Geoffrey Campbell

PhD (ANU), GDip Internet and Web Comp (RMIT University)
Long Term Campus Visitor

Research interests

In 2019 I wrote a paper generalizing the q-binomial theorem into a Euclidean n-space formula with a view to formalising an approach to vector partition theory. (See https://arxiv.org/abs/1906.07526) The substance of an updated version of this paper has led to my writing the below detailed book monograph on Vector Partitions and the Visible Points, which has since been renamed in its present draft version to Partitions, Visible Point Vectors and Ramanujan Functions.

In late 2020 I signed with Chapman and Hall Publishers to write a book monograph presently entitled Partitions, Visible Point Vectors and Ramanujan Functions. My motivation was and is to cover classical Integer Partitions, partitions in Statistical Mechanics Solved Models such as the Hard Hexagon, and bring this to a research level with the Visible Point Vector identities I found some years ago. There are self-contained chapters at research level. Along the way, we cover contemporary theory involving Plane Partitions, Ono's exact Partition Formula, Polylogarithms, generalized Euler Sums and Mordell-Wittam-Tornheim ensembles. I expect the book will be 300 to 400 pages and be completed prior to December 2022.

My high level table of contents, from early 2021 is updated here to February 2022, showing how the book is progressing to date:

Partitions, Visible Point Vectors and Ramanujan Functions

Preface
            1. About this book and its academic context
            2. The author’s vision and motivations
            3. Intended audience
Chapter 1. Historical background - The range of literature
            1. Partitions as grown from the Ramanujan works
            2. Andrews’ ”The Theory of Partitions” and ”Integer Partitions” with Eriksson
            3. The 2021 book ”G. E. Andrews 80 Years of Combinatory Analysis”
            4. Basic Hypergeometric Series
            5. Lattice Sums in Chemistry molecular structures
            6. Polylogarithms and computational research related results
            7. Partition theory in Statistical Mechanics and Theoretical Physics
Chapter 2. General introduction and context for partitions
Chapter 3. Integer partitions and their generating functions
            1. Preview
            2. Euler’s approach to partitions
            3. Euler’s partition identity
            4. Euler’s pentagonal number theorem
            5. The q-binomial Theorem
            6. The Jacobi Triple Product
            7. Two Identities from Gauss
            8. The Watson Quintuple Product
            9. The Heine fundamental transformation
            10. The q-analogues of Gauss’s Theorem and Kummer’s Theorem
            11. Euler-Pair theorem
            12. Rogers-Ramanujan identities
Chapter 4. Congruence properties of partitions
            1. Ferrers graphs
            2. Durfee squares
Chapter 5. Gaussian polynomials
            1. Combinatorial interpretations
                        1.1. Balls into bins
                        1.2. Reflection
            2. Analogs of Pascal’s identity
            3. q-binomial theorem analogue
            4. Gaussian polynomials in the Theory of Partitions
Chapter 6. Plane partitions
            1. Definitions of Plane Partitions
            2. Lozenge tilings of a hexagon.
            3. Generating functions for plane partitions
            4. Exercises
Chapter 7. Formulas for partition functions
            1. Asymptotic formulas for integer partitions
                        1.1. The Circle Method of Hardy and Ramanujan
                        1.2. Refinements due to Rademacher, and Szekeres-Richmond
            2. Ono’s exact formula for partition functions
            3. Exercises
Chapter 8. The partition function in Statistical Mechanics
            1. Baxter’s Hard Hexagon Model solved exactly by Rogers-Ramanujan identities
            2. Hard Hexagon Model Regime II identities from Baxter
            3. Outline of proofs of Baxter’s Regime II Conjectures
            4. Transforms between Baxter’s Regime II and Regime III.
            5. The partition function of the Hard Hexagon Model
                       5.1. Solution
            6. Exercises
Chapter 9. Vector partitions and their generating function identities
            1. Defining Vector Grids and resulting Partition Grids
            2. A theorem for 2D partitions into exactly two parts.
            3. Visible Point Vector partitions and their generating functions
            4. Exercises
Chapter 10. Integer Partitions to Vector Partitions
            1. Introduction
            2. Setting up a higher dimensional approach to q-binomial theorem
            3. Proof of the n-space q-binomial theorem.
            4. Finite Product 2D, 3D and nD cases of q-binomial extensions.
            5. Exercises
Chapter 11. Weighted Vector Partitions as hybrid n-space variations
            1. Examples of hybrid variations for identities generating weighted vector partitions
            2. Proof of the n-space hybrid q-binomial theorem.
            3. Exercises
Chapter 12. Functional Equations for n-space Vector Partitions
            1. An n-space q-binomial functional equation.
            2. Exercises
Chapter 13. Binary and n-ary Partitions and their Vector Generalizations
            1. Introductory Elementary Ideas for Binary Partitions.
Chapter 14. Visible Point Vector Identities in the first Hyperquadrant
            1. Introducing the VPV identities.
            2. Deriving the 2D first quadrant VPV identity.
            3. Deriving the n-dimensional first hyperquadrant VPV identity.
            4. Diversionary note on the abc Conjecture.
            5. Exercises
Chapter 15. Visible Point Vector Identities in Hyperpyramid lattices
            1. VPV identities in square hyperpyramid regions.
            2. Deriving 2D VPV identities in square hyperpyramid regions.
            3. Deriving 3D VPV identities in square hyperpyramid regions.
            4. VPV identities in nD square hyperpyramid regions.
            5. Exercises
Chapter 16. Polylogarithms, Euler Sums and Mordell-Wittam-Tornheim functions.
            1. Early history of the Dilogarithm
            2. The Trilogarithm function
            3. The Polylogarithm function
            4. Mordell-Tornheim-Witten ensembles
            5. Finite Euler Sums
            6. Exercises
Chapter 17. Visible Point Vector identities related to particular Euler sum values
            1. Polylogarithms near trivial zeroes of the Riemann zeta function
            2. Exercises
Chapter 18. Visible Point Vector Identities in Skewed Hyperpyramid lattices
            1. Visible Point Vector identities in asymmetric hyperpyramid regions.
                       1.1. A more general hyperpyramid theorem.
            2. Exercises
Chapter 19. The Ramanujan trigonometric function and visible point identities
            1. Introductory remarks
            2. Dirichlet series generating functions
            3. Proof of a Theorem
            4. A new Jordan Totient generating function, and some related results
            5. Further multidimensional formulae
            6. Application of Jacobi theta series to the generalized summations.
            7. Exercises
Chapter 20. Other Non-weighted n-space Vector Partition Theorems
            1. A binary partition 2 -space variation of extended q-binomial theorem.
            2. Exercises
Chapter 21. Determinants, Bell Polynomial Expansions for Vector Partitions
            1. Some reference cases of the determinants in this book.
            2. Calculus of Determinant Evaluation
            3. Exercises
Chapter 22. The 2D and 3D Light Diffusion Models
            1. The 2D Light Diffusion Model in the first quadrant.
            2. Partition Grids for 2D Vector partitions
            3. Partition Grids for unweighted 2D VPVs
            4. Exercises
Chapter 23. Partition Grids for unweighted 2D VPVs II
            1. Exercises
Chapter 24. Partition Grids for unweighted 2D VPVs III
            1. Exercises
Chapter 25. Partition Grids for weighted 2D VPVs IV
            1. Exercises
Chapter 26. The 3D Light Diffusion Model in the first hyperquadrant.
            1. Exercises
Bibliography

