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The Australian National University

Dr Geoffrey Campbell

PhD (ANU); Grad Dip Internet & Web (RMIT Uni)
ANU College of Science
T: +61 408 316 592

Areas of expertise

  • Algebra And Number Theory 010101
  • Combinatorics And Discrete Mathematics (Excl. Physical Combinatorics) 010104
  • Real And Complex Functions (Incl. Several Variables) 010111
  • Pure Mathematics Not Elsewhere Classified 010199
  • Numerical And Computational Mathematics Not Elsewhere Classified 010399
  • Earth Sciences Not Elsewhere Classified 049999

Research interests

  • q series identities (Basic hypergeometric series),
  • Tiling patterns arising from theory of quasicrystals,
  • Dirichlet series analogues of q series,
  • Dirichlet series and Riemann zeta functions,
  • Theory of Partitions,
  • Arithmetical functions and divisor functions,
  • Visible Point Vector (vpv) identities (includes lattice points in regions),
  • Ordinary hypergeometric series,
  • Number Theory in general including Diophantine equations,
  • Voting Methodologies,
  • Combinatorial objects in Enumerative Word problems,
  • Schur functions, and the Google search algorithms.

Researcher's projects

I have two research monographs and quite a few number theory research papers as work in progress.

Monograph 1: Vector Partitions

In my papers of the 1990s up to 2000 I introduced new combinatorial identities - these are cited in the WolframMathWorld online encyclopaedia under 'Visible Point Vector Identities'. See http://mathworld.wolfram.com/VisiblePointVectorIdentity.html. Many of the papers either appeared out of chronology or were accepted by referees but later withdrawn as there were page charges imposed. The work has progressed beyond the papers themselves and recent research on computational identities by Professor Jon Borwein at University of Newcastle can be applied to give new and interesting results. Also the identities appear to have applications in Vector Partition theory extending the works of Professor George E Andrews on integer partitions.

Monograph 2: Dirichlet Series Analogues of q Series

I discovered a transform that maps the classical basic hypergeometric series identities onto classes of Dirichlet summations involving Riemann zeta functions and divisor functions. In my 2006 paper (see http://www.springerlink.com/content/1nk5046814024n36/) I also discovered that the coefficients in the new identities had interpretations in terms of tiling patterns known to the theory of quasicrystals. There is a team in Germany led by Professor Michael Baake specialising in these quasicrystal tiling researches.

The theory underlying the new Dirichlet series analogues, goes into the theory of aperiodic tiling patterns such as those that occur in the seminal works of the 2011 Nobel Prize works of Professor Dan Shechtman, who discovered quasicrystals in nature. This was a paradigm shift in the science of crystallography.

An extended list of new Dirichlet series analogue identities resultant from the transform has not yet been published, but will be included in this monograph. The quasicrystal tilings that enumerate coefficients in the new Dirichlet series analogues of the q series are dependent on the theory underlying Coincidence Site Lattices. My colloquium in 2011 highlighted some of these tiling patterns arising from the Dirichlet series analogues of the basic hypergeometric sums. (see http://www.amsi.org.au/index.php/events-mainmenu/agr-events/737-drichlet-analogues-of-q-series-and-some-tiling-patterns-from-quasicrystals)

Features of this are:-

• The concept of a Coincidence Site Lattice (CSL) arises in the crystallography of grain and twin boundaries.
• Different domains of a crystal across a boundary are related by having a sublattice (of full rank) in common. This is the CSL.
• It can be viewed as the intersection of a lattice with a rotated copy of itself, where the points in common form a sublattice of finite index.
• Until recently, CSLs have been investigated only for true lattices or for crystallographic packings, for example cubic or hexagonal crystals.

Note: I gave colloquia at LaTrobe University on these two monograph subject areas in March and September 2011.


Papers arising from LinkedIn Number Theory Group

As manager of the online LinkedIn Number Theory Group, I have about 30 papers arising from discussions in that forum, many papers of which I am co-authoring with members of the group. As these papers are drafted and settle, they will appear below with a link to the arxive or to the Journal or other publication point.

Publications

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Updated:  16 December 2017 / Responsible Officer:  Director (Research Services Division) / Page Contact:  Researchers