## Projects in Mathematics

Propagation via nonlinear partial differential equations

Project Title: Propagation via nonlinear partial differential equations

Principal investigator: Prof. Yihong Du (UNE)

Other participants:
Dr Maolin Zhou (UNE), Prof. Bendong Lou (Shanghai Normal Univ), Prof Xing Liang (Univ of Sci and Tech of China), Prof Mingxin Wang (Harbin Inst. Tech.), Dr Wenjie Ni (UNE), Dr Ting-Ying Chang (UNE), Prof Fang Li (Zhong Shan Univ), Prof Rui Peng (Jiangsu Normal Univ.).

Funding body: Australian Research Council (DP190103757, 2019-2021)

Project description:
Propagation occurs in various forms, such as the spreading of invasive species, infectious diseases or cancer cells, or the progression of the healing front of a wound.

The goal of this project is to develop new mathematics to better understand these propagation phenomena. Although the sources of propagation are diverse, the basic phenomena have been observed to follow certain rules that can be captured by mathematical models of nonlinear partial differential equations (PDEs) or systems of such equations. Thus it is possible for a mathematical theory based on a set of such equations to provide vital insights and useful predictions for the spreading of a variety of subjects, such as the propagation of nerve impulses, and the spreading of invasive species or cancerous cells.

This topic has attracted the efforts of numerous first rate mathematicians in the past several decades, and extensive research in this area is currently conducted by several top tier research groups around the world. Yet, with the advances of the relevant sciences, more and more demanding questions arise in this area, with many important and basic ones waiting to be answered.

The specific aims of this proposal focus on the following three tasks:

• Complete the basic theory on free boundary models of propagation by answering several important questions left open in previous work;
• Develop a mathematical theory for propagation models with free boundary and nonlocal diffusion;
• Develop a mathematical theory for propagation over graphs/networks.

This project is a continuation of the ARC funded project (DP150101867, 2015-2018).

Related publications:

1. Y. Du and Wenjie Ni, Analysis of a West Nile virus model with nonlocal diffusion and free boundaries, preprint, 2019.
2. Y. Du, Fang Li and Maolin Zhou, Semi-wave and spreading speed of the nonlocal Fisher-KPP equation with free boundaries, preprint (arXiv: 1909.03711).
3. Yuanyang Hu, Xinan Hao, Xianfa Song and Y. Du, A free boundary problem for spreading under shifting climate, to appear in J. Diff. Equations. (arXiv: 1908.04041)
4. Y. Du, Bendong Lou, Rui Peng and Maolin Zhou, The Fisher-KPP equation over simple graphs: Varied persistence states in river networks, J. Math. Biol, (published online Jan. 2020). (arXiv:1809.06961)
5. Y. Du, Fernando Quiros and Maolin Zhou, Logarithmic corrections in Fisher-KPP type porous medium equations, J. Math. Pure Appl., (published online Dec. 2019) (arXiv:1806.02022)
6. Zhiguo Wang, Hua Nie and Y. Du, Asymptotic spreading speed for the weak competition system with a free boundary, DCDS-A, 39(2019), 5223-5262.
7. Maud El-Hachem, Scott W. McCue, Wang Jin, Yihong Du, Matthew J. Simpson, Revisiting the Fisher-KPP equation to interpret the spreading-extinction dichotomy, Proc. Royal Soc. A, 475 (2019), no. 2229, article id: 20190378.
8. Zhiguo Wang, Hua Nie and Y. Du, Spreading speed for a West Nile virus model with free boundary, J. Math. Biol., 79(2019), 433-466.
9. Jiafeng Cao, Y. Du, Fang Li and Wan-Tong Li, The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries, J. Functional Anal., 277 (2019), 2772-2814.
10. Weiwei Ding, Y. Du and Xing Liang, Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 2: Spreading speed, AIHP Analyse non Lineaire, 36 (2019), 1539-1573.
Propagation Described by Partial Differential Equations with Free Boundary

Project title: Propagation Described by Partial Differential Equations with Free Boundary

Principal investigator: Prof. Yihong Du (UNE)

Other participants:
Dr Maolin Zhou (UNE), Dr Weiwei Ding (UNE), Prof. Bendong Lou (Shanghai Normal Univ), Prof Xing Liang (Univ of Sci and Tech of China)

Funding body:   Australian Research Council (2015-2018)

