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Two New Roles in Norwich

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People, Places, and Mathematics

Part of the book series: Springer Biographies ((SPRINGERBIOGS))

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Abstract

During the 1990s the Physics part of the School of Mathematics and Physics was closed, primarily because of falling student intakes. The remaining School of Mathematics was not without its troubles, and by the late 1990s was in an extremely difficult financial position. I was largely inattentive to all this, concentrating on my own teaching and research roles. This only changed when there was a discussion about who should take over as Head of School. The process was managed by the Dean of Science, who would take soundings and if necessary hold a vote.

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Notes

  1. 1.

    The notion of threshold concepts was brought into education through the work of Meyer and Land [201], who described it as ‘passing through a portal, from which a new perspective opens up, allowing things formerly not perceived to come into view. This permits a new and previously inaccessible way of thinking about something.’ The identification of such concepts is a real difficulty in using these ideas, but Mathematics is certainly filled with them.

  2. 2.

    Anatole Katok was one of the major figures in the development of dynamics in the second half of the twentieth Century. Several obituaries detailing something of his contribution and life have appeared, including one by his former student Hasselblatt [146] in Ergodic Theory and Dynamical Systems (one of the journals he was involved in founding) and a short note in the Moscow Mathematics Journal [4].

  3. 3.

    Andrew was a Chemist who worked in the development of anti-cancer drugs, cisplatin in particular [139].

  4. 4.

    Nicolas d’Oresme was a fourteenth Century philosopher who made contributions in multiple areas. His (tenuous) connection with ergodic theory arises because he wrote about the idea of a model of the solar system in which the periods of the orbits of different planets could be incommensurable, which with an optimistic eye presages the concept of unique ergodicity.

  5. 5.

    This contrast between extremely bad behaviour at one end of the figure repeated on page 258 and (relatively) regular behaviour at the other raises the question of whether there is a different point of view that might put them on the same footing. This would be a notion of dynamical zeta function defined on the adeles, where the singularities that are too close together on the complex plane are teased apart and Tauberian relationships between analytic behaviour and growth rates are recovered.

  6. 6.

    They are extreme because the approach of Kitchens and Schmidt in part uses character theory to relate an algebraic dynamical system defined by a countable group \(\varGamma \) acting by automorphisms of a compact abelian group X to the structure of a \(\mathbb {Z}[\varGamma ]\) module for the dual group of X. Here \(\mathbb {Z}[\varGamma ]\) denotes the ‘integral group ring’ of the acting group \(\varGamma \). For actions of \(\mathbb {Z}^d\), this group ring is a ring of reasonable ‘size’: A notion of size called the ‘Krull dimension’ is finite. The same holds for the integral group ring of the additive group \(\mathbb {Q}^d\), meaning that in this context algebraic \(\mathbb {Q}^d\) actions were a modest next step in understanding. However, it is a real step because \(\mathbb {Z}[\mathbb {Z}^d]\) is Noetherian (another ‘smallness’ property a ring may have) while \(\mathbb {Z}[\mathbb {Q}^d]\) is not. In contrast \(\mathbb {Q}^{\times }\) is an enormous group, and its integral group ring has infinite Krull dimension.

  7. 7.

    My difficulty was related to the following kind of problem. Counting the number of solutions to an equation like \((2^n-1)g=g\) in a finite abelian group G for all \(n\geqslant 1\) cannot distinguish between a factor of G that looks like \(\mathbb {Z}/p^2\mathbb {Z}\) or a factor that looks like \(\mathbb {Z}/p\mathbb {Z}\oplus \mathbb {Z}/p\mathbb {Z}\) for a prime p if p has the strange property that whenever p divides \(2^n-1\) we also have \(p^2\) dividing \(2^n-1\). It is known that the primes 1093 and 3511 have this property, and no others smaller than \(4\times 10^{12}\) were known at the time (the search has since been extended to \(6.7\times 10^{15}\)). A standard sort of heuristic argument, based on the optimistic belief that certain arithmetically defined quantities behave more or less randomly unless there is an obvious reason for them not to do so, suggests there should be infinitely many such primes. They are called Wieferich primes as a result of his proof that if \(x,y,z\) are integers with \(x^p+y^p+z^p=0\) with the product xyz not divisible by p, then p must be a Wieferich prime [352].

  8. 8.

    An ‘equation’ in a finite group means something like \(x^2y^3x^{-1}z=e\), where e denotes the identity. The ‘solution’ is to identify the triples \((x,y,z)\) with \(x,y,z\) in the group that satisfy the equation. In an abelian group we would conventionally write this in additive rather than multiplicative notation, so it would become \(2x+3y-x+z=0\), which simplifies a little to \(x+3y-z=0\). Several different equations can be studied at the same time, giving ‘simultaneous’ equations. The question thrown up was this: If you are told the number of solutions to all such simultaneous equations, can you work out the structure of the group?

  9. 9.

    The specific statement I needed was this: If G is a finite abelian group and p is a prime, then the number of solutions to the equations \(pg=e,p^2g=e,\dots ,p^ng=e\) determine the p-primary component of G if and only if the number of elements in G is not divisible by \(p^{2n+2}\). The ‘if and only if’ means that this bound—the \((2n+2)\)—is the exact answer, and one cannot do better.

  10. 10.

    Tony graduated in 2010, with external examiner Prof. Samir Siksek from the University of Warwick [129].

  11. 11.

    Sawian graduated in 2010, with external examiner Prof. Oliver Jenkinson, Queen Mary University of London [160].

  12. 12.

    Apisit graduated in 2010, with external examiner Prof. Franco Vivaldi, Queen Mary University of London [239].

  13. 13.

    It is of course quite possible that I imagined these pleasant visits as part of the Bielefeld-Verschwörung.

  14. 14.

    https://www.uea.ac.uk/library/the-independent-climate-change-email-review.

  15. 15.

    https://publications.parliament.uk/pa/cm200910/cmselect/cmsctech/387/387i.pdf.

  16. 16.

    https://www.legislation.gov.uk/ukpga/2000/36/contents.

  17. 17.

    Transcript from https://www.cfoi.org.uk/.

  18. 18.

    The repulsion concept here is related to a phenomenon that is widely observed but only understood in rare instances. The simplest example is Catalan’s conjecture that the only consecutive powers are 8 and 9. More formally the conjecture is that the only integer solutions to the equation \(x^a-y^b=1\) with \(a,b>1\) and \(x,y>0\) are \(x=3,a=2,y=2,b=3\) corresponding to \(3^2-2^3=1\). This was proved by Mihăilescu [203] in 2004. More generally, Pillai conjectured in 1945 that for any \(k\geqslant 1\) the equation \(x^a-y^b=k\) has only finitely many integer solutions. Equivalently, powers ‘repel’ each other in that different powers of integers cannot be close together infinitely open. This problem remains open, with active research on more constrained versions.

  19. 19.

    Robert graduated in 2015, with external examiner Prof. Sanju Velani, University of York [261].

  20. 20.

    Matthew graduated in 2013, with external examiner Prof. Chris Smyth, University of Edinburgh [283].

  21. 21.

    Stefanie graduated in 2015, with external examiner Prof. Oliver Jenkinson, Queen Mary University of London [369].

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Ward, T. (2023). Two New Roles in Norwich. In: People, Places, and Mathematics. Springer Biographies. Springer, Cham. https://doi.org/10.1007/978-3-031-39074-6_12

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