News and opinion articles

• Research and development cuts short-change the nation, The Saturday Paper, September 28, 2024, John Hewson; HTML
• What we miss by not funding maths, Future Campus, June 18, 2024, Jan de Gier; HTML
• If Australia wants innovation it should value expertise more highly, The Australian, December 17, 2021, Jan de Gier; HTML
• New mathematical record: what’s the point of calculating pi?, The Guardian, Aug 17, 2021, Donna Lu; HTML
• Zero-knowledge proofs: validation without information, The Age Examine, Aug 3, 2021; PDF
• Brainstorming MATRIX wins $US600,000 mathematics grant, The Australian, Feb 2, 2021, Jill Rowbotham; HTML
• Calculus ignored even though evidence is we rely on it, Australian Financial Review, Dec 7, 2020, Jan de Gier and Asha Rao; HTML
• Heroines of mathematics, Pursuit 2019, Lito Vilisoni Wilson; HTML
• Prove it – with maths, Pursuit 2019, Daryl Holland; HTML
• The proof behind the man who knew infinity, Pursuit 2016, Jan de Gier and Michael Wheeler; HTML
• Melbourne traffic: Trams push cars out of the slow lane on Smith Street, Collingwood, The Age, Jul 17, 2016, Adam Carey; HTML
• Maths researchers enter the MATRIX to put Australia on the map, Australian Financial Review, Jul 4, 2016, Tim Dodd; HTML
• Discovery for discovery’s sake pays the biggest dividends, Australian Financial Review, Sep 4, 2015, Jan de Gier and Tony Guttmann; HTML
• Research on the roads: the trouble with traffic, International Innovation 180 (2015); HTML, PDF
• Trams that never stop at traffic lights could be part of Melbourne’s people-moving future, ABC News, 13 February 2015, Loretta Florence; HTML
• Melbourne trams may never have to stop at traffic lights, under VicRoads plan The Age, 13 February 2015, Marissa Calligeros; HTML
• Applying physics to better traffic flow, The Australian, Australian IT, 17 January 2012, Jennifer Foreshew; PDF.
• Going places: why better traffic lights make better sense, The Conversation (19 December 2011), J. de Gier and T.M. Garoni; HTML.
• Ending traffic jams, Voice 7(12) (2011), Sally Sherwen; HTML.
• Conference registration deadlines (2010), J. de Gier, and J. Links; PDF
• Maths Matters: Back to the future, Austms Gazette 35(2) (2008), 79–83, J. de Gier; PDF

Current Research

I am interested in solvable lattice models, an area of mathematical physics and statistical mechanics which offers exciting research possibilities in pure as well as applied mathematics. The study of solvable lattice models uses a variety of techniques, ranging from algebraic concepts such as the Yang-Baxter equation, Hecke algebras and quantum groups to analytic methods such as complex analysis and elliptic curves. Due to this wide variety of methods, the study of solvable lattice models often produces unexpected links between different areas of research.

Solvable lattice models provide useful frameworks for modeling real world phenomena. Examples of solvable lattice models that are widely used in applications are quantum spin chains and ladders as models for metals and superconductivity, exclusion processes as models for traffic and fluid flow, more general stochastic processes to model random phenomena and random tilings as models for quasicrystals.

Integrable stochastic processes

I have a longstanding interest in (quantum) stochastic particle models such as exclusion processes. Many interesting quantities for these processes can be computed using integrable models. These include non-trivial matrix product stationary states and universal current distribution functions such as the Gaussian and the Tracy-Widom distributions.

Multivariable polynomials

I also study connections between enumerative combinatorics & statistical mechanics on the one hand, and (symmetric) polynomials and representation theory on the other. Schur and Macdonald polynomials are important classes of polynomials and we have developed a new framework to study and generalise such polynomials using solvable models and matrix product methods.