Project description:
Front propagation appears in many branches of sciences; examples include the propagation of nerve impulses, the propagation of premixed flames, the spreading of advantageous genes, and the spreading of an invasive species. Although the sources of propagation are diverse, the basic phenomena have been observed to follow certain rules that can be captured by mathematical models of nonlinear partial differential equations (PDEs) or systems of such equations. Thus it is possible for a mathematical theory based on a set of such equations to provide vital insights and useful predictions for the spreading of a variety of subjects arising in different areas of sciences. This topic has been investigated by many leading mathematicians in the past several decades, and extensive research in this area is conducted by several first rate research groups around the world. With the advances of the relevant sciences, more and more demanding questions arise in this area, and many important and basic questions still remain open.  The aim of this project is to develop new mathematics for better applications and deeper insights into the propagation phenomena. We modify existing models by introducing a free boundary to represent the spreading front, and so the partial differential equations are satisfied over a varying spatial domain whose boundary evolves with time, which need to be solved together with the partial differential equation. This makes the model more realistic but at the same time makes the mathematical treatment of the model much more difficult to handle. This research is mostly theoretical though numerical simulations are also needed from time to time.

Related publications:

1. Wendi Bao, Y. Du, Zhigui Lin and Huaiping Zhu, Free boundary models for mosquito range movement driven by climate warming, J. Math. Biology, 76 (2018), 841-875.
2. Y. Du, Lei Wei and Ling Zhou, Spreading in a shifting environment modelled by the diffusive logistic equation with a free boundary, J. Dyn. Diff. Equations, 30(2018), 1389-1426
3. Y. Du, Bendong Lou and Maolin Zhou, Spreading and vanishing for nonlinear Stefan problems in high space dimensions, J. Elliptic and Parabolic Equations, 2 (2016), 297-321.
4. Chengxia Lei, Hua Nie, Wei Dong and Y. Du, Spreading of two competing species governed by a free boundary model in a shifting environment, J. Math. Anal. Appl., 462(2018), 1254-1282.
5. Y. Du and Chang-Hong Wu, Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries, Cal. Var. PDE, 57:52(2018), 36 pages.
6. Y. Du, Mingxin Wang and Maolin Zhou, Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math. Pure Appl., 107(2017), 253-287.
7. Weiwei Ding, Y. Du and Xing Liang, Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 1: Continuous initial functions, J. Diff. Eqns., 262(2017), 4988-5021.
8. Y. Du, Hiroshi Matsuzawa and Maolin Zhou, Spreading speed and profile for nonlinear Stefan problems in high space dimensions, J. Math. Pures Appl., 103(2015), 741-787.
9. Y. Du and Xing Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann. Inst. Henri Poincare Anal. Non Lineaire, 32(2015), 279-305.
10. Y. Du and Bendong Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17(2015), 2673-2724.
Nonlinear free boundary problems: propagation and regularity

Project title: Nonlinear free boundary problems: propagation and regularity

Principal investigator: Dr Maolin Zhou (UNE)

Funding body:   Australian Research Council (2017-2020)

Project description:
Nonlinear free boundary problems arise from many applied fields, and pose great challenges to the theory of nonlinear partial differential equations, as the underlying domain of the solution to such problems has to be solved together with the solution itself. This project aims to completely understand the propagation profile and regularity of two important classes of free boundary problems, which would greatly enhance the existing theory of partial differential equations, and extend its applications to situations not covered before.

Related publications:

1. Y.Du, B.Lou and M.Zhou, Spreading and vanishing for nonlinear Stefan problems in high space dimensions, J. Elliptic and Parabolic Equations, 2 (2016), 297-321.
2. Y.Du, M.Wang and M.Zhou, Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math. Pures Appl. (9) 107(2017), 253-287.
3. W.Lei, G.Zhang and M.Zhou, Long time behavior for solutions of the diffusive logistic equation with advection and free boundary, Calculus of Variations and PDEs 55 (2016), no. 4, 1-34.
4. X.Chen, B.Lou, M.Zhou and T.Giletti, Long time behavior of solutions of a reaction diffusion equation on unbounded intervals with Robin boundary conditions, Ann. Inst. H. Poincare Anal. Non Lineaire 33 (2016), no. 1, 67-92.
5. H.Gu, B.Lou and M.Zhou, Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries,  J. Funct. Anal. 269 (2015), no. 6, 1714-1768.
6. T.Giletti, L.Monsaingeo and M.Zhou, A