People_
Professor Nalini Joshi
AO FAA FRSN FAustMS
Address
F07 - Carslaw Building
The University of Sydney
Websites
Payne-Scott Professor Nalini Joshi AO is Chair of Applied Mathematics at the University of Sydney, and a Georgina Sweet Australian Laureate Fellow.
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Nalini's research focuses on ways to describe new special functions that arise as models in large classes of applications.
If each function was a number, they would be like the transcendental number π, which cannot be expressed as a finite combination of simpler numbers, yet they appear everywhere in applications.
These special, universal functions describe a wide range of pheneomena, including:
interaction of heavy atomic particles
quantum gravity
distribution of bus arrival times in Cuernavaca, the New York subway system and flight boarding times
- the distribution of large prime numbers;
- collisions of atomic particles;
- aircraft boarding times;
- bus arrival times;
- the New York subway system;
- electric fields in electrolyte solutions;
- magnetic fields and waves in plasma physics;
- water waves in the oceans;
- stationary solutions of Einstein’s equations of general relativity;
- cloud formations in the atmosphere.
Nalini describes her research as developing methods to solve mathematical puzzles that allow her to discover and describethese functions. The puzzles involve non-linear differential and difference equations, studied through a wide variety of lenses: including analysis, algebra, algebraic geometry, topology, and asymptotic methods in limits.
The equations she studies are called integrable systems. When they involve one independent variable, they are the Painlevé equations, while in two or more variables, the equations are soliton equations.
The mathematical tools involve a panorama of different perspectives. Instead of describing solutions as functions of an independent variable like time, they can be tracked by curves that go through initial values. The first perspective is like pointing a telescope to one point in the sky at night and taking pictures while time is changing. The second perspective is like tracking one star as it follows circular arcs of light in the sky at night.
The second perspective is called a space of initial values. It turns out that the geometry of this space gives us a great deal of information about the global nature of all possible solutions of the integrable systems and leads to some deep algebraic descriptions of the solutions, given by reflection groups.
The discrete Painlevé equations turn out to be given by translations (or walks) on lattices created by affine reflection groups. The results are very pretty and beautiful.
Nalini's research papers and books can be found on her publication page or her OrCiD page.
Specific research areas: Integrable systems, the Painlevé equations, discrete Painlevé equations, lattice equations, geometric asymptotics, nonlinear dynamics, nonlinear waves, perturbation theory.
See Nalini's course on integrable systems. Nalini has several open PhD projects in this area, which propose to extend the mathematical toolbox to describe solutions of non-linear differential and discrete equations.
Timetable
Fellow of the Australian Academy of Science (elected 2008)
ARC Georgina Sweet Australian Laureate Fellowship (2012-18)
AFR Westpac 100 Women of Influence (2016)
Payne-Scott Professorial Distinction (2016)
Eureka award for Outstanding Mentor of Young Researchers (2018)
Bragg fellowship, Royal Institution of Australia (2019)
NSW Premier's Prize for Excellence in Mathematics, Earth Science, Chemistry or Physics (2019)
The George Szekeres Medal of the Australian Mathematical Society (2020)
The ANZIAM Medal (2021)
(University of Kyushu) I hold an Australian Research Council grant with Professor Kenji Kajiwara, who is a member of the Centre for Mathematics in Industry, University of Kyushu. | |
(The University of Leeds) I collaborate with Professor Frank W. Nijhoff, who is the Professor of Mathematical Physics. Our recent book ``Discrete Systems and Integrability'', co-authored together with Professor Jarmo Hietarinta from the University of Turku in Finland, was published by Cambridge University Press. | |
(The University of Leeds) Visiting Professor |
| Project title | Research student |
|---|---|
| Billiard Dynamics in Projective Spaces Over Fields with Finite Characteristics | Harry HIATT |
Selected publications
Discrete Painleve Equations
(American Mathematical Society, 2019)
Discrete Systems and Integrability
(Cambridge University Press, 2016)
Publications
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Books
- Joshi, N. (2019). Discrete Painleve Equations. USA: American Mathematical Society. [More Information]
- Hietarinta, J., Joshi, N., Nijhoff, F. (2016). Discrete Systems and Integrability. Cambridge: Cambridge University Press. [More Information]
Book Chapters
- Gubbiotti, G., Joshi, N., Tran, D., Viallet, C. (2020). Complexity and Integrability in 4D Bi-rational Maps with Two Invariants. In Frank Nijhoff, Yang Shi, Da-Jun Zhang (Eds.), Asymptotic, Algebraic and Geometric Aspects of Integrable Systems, (pp. 17-36). Cham: Springer. [More Information]
- Kruskal, M., Joshi, N., Halburd, R. (2004). Analytical And Asymptotic Methods For Nonlinear Singularity Analysis: A Review And Extensions Of Tests For The Painlevé Property. In Basil Grammaticos, Yvette Kosmann-Schwarzbach, K. M. Tamizhmani (Eds.), Integrability of Nonlinear Systems, (pp. 175-208). Berlin: Springer.