For example, we have developed new explicit expressions for classes of multi-variable polynomials \( f_\mu(x_1,\ldots,x_n) \) indexed by compositions , such as Macdonald polynomials, in the following matrix product form:
\[ f_\mu(x_1,\ldots,x_n) := \rho\big( A_{\mu_1}(x_1) \cdots A_{\mu_n}(x_n) \big),\]
where $\(\rho\) is a linear form and the linear operators \( A_i(x) \) obey the Zamolodchikov–Faddeev (ZF) algebra
\[ \check{R}(x/y)\cdot\big[\mathbb{A}(x)\otimes \mathbb{A}(y)\big] = \big[\mathbb{A}(y)\otimes \mathbb{A}(x)\big], \]
where \(\check{R}(x)\) is the R-matrix of an appropriate Hopf algebra or quantum group.

Traffic modelling

I was involved in a traffic modelling project in collaboration with researchers at Monash and VicRoads. Check out CEASAR, our traffic network simulator developed in the ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS).

Available on arXiv.

ORCID https://orcid.org/0000-0001-5415-3318.
Google scholar.

Research papers

1. Symmetric functions from the six-vertex model in half-space, Alexandr Garbali, J. de Gier, W. Mead and M. Wheeler, Ann. Henri Poincaré (2024). DOI; arXiv:2312.14348
2. Transition probability and total crossing events in the multi-species asymmetric exclusion process, J. de Gier, W. Mead and M. Wheeler, J. Phys. A: Math. Theor. 56 (2023), 255204. DOI; arXiv:2109.14232
3. Limiting current distribution for a two species asymmetric exclusion process, Z. Chen, J. de Gier, I. Hiki, T. Sasamoto and M. Usui, Commun. Math. Phys. 395 (2022), 59-142. DOI; Online access; arXiv:2104.00026
4. Limit shapes for the asymmetric five vertex model, J. de Gier, R. Kenyon and S.S. Watson, Commun. Math. Phys. 385 (2021), 793-836. DOI; Online access; arXiv:1812.11934
5. The \( R\)-matrix of the quantum toroidal algebra \( U_{q,t}(\overset{..}{gl}_1)\) in the Fock module, Alexandr Garbali and Jan de Gier, Commun. Math. Phys. 384 (2021), 1971-2008. DOI; Online access; arXiv:2004.09241
6. Integrable stochastic dualities and the deformed Knizhnik–Zamolodchikov equation, Z. Chen, J. de Gier and M. Wheeler, Int. Math. Res. Not. 19 (2020), 5872-5925. DOI; arXiv:1709.06227
7. Kardar-Parisi-Zhang Universality of the Nagel-Schreckenberg Model, J. de Gier, A. Schadschneider, J. Schmidt and G.M. Schütz, Phys. Rev. E 100 (2019), 052111. DOI; arXiv:1907.00636
8. T-Q relations for the integrable two-species asymmetric simple exclusion process with open boundaries, X. Zhang, F. Wen and J. de Gier, J. Stat. Mech (2019), 014001. DOI; arXiv:1808.05880
9. Exact confirmation of 1D nonlinear fluctuating hydrodynamics for a two-species exclusion process, Z. Chen, J. de Gier, I. Hiki and T. Sasamoto, Phys. Rev. Lett. 120 (2018), 240601. DOI; arXiv:1803.06829
10. Behaviour of traffic on a link with traffic light boundaries, 116-138, L. Zhang, C. Finn, T.M. Garoni and J. de Gier, Physica A 503 (2018).
11. A curious mapping between supersymmetric quantum chains, Gyorgy Z. Feher, Alexandr Garbali, Jan de Gier and Kareljan Schoutens, Proceedings of the workshop Integrability in Low-dimensional Quantum Systems, 2017 MATRIX Annals.
12. Dynamical universality class of the Nagel–Schreckenberg and related models, A. Schadschneider, J. Schmidt, J. de Gier and G.M. Schütz, in Traffic and Granular Flow ’17, ed. S.H. Hamdar, (Springer Nature Switzerland AG).
13. A new generalisation of Macdonald polynomials, A. Garbali, J. de Gier and M. Wheeler, Commun. Math. Phys. 352 (2017), 773–804. DOI; arXiv:1605.07200
14. Finite-size corrections for universal boundary entropy in bond percolation, SciPost Phys. 1, 012 (2016), J. de Gier, J.L. Jacobsen and A. Ponsaing.
15. Koornwinder polynomials and the stationary multi-species asymmetric exclusion process with open boundaries, J. Phys. A: Math. Theor. 49 (2016), 444002, L. Cantini, A. Garbali, J. de Gier and M. Wheeler.
16. Integrable supersymmetric chain without particle conservation, J. Stat. Mech. (2016) 023104, J. de Gier, G.Z. Feher, B. Nienhuis and M. Rusaczonek.