Other projects I have:

In 2006 I published an introductory paper for Dirichlet series analogues of q-series, which led me to arithmetical function identities encoding so-named quasicrystals. I give new analogue summations for classical q-series and hypergeometric series summations in terms of Riemann Zeta functions and Jordan Totient functions. This area of research is still being developed, and I am drafting a monograph on this topic as well, all of it sequel to the 2006 paper.

I therefore can say I have research interests in the Theory of Higher Dimensional Partitions, Aperiodic Order, Dynamical Systems, Combinatorics, Discrete Geometry, Number Theory, Quasicrystal tilings and their Dirichlet series functions, and Mathematical Physics, and areas where these theories may overlap.

I am also Manager of the almost 40,000 member LinkedIn Number Theory Group located at https://www.linkedin.com/groups/4510047/. I have posted many brief mathematical problems and news stories at graduate and research levels over the past seven years in that forum. An informative post in that group is http://bit.ly/3a6Wyhc.

I am also an Administrator for the Facebook Group Classical Mathematics presently with about 10,000 members. A typical post in that group is http://bit.ly/2NFtbLn.

Groups

I am a published poet as well as a mathematics person.

POEMS: 
In 2010 I published Words in Common, which is a collection over decades of journal/magazine/anthology published poems partly funded by the Australia Council for the Arts many years ago. The book was officially launched by Professor Kevin Brophy from University of Melbourne, Creative Writing Department. It was edited by Associate Professor Trevor Code from Deakin University.

See a link with excerpts from the poems at http://bit.ly/3jAvlb8.
See a review of Words In Common by Canberra poet Michael Byrne at http://bit.ly/2MRlv8y.