- Joshi, N. (2003). Hunting mathematical butterflies. In Ball, Akhmediev (Eds.), Nonlinear Dynamics: from Lasers to Butterflies, (pp. 77-114). USA: World Scientific Publishing.
Journals
- Joshi, N., Roffelsen, P. (2025). On the crystal limit of the q-difference sixth Painlevé equation. Journal of Nonlinear Science, 35(1), Article number 31-30 pages. [More Information]
- Joshi, N., Lasic Latimer, T. (2023). Asymptotic Behaviours of q-orthogonal Polynomials from a q-Riemann Hilbert Problem. Constructive Approximation, 58(1), 151-179. [More Information]
- Joshi, N., Lasic Latimer, T. (2023). Asymptotics of discrete q-Freud I I orthogonal polynomials from the q-Riemann Hilbert problem. Nonlinearity, 36(8), 3969-4006. [More Information]
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Edited Journals
- Joshi, N., Noumi, M., Sakai, H., Viallet, C. (2013). Journal of Nonlinear Mathematical Physics. Journal Of Nonlinear Mathematical Physics, 20(Supplement 1).
Conferences
- Joshi, N., Atkinson, J., Howes, P., Nakazono, N. (2012). Extension of a one-dimensional reduction of the Q4 mapping to a discrete Painleve equation. The Japan Society for Industrial and Applied Mathematics 2012.
- Brown, N., Bower, M., Skalicky, J., Wood, L., Donovan, D., L'och, B., Bloom, W., Joshi, N. (2010). A professional development framework for teaching in higher education. 33rd Higher Education Research and Development Society of Australasia International Conference: HERDSA 2010 Reshaping Higher Education, Milperra: Higher Education Research and Development Society of Australasia.
- Joshi, N. (2005). Asymptotics for Extended Cellular Automata. Recent Trends in Exponential Asymptotics, Kyoto: RIMS.
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2025
- Joshi, N., Roffelsen, P. (2025). On the crystal limit of the q-difference sixth Painlevé equation. Journal of Nonlinear Science, 35(1), Article number 31-30 pages. [More Information]
2023
- Joshi, N., Lasic Latimer, T. (2023). Asymptotic Behaviours of q-orthogonal Polynomials from a q-Riemann Hilbert Problem. Constructive Approximation, 58(1), 151-179. [More Information]
- Joshi, N., Lasic Latimer, T. (2023). Asymptotics of discrete q-Freud I I orthogonal polynomials from the q-Riemann Hilbert problem. Nonlinearity, 36(8), 3969-4006. [More Information]
- Heu, V., Joshi, N., Radnovic, M. (2023). Global asymptotics of the sixth Painleve equation in Okamoto’s space. Forum of Mathematics, Sigma, 11, e17 - 1-e17 - 38. [More Information]
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2021
- Joshi, N., Kajiwara, K., Masuda, T., Nakazono, N. (2021). Discrete power functions on a hexagonal lattice I: derivation of defining equations from the symmetry of the Garnier system in two variables. Journal of Physics A: Mathematical and Theoretical, 54(33), 335202. [More Information]
- Joshi, N., Lasic Latimer, T. (2021). On a class of q-orthogonal polynomials and the q-Riemann-Hilbert problem. Proceedings of the Royal Society A, 477(2254), 20210452. [More Information]
- Joshi, N., Roffelsen, P. (2021). On the Riemann-Hilbert Problem for a q-Difference Painleve Equation. Communications in Mathematical Physics, 384(1), 549-585. [More Information]
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2020
- Gubbiotti, G., Joshi, N., Tran, D., Viallet, C. (2020). Bi-rational maps in four dimensions with two invariants. Journal of Physics A: Mathematical and Theoretical, 53(11), Art. 115201 - 1-Art. 115201 - 24. [More Information]