17. Matrix product and sum rule for Macdonald polynomials, Proceedings of the 28-th International Conference on Formal Power Series and Algebraic Combinatorics (2016), L. Cantini, J. de Gier and M. Wheeler; HTML
18. A summation formula for Macdonald polynomials, Lett. Math. Phys. 106 (2016), 381–394, J. de Gier and M. Wheeler.
19. Matrix product formula for Macdonald polynomials, J. Phys. A: Math. Theor. 48 (2015), 384001, L. Cantini, J. de Gier and M. Wheeler.
20. Exclusion in a priority queue, J. Stat. Mech. (2014), P07014, J. de Gier and C. Finn.
21. Traffic disruption and recovery in road networks, Physica A 401 (2014), 82-102, L. Zhang, T.M. Garoni and J. de Gier.
22. The critical fugacity for surface adsorption of SAW on the honeycomb lattice is 1+√2, Comm. Math. Phys. 326 (2014), 727–754, N.R. Beaton, M. Bousquet-Mélou, J. de Gier, H. Duminil-Copin and A.J. Guttmann.
23. Discrete holomorphicity and integrability in loop models with open boundaries, J. Stat. Mech. (2013), P02029, J. de Gier, A. Lee and J. Rasmussen.
24. A comparative study of Macroscopic Fundamental Diagrams of arterial road networks governed by adaptive traffic signal systems, L. Zhang, T.M. Garoni and J. de Gier, Transportation Research B: Methodological 49 (2013), 1–23.
25. Off-critical parafermions and the winding angle distribution of the O(n) model, J. Phys. A: Math. Theor. 45 (2012), 275002, A. Elvey Price, J. de Gier, A.J. Guttmann and A. Lee.
26. Deformed Kazhdan-Lusztig elements and Macdonald polynomials, J. Alg. Comb. Theory A 119 (2012), 183-211, J. de Gier, A. Lascoux and M. Sorrell.
27. Relaxation rate of the reverse biased asymmetric exclusion process, J. Phys. A 44 (2011), 405002, J. de Gier, C. Finn and M. Sorrell.
28. Current large deviation function for the open asymmetric simple exclusion process, Phys. Rev. Lett. 107 (2011), 010602, J. de Gier and F.H.L. Essler.
29. Traffic flow on realistic road networks with adaptive traffic lights, J. Stat. Mech. (2011), P04008, J. de Gier, T.M. Garoni and O. Rojas.
30. Separation of variables for symplectic characters, Lett. Math. Phys. 97 (2011), 61-83, J. de Gier, and A. Ponsaing.
31. Factorised solutions of Temperley-Lieb qKZ equations on a segment, Adv. Theor. Math. Phys. 14 (2010), 795-877, J. de Gier and P. Pyatov.
32. Combinatorics of Kazhdan-Lusztig elements: Factorisation and fully packed loop models, in Combinatorial representation theory Oberwolfach Reports 7 (2010), 832-835, J. de Gier.
33. Autocorrelations in the totally asymmetric simple exclusion process and Nagel-Schreckenberg model, Phys. Rev. E 82 (2010), 021107, J. de Gier, T.M. Garoni and Z. Zhou.
34. Exact spin quantum Hall current between boundaries of a lattice strip , Nucl. Phys. B 838 (2010), 371-390, J. de Gier, B. Nienhuis and A. Ponsaing.
35. Exact finite size groundstate of the O(n=1) loop model with open boundaries, J. Stat. Mech. (2009), P04010, 26 pp., J. de Gier, A. Ponsaing and K. Shigechi.
36. Fully packed loop models on finite geometries, Polygons, polyominoes and polycubes, Lecture Notes in Physics 775 (2009), ed. A.J. Guttmann, Ch. 13, 30pp., J. de Gier.
37. Punctured plane partitions and the q-deformed Knizhnik–Zamolodchikov and Hirota equations, J. Alg. Comb. Theory A 116 (2009), 772–794, J. de Gier, P. Pyatov and P. Zinn-Justin.
38. The two-boundary Temperley-Lieb algebra, J. Algebra 321 (2009), 1132–1167, J. de Gier and A. Nichols.
39. Slowest relaxation mode of the partially asymmetric exclusion process with open boundaries, J. Phys. A 41 (2008), 485002, 25pp., J. de Gier and F.H.L. Essler.
40. The Razumov-Stroganov conjecture: Stochastic processes, loops and combinatorics, J. Stat. Mech. (2007), N02001, 6pp., J. de Gier.
41. Exact spectral gaps of the asymmetric exclusion process with open boundaries, J. Stat. Mech. (2006), P12011, 45 pp., J. de Gier and F.H.L. Essler.
42. Bethe Ansatz solution of the asymmetric exclusion process with open boundaries, Phys. Rev. Lett. 95 (2005), 240601, 4pp., J. de Gier and F.H.L. Essler.
43. Magic in the spectra of the XXZ quantum chain with boundaries at \(\Delta=0\) and \(\Delta=-1/2\), Nucl. Phys. B 729 (2005), 387-418, J. de Gier, A. Nichols, P. Pyatov and V. Rittenberg.