A SELECTION OF MY MATHEMATICAL PAPERS:

27            CAMPBELL, G. B. An interview with Rodney James Baxter, Aust. Math. Soc. Gazette, Volume 47, No1, pp24-32, March 2020. (https://austms.org.au/wp-content/uploads/2020/07/471Web.pdf)

26.           CAMPBELL, G. B. Some n-space q-binomial theorem extensions and similar identities, arXiv:1906.07526v1 [math.NT], Jun 2019. (https://arxiv.org/abs/1906.07526)

25            CAMPBELL, G. B. and ZUJEV, A. The series that Ramanujan misunderstood, arXiv:1610.03693v1 [math.NT], Oct 2016. (https://arxiv.org/abs/1610.03693v1)

24            CAMPBELL, G. B. and ZUJEV, A. On integer solutions to x5 - (x+1)5 - (x+2)5 + (x+3)5 = 5m + 5n, arXiv:1603.00080v1 [math.NT], Feb 2016. (https://arxiv.org/abs/1603.00080v1)

23            CAMPBELL, G. B. and ZUJEV, A. Some equations with features of digit reversal and powers, arXiv:1602.06320v1 [math.NT], Feb 2016. (https://arxiv.org/abs/1602.06320v1)

22            CAMPBELL, G. B. and ZUJEV, A. Gaussian integer solutions for the fifth power taxicab number problem, arXiv:1511.07424v1 [math.NT], Nov 2015. (https://arxiv.org/abs/1511.07424v1)

21            CAMPBELL, G. B. and ZUJEV, A. Variations on Ramanujan's nested radicals, arXiv:1511.06865v1 [math.NT], Nov 2015. (https://arxiv.org/abs/1511.06865v1)

20            CAMPBELL, G. B. and ZUJEV, A. A diophantine sum with factorials, arXiv:1510.03056v2 [math.NT], Oct 2015. (https://arxiv.org/abs/1510.03056v2)

19            CAMPBELL, G. B. The q-Dixon sum Dirichlet series analogue, arXiv:1302.2664v1, Feb 2013. (https://arxiv.org/abs/1302.2664v1)

18            CAMPBELL, G. B. Ramanujan and Eckford Cohen totients from Visible Point Identities, arXiv:1212.2818v1 [math.NT], Dec 2012. (https://arxiv.org/abs/1212.2818v1)

17            CAMPBELL, G. B. D-analogues of q-shifted factorial and the q-Kummer sum, arXiv:1212.2248v1 [math.NT], Dec 2012. (https://arxiv.org/abs/1212.2248v1)

16            CAMPBELL, G. B. Polylogarithm approaches to Riemann Zeta function zeroes, arXiv:1212.2246v1 [math.NT], Dec 2012. (https://arxiv.org/abs/1212.2246v1)

15            CAMPBELL, G. B. Dirichlet series analogues of q-shifted factorial and the q-Kummer sum, Research paper 2003-6, Department of Mathematics, LaTrobe University, 2003.

14            CAMPBELL, G. B. An Euler Product transform applied to q series, Ramanujan J (2006) 12:267-293. (https://doi.org/10.1007/s11139-006-0078-y)

13            CAMPBELL, G. B. A New Class of Identities akin to q-Series in Several Variables, Research paper no (to be determined), Centre for Mathematics and its applications, The Australian National University, 1998.

12            CAMPBELL, G. B. Combinatorial Identities in Number Theory related to q-series and Arithmetical functions, Bull. Austral. Math. Soc., Vol. 58, (1998) pp345-347.

11            CAMPBELL, G. B. On generating functions for vector partitions, Research paper no 55-97, Centre for Mathematics and its applications, The Australian National University, 1997.

10            CAMPBELL, G. B. Visible point vector summations from hypercube and hyperpyramid lattices, Internat. J. Math. & Math. Sci., Vol 21, No 4, 741-748, 1998. (https://www.researchgate.net/publication/26536267_Visible_point_vector_summations_from_hypercube_and_hyperpyramid_lattices)

9              CAMPBELL, G. B. Infinite products over hyperpyramid lattices, Internat. J. Math. & Math. Sci., Vol 23, No 4, 2000, 271-277. (http://downloads.hindawi.com/journals/ijmms/2000/108918.pdf)

8              CAMPBELL, G. B. A closer look at some new identities, Internat. J. Math. & Math. Sci., Vol 21, No 3, 1998, pp581-586. (https://www.researchgate.net/publication/26536244_A_closer_look_at_some_new_identities)

7              CAMPBELL, G. B. Infinite products over visible lattice points, Internat. J. Math. & Math. Sci., Vol 17, No 4, 1994, 637-654. (http://downloads.hindawi.com/journals/ijmms/1994/705467.pdf)

6              CAMPBELL, G. B. A new class of infinite product, and Euler's totient, Internat. J. Math. & Math. Sci., Vol 17, No 4, 1994, 417-422.

5              CAMPBELL, G. B. Formulae with functions exhibiting self-similarity, Research Paper preprint series, Centre for Mathematics and its Applications, The Australian National University, 1993.

4              CAMPBELL, G. B. A generalised formula of Hardy, Int. J. Math. Math. Sci., Vol 17, No 2, 1994, 369-378.

3              CAMPBELL, G. B. Dirichlet summations and products over primes, Internat. J. Math. & Math. Sci., Vol 16, No 2, 1993, 359-372.

2              CAMPBELL, G. B. Multiplicative functions over Riemann zeta function products, J. Ramanujan Soc. 7 No. 1, 1992, 52-63.

1              CAMPBELL, G. B. Generalization of a formula of Hardy, La Trobe University preprints no 79-5, 1979 (written whilst a young student.)