- Gubbiotti, G., Joshi, N., Tran, D., Viallet, C. (2020). Complexity and Integrability in 4D Bi-rational Maps with Two Invariants. In Frank Nijhoff, Yang Shi, Da-Jun Zhang (Eds.), Asymptotic, Algebraic and Geometric Aspects of Integrable Systems, (pp. 17-36). Cham: Springer. [More Information]
- Joshi, N. (2020). Discrete Painlevé equations. Notices of the American Mathematical Society, 67(6), 797-805. [More Information]
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2019
- Joshi, N., Radnovic, M. (2019). Asymptotic behaviour of the third Painleve transcendents in the space of initial values. Transactions of the American Mathematical Society, 372(9), 6507-6546. [More Information]
- Joshi, N. (2019). Discrete Painleve Equations. USA: American Mathematical Society. [More Information]
- Joshi, N., Lustri, C. (2019). Generalized solitary waves in a finite-difference Korteweg-de Vries equation. Studies in Applied Mathematics, 142(3), 359-384. [More Information]
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2018
- Joshi, N., Radnovic, M. (2018). Asymptotic behaviour of the fifth Painleve transcendents in the space of initial values. Proceedings of the London Mathematical Society, 3(116), 1329-1364. [More Information]
- Joshi, N., Liu, Q. (2018). Asymptotic behaviours given by elliptic functions in PI-PV. Nonlinearity, 31(8), 3726-3747. [More Information]
2017
- Joshi, N., Nakazono, N. (2017). Elliptic Painleve equations from next-nearest-neighbor translations on the E8(1) lattice. Journal of Physics A: Mathematical and Theoretical, 50, 1-17. [More Information]
- Joshi, N., Kajiwara, K., Masuda, T., Nakazono, N., Shi, Y. (2017). Geometric description of a discrete power function associated with the sixth Painlevé equation. Proceedings of the Royal Society A, A473 (Art. 20170312), 1-19. [More Information]
- Joshi, N., Takei, Y. (2017). On stokes phenomena for the alternate discrete PI equation. Trends in Mathematics, Part F2, 369-381. [More Information]
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2016
- Joshi, N., Roffelsen, P. (2016). Analytic solutions of q-P(A1) near its critical points. Nonlinearity, 29(12), 3696-3742. [More Information]
- Joshi, N., Radnovic, M. (2016). Asymptotic Behavior of the Fourth Painleve Transcendents in the Space of Initial Values. Constructive Approximation, 44(2), 195-231. [More Information]
- Hietarinta, J., Joshi, N., Nijhoff, F. (2016). Discrete Systems and Integrability. Cambridge: Cambridge University Press. [More Information]
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2015
- Joshi, N., Nakazono, N., Shi, Y. (2015). Lattice equations arising from discrete Painlev systems. I. (A 2 + A 1)(1) and ( A 1 + A 1 ) ( 1 ) cases. Journal of Mathematical Physics, 56(9), 092705-1-092705-25. [More Information]
- Joshi, N. (2015). Quicksilver Solutions of a q-Difference First Painleve Equation. Studies in Applied Mathematics, 134(2), 233-251. [More Information]
- Joshi, N., Lustri, C. (2015). Stokes phenomena in discrete Painlevé I. Proceedings of the Royal Society A, 471(2177), 1-22. [More Information]
2014
- Joshi, N., Nakazono, N., Shi, Y. (2014). Geometric reductions of ABS equations on an n-cube to discrete Painlevé systems. Journal of Physics A: Mathematical and Theoretical, 47(50), 1-16. [More Information]
- Howes, P., Joshi, N. (2014). Global Asymptotics of the Second Painlev Equation in Okamoto's Space. Constructive Approximation, 39(1), 11-41. [More Information]
2013
- Joshi, N., Noumi, M., Sakai, H., Viallet, C. (2013). Journal of Nonlinear Mathematical Physics. Journal Of Nonlinear Mathematical Physics, 20(Supplement 1).