44. One-boundary Temperley-Lieb algebras in the XXZ and loop models, JSTAT (2005), P03003, 30pp., A. Nichols, V. Rittenberg and J. de Gier.
45. Brauer loops and the commuting variety, J. Stat. Mech. (2005), P01006, 10pp., J. de Gier and B. Nienhuis.
46. Loops, matchings and alternating-sign matrices, Discr. Math. 298 (2005), 365–388, arXiv:math.CO/0211285, J. de Gier.
47. Refined Razumov-Stroganov conjectures for open boundaries, J. Stat. Mech. (2004), P09009, 14pp., J. de Gier and V. Rittenberg.
48. The raise and peel model of a fluctuating interface, J. Stat. Phys. 114 (2004) 1-35, J. de Gier, B. Nienhuis, P. A. Pearce and V. Rittenberg.
49. Exact expressions for correlations in the ground state of the dense O(1) loop model, J. Stat. Mech. (2004), P09010, 24pp., S. Mitra, B. Nienhuis, J. de Gier and M.T. Batchelor.
50. Nonequilibrium stationary states and equilibrium models with long range interactions, J. Phys. A 37 (2004) 4303-4320, R. Brak, J. de Gier, and V. Rittenberg.
51. Bethe Ansatz for the Temperley-Lieb loop model with open boundaries, J. Stat. Mech. (2004), P03002, 27pp., J. de Gier and P. Pyatov.
52. Magnetization plateaux in Bethe Ansatz solvable spin-S ladders, Phys. Rev. B 68 (2003), 024418 (1-8), M. Maslen, M.T. Batchelor and J. de Gier.
53. Stochastic processes and conformal invariance, Phys. Rev. E 67 (2003) 016101-016104,J. de Gier, B. Nienhuis, P. A. Pearce and V. Rittenberg.
54. Temperley-Lieb stochastic processes, J. Phys. A 35 (2002) L661-L668, P. A. Pearce, V. Rittenberg, J. de Gier and B. Nienhuis.
55. The rotor model and combinatorics, Int. J. Mod. Phys. B 16 (2002) 1883-1889, M.T. Batchelor, J. de Gier and B. Nienhuis.
56. The XXZ chain at \(\Delta=- 1/2\): Bethe roots, symmetric functions and determinants, J. Math. Phys. 43 (2002), 4135-4146, J. de Gier, M.T. Batchelor, B. Nienhuis and S. Mitra.
57. Six – Vertex model with domain wall boundary conditions. Variable inhomogeneities., J. Phys. A 34 (2001) 8135-8144, J. de Gier and V. Korepin.
58. Exactly solvable su(n) mixed spin ladders, J. Stat. Phys. 102 (2001) 559-566, presented at the Baxter Revolution in Mathematical Physics Conference , (2000), M.T. Batchelor, J. de Gier and M. Maslen.
59. Exact stationary state for a deterministic high speed traffic model with open boundaries, J. Phys. A 34 (2001) 3707-3720, J. de Gier.
60. The quantum symmetric XXZ chain at \(\Delta=-1/2\), alternating sign matrices and plane partitions, J. Phys. A 34 (2001) L265-L270, M.T. Batchelor, J. de Gier and B. Nienhuis.
61. Magnetization plateaus in a solvable 3-leg spin ladder, Phys. Rev. B 62 (2000) R3584-R3587, J. de Gier and M.T. Batchelor.
62. Phase diagram of the su(8) quantum spin tube, Phys. Rev. B61 (2000) 15196-15202, J. de Gier, M.T. Batchelor and M. Maslen.
63. Exactly solvable quantum spin ladders associated with the orthogonal and symplectic Lie algebras, J. Phys. A 33 (2000) L97-L101, M.T. Batchelor, J. de Gier, J. Links and M. Maslen.
64. Exact stationary state for an asymmetric exclusion process with fully parallel dynamics, Phys. Rev. E 59 (1999) 4899-4911, J. de Gier and B. Nienhuis.
65. Bethe Ansatz solution of a decagonal rectangle triangle random tiling, J. Phys. A 31 (1998) 2141-2154, J. de Gier and B. Nienhuis.
66. Solvable rectangle triangle random tilings, Proceedings of the 6th International Conference on Quasicrystals, eds. S. Takeuchi and T. Fujiwara, World Scientific (1998) 91-94, J. de Gier and B. Nienhuis.
67. Integrability of the square-triangle random tiling, Phys. Rev. E 55 (1997) 3926-3933, J. de Gier and B. Nienhuis.
68. The exact solution of an octagonal rectangle-triangle random tiling, J. Stat. Phys. 87 (1997) 415-437, J. de Gier and B. Nienhuis.
69. Exact solution of an octagonal random tiling model, Phys. Rev. Lett. 76 (1996) 2918-2921, J. de Gier and B. Nienhuis.
70. Operator spectrum and exact exponents of the fully packed loop model, J. Phys.A: Math. Gen. 29 (1996) 6489-6504, J. Kondev, J. de Gier and B. Nienhuis.

