (born 11 May 1942)^[1] is a British mathematician, a professor emeritus of mathematics at University College London.^[2]

Peter McMullen
Born	(1942-05-11) 11 May 1942 (age 82)
Nationality	British
Alma mater	Trinity College, Cambridge
Known for	Upper bound theorem, McMullen problem
Scientific career
Fields	Discrete geometry
Institutions	Western Washington University (1968–1969) University College London

Education and career

McMullen earned bachelor's and master's degrees from Trinity College, Cambridge, and studied at the University of Birmingham, where he received his doctorate in 1968.^[3] He taught at Western Washington University from 1968 to 1969.^[4] In 1978 he earned his Doctor of Science at University College London where he still works as a professor emeritus. In 2006 he was accepted as a corresponding member of the Austrian Academy of Sciences.^[5]

Contributions

McMullen is known for his work in polyhedral combinatorics and discrete geometry, and in particular for proving what was then called the upper bound conjecture and now is the upper bound theorem. This result states that cyclic polytopes have the maximum possible number of faces among all polytopes with a given dimension and number of vertices.^[6] McMullen also formulated the g-conjecture, later the g-theorem of Louis Billera, Carl W. Lee, and Richard P. Stanley, characterizing the f-vectors of simplicial spheres.^[7]

The McMullen problem is an unsolved question in discrete geometry named after McMullen, concerning the number of points in general position for which a projective transformation into convex position can be guaranteed to exist. It was credited to a private communication from McMullen in a 1972 paper by David G. Larman.^[8]

He is also known for his 1960s drawing, by hand, of a 2-dimensional representation of the Gosset polytope 4₂₁, the vertices of which form the vectors of the E8 root system.^[9]

Awards and honours

McMullen was invited to speak at the 1974 International Congress of Mathematicians in Vancouver; his contribution there had the title Metrical and combinatorial properties of convex polytopes.^[10]

He was elected as a foreign member of the Austrian Academy of Sciences in 2006.^[11] In 2012 he became an inaugural fellow of the American Mathematical Society.^[12]

Selected publications

Research papers

• McMullen, P. (1970), "The maximum numbers of faces of a convex polytope", Mathematika, 17 (2): 179–184, doi:10.1112/s0025579300002850, MR 0283691, S2CID 122025424.
• —— (1975), "Non-linear angle-sum relations for polyhedral cones and polytopes", Mathematical Proceedings of the Cambridge Philosophical Society, 78 (2): 247–261, Bibcode:1975MPCPS..78..247M, doi:10.1017/s0305004100051665, MR 0394436, S2CID 63778391.
• —— (1993), "On simple polytopes", Inventiones Mathematicae, 113 (2): 419–444, Bibcode:1993InMat.113..419M, doi:10.1007/BF01244313, MR 1228132, S2CID 122228607.

Survey articles

• ——; Schneider, Rolf (1983), "Valuations on convex bodies", Convexity and its applications, Basel: Birkhäuser, pp. 170–247, MR 0731112. Updated as "Valuations and dissections" (by McMullen alone) in Handbook of convex geometry (1993), MR1243000.

Books

• ——; Shephard, Geoffrey C. (1971), Convex Polytopes and the Upper Bound Conjecture, Cambridge University Press.
• ——; Schulte, Egon (2002), Abstract regular polytopes, Encyclopedia of Mathematics and its Applications, vol. 92, Cambridge: Cambridge University Press, doi:10.1017/CBO9780511546686, ISBN 0-521-81496-0, MR 1965665, S2CID 115688843.

References

1. Peter McMullen, Peter M. Gruber, retrieved 2013-11-03.
2. UCL IRIS information system, accessed 2013-11-05.
3. McMullen, Peter; Schulte, Egon (12 December 2002), Abstract and regular polytopes, ISBN 9780521814966, retrieved 2022-05-11
4. Peter McMullen Collection, 1967-1968, Special Collections, Wilson Library, Western Washington University, retrieved from worldcat.org 2013-11-03.
5. "Austrian Academy of Sciences: Peter McMullen". Retrieved 2022-05-11.
6. Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Springer, p. 254, ISBN 9780387943657, Finally, in 1970 McMullen gave a complete proof of the upper-bound conjecture – since then it has been known as the upper bound theorem. McMullen's proof is amazingly simple and elegant, combining to key tools: shellability and h-vectors.
7. Gruber, Peter M. (2007), Convex and discrete geometry, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 336, Berlin: Springer, p. 265, ISBN 978-3-540-71132-2, MR 2335496, The problem of characterizing the f-vectors of convex polytopes is ... far from a solution, but there are important contributions towards it. For simplicial convex polytopes a characterization was proposed by McMullen in the form of his celebrated g-conjecture. The g-conjecture was proved by Billera and Lee and Stanley.
8. Larman, D. G. (1972), "On sets projectively equivalent to the vertices of a convex polytope", The Bulletin of the London Mathematical Society, 4: 6–12, doi:10.1112/blms/4.1.6, MR 0307040
9. "A picture of the E8 root system". American Institute of Mathematics. Retrieved 2022-05-11.
10. ICM 1974 proceedings Archived 2017-12-04 at the Wayback Machine.
11. Awards, Appointments, Elections & Honours, University College London, June 2006, retrieved 2013-11-03.
12. List of AMS fellows, retrieved 2013-11-03.