- Joshi, N., Noumi, M., Sakai, H., Viallet, C. (2013). Preface to special issue on the geometry of the painlevé equations. Journal Of Nonlinear Mathematical Physics, 20. [More Information]
- Atkinson, J., Joshi, N. (2013). Singular-Boundary Reductions of Type-Q ABS Equations. International Mathematics Research Notices, 7, 1451-1481. [More Information]
2012
- Joshi, N., Shi, Y. (2012). Exact solutions of a q-discrete second Painlevé equation from its iso-monodromy deformation problem. II. Hypergeometric solutions. Proceedings of the Royal Society A, 468(2146), 3247-3264. [More Information]
- Joshi, N., Atkinson, J., Howes, P., Nakazono, N. (2012). Extension of a one-dimensional reduction of the Q4 mapping to a discrete Painleve equation. The Japan Society for Industrial and Applied Mathematics 2012.
- Atkinson, J., Joshi, N. (2012). The Schwarzian variable associated with discrete KdV-type equations. Nonlinearity, 25(6), 1851-1866. [More Information]
2011
- Joshi, N., Shi, Y. (2011). Exact solutions of a q-discrete second Painlevé equation from its iso-monodromy deformation problem: I. Rational solutions. Proceedings of the Royal Society A, 467, 3443-3468. [More Information]
- Duistermaat, J., Joshi, N. (2011). Okamoto's Space for the First Painlevé Equation in Boutroux Coordinates. Archive for Rational Mechanics and Analysis, 202(3), 707-785. [More Information]
- Wood, L., Vu, T., Bower, M., Brown, N., Skalicky, J., Donovan, D., L'och, B., Joshi, N., Bloom, W. (2011). Professional development for teaching in higher education. International Journal of Mathematical Education in Science and Technology, 42(7), 997-1009. [More Information]
2010
- Brown, N., Bower, M., Skalicky, J., Wood, L., Donovan, D., L'och, B., Bloom, W., Joshi, N. (2010). A professional development framework for teaching in higher education. 33rd Higher Education Research and Development Society of Australasia International Conference: HERDSA 2010 Reshaping Higher Education, Milperra: Higher Education Research and Development Society of Australasia.
- Butler, S., Joshi, N. (2010). An inverse scattering transform for the lattice potential KdV equation. Inverse Problems, 26(115012), 1-28. [More Information]
- Kassotakis, P., Joshi, N. (2010). Integrable Non-QRT Mappings of the Plane. Letters in Mathematical Physics, 91(1), 71-81. [More Information]
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2009
- Joshi, N., Spicer, P. (2009). Direct "Delay" Reductions of the Toda Hierarchy. Journal of the Physical Society of Japan, 78(9), 094006-1-094006-5. [More Information]
- Joshi, N. (2009). Direct 'delay' reductions of the Toda equation. Journal of Physics A: Mathematical and Theoretical, 42(2), 1-8. [More Information]
- Joshi, N., Morrison, T. (2009). Existence and uniqueness of Tronquee solutions of the fourth-order Jimbo-Miwa second Painleve equation. Proceedings of the American Mathematical Society, 137(6), 2005-2014. [More Information]
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2008
- Joshi, N., Morrison, T. (2008). New exact solutions of spatially and temporally varying reaction-diffusion equations. Analysis and Applications, 6(4), 371-381. [More Information]
- Field, C., Joshi, N., Nijhoff, F. (2008). q-difference equations of kd V type and Chazy-type second degree difference equations. Journal of Physics A: Mathematical and Theoretical, 41, 332005-332018. [More Information]
2007
- Hay, M., Hietarinta, J., Joshi, N., Nijhoff, F. (2007). A Lax pair for a lattice modified KdV equation, reductions to q-Painleve equations and associated Lax pairs. Journal of Physics A: Mathematical and Theoretical, 40(2), F61-F73. [More Information]
- Joshi, N., Kitaev, A., Treharne, P. (2007). On the linearization of the Painleve' III-VI equations and reductions of the three-wave resonant system. Journal of Mathematical Physics, 48(10), 103512-1-103512-42. [More Information]
- Joshi, N., Ormerod, C. (2007). The general theory of linear difference equations over the max-plus semi-ring. Studies in Applied Mathematics, 118(1), 85-97. [More Information]
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2006
- Ramani, A., Joshi, N., Grammaticos, B., Tamizhmani, T. (2006). Deconstructing an integrable lattice equation. Journal of Physics A: Mathematical and General, 39(8), L145-L149. [More Information]
- Joshi, N., Grammaticos, B., Tamizhmani, T., Ramani, A. (2006). From integrable lattices to non-QRT mappings. Letters in Mathematical Physics, 78(1), 27-37. [More Information]
- Joshi, N., Pickering, A. (2006). Generalized Halphen systems. Proceedings of the Royal Society of Edinburgh Section A (Mathematics), 136A (6), 1287-1301. [More Information]
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2005
- Joshi, N. (2005). Asymptotics for Extended Cellular Automata. Recent Trends in Exponential Asymptotics, Kyoto: RIMS.