Editorial publications

• • 2019-20 MATRIX Annals, D. Wood, J. de Gier, C. Praeger and Terence Tao; MATRIX; Springer.
  • 2018 MATRIX Annals, D. Wood, J. de Gier, C. Praeger and Terence Tao; MATRIX; Springer.
  • 2017 MATRIX Annals, D. Wood, J. de Gier, C. Praeger and Terence Tao; MATRIX; Springer.
  • 2016 MATRIX Annals, D. Wood, J. de Gier, C. Praeger and Terence Tao; MATRIX; Springer.
  • Counting Complexity: An international workshop on statistical mechanics and combinatorics (in honour of Tony Guttmann’s 60th birthday), J. Phys: Conf. Series 42 (2006), eds. J. de Gier and Ole Warnaar; HTML.
  • AustMS Gazette 33 (2006), 5 issues, J. de Gier and S.O. Warnaar; HTML.
  • AustMS Gazette 32 (2005), 5 issues, J. de Gier and S.O. Warnaar; HTML.
  • AustMS Gazette 31 (2004), 5 issues, J. de Gier and S.O. Warnaar; HTML

Professor Jan de Gier
School of Mathematics and Statistics
The University of Melbourne
VIC 3010
Australia
Tel: +61-(0)3-834-49709
jdgier@unimelb.edu.au