- Gordoa, P., Joshi, N., Pickering, A. (2005). Backlund transformations for fourth Painleve hierarchies. Journal of Differential Equations, 217(1), 124-153. [More Information]
- Joshi, N., Lafortune, S. (2005). How to detect integrability in cellular automata. Journal of Physics A: Mathematical and General, 38(28), L499-L504. [More Information]
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2004
- Kruskal, M., Joshi, N., Halburd, R. (2004). Analytical And Asymptotic Methods For Nonlinear Singularity Analysis: A Review And Extensions Of Tests For The Painlevé Property. In Basil Grammaticos, Yvette Kosmann-Schwarzbach, K. M. Tamizhmani (Eds.), Integrability of Nonlinear Systems, (pp. 175-208). Berlin: Springer.
- Joshi, N., Kajiwara, K., Mazzocco, M. (2004). Generating Function Associated With The Determinant Formula For The Solutions Of The Painlevé II Equation. Asterisque, 297(2004), 67-78.
- Joshi, N., Nijhoff, F., Ormerod, C. (2004). Lax pairs for ultra-discrete Painlevé cellular automata. Journal of Physics A: Mathematical and General, 37(2004), L559-L565. [More Information]
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2003
- Gordoa, P., Joshi, N., Pickering, A. (2003). A new technique in nonlinear singularity analysis. Publications of the Research Institute for Mathematical Sciences, 39, 435-449. [More Information]
- Maruno, K., Ohta, Y., Joshi, N. (2003). Exact localized solutions of quintic discrete nonlinear Schrodinger equation. Physics Letters. Section A: General, Atomic and Solid State Physics, 311(2-3), 214-220. [More Information]
- Joshi, N., Mazzocco, M. (2003). Existence and uniqueness of tri-tronquee solutions of the second Painlevé hierarchy. Nonlinearity, 16(2), 427-439. [More Information]
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2002
- HONE, A., Joshi, N., Kitaev, A. (2002). An entire function defined by a nonlinear recurrence relation. Journal of The London Mathematical Society, 66(2), 377-387. [More Information]
- CRESSWELL, C., Joshi, N. (2002). Consistent composition of Backlund transformations produces confined maps. Letters in Mathematical Physics, 61(1), 1-14. [More Information]
2001
- Gordoa, P., Joshi, N., Pickering, I. (2001). Mappings preserving locations of movable poles: II. The third and fifth Painlevé equations. Nonlinearity, 14(3), 567-582.
- Gordoa, P., Joshi, N., Pickering, I. (2001). On a Generalized 2 + 1 Dspersive Water Wave Hierarchy. Publications of the Research Institute for Mathematical Sciences, 37, 327-347.
- Joshi, N., Kitaev, A. (2001). On Boutroux's Tritronquée Solutions of the First Painlevé Equation. Studies in Applied Mathematics, 107(3), 253-291.
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1999
- Cresswell, C., Joshi, N. (1999). The discrete first, second and thirty-fourth Painleve hierarchies. Journal of Physics A: Mathematical and General, 32(4), 655-669.
- Cresswell, C., Joshi, N. (1999). The discrete Painleve I hierarchy. Symmetries and Integrability of Difference Equations, UK: Cambridge University Press. [More Information]
Selected Grants
2020
- Dynamics on space-filling shapes, Joshi N, Australian Research Council (ARC)/Discovery Projects (DP)
2019
- Geometric analysis of nonlinear systems, Joshi N, Radnovic M, Australian Research Council (ARC)/Discovery Projects (DP)
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In the media
- Interview with Charlie Pickering on the Weeklyhttps://www.youtube.com/watch?reload=9&v=I7-MAdUPGhQ
- Answering a question about time travel on ABC Q&A with Professor Brian Coxhttps://www.abc.net.au/news/2014-10-20/is-time-travel-possible-two-scientists-give-their-views-on-qanda/5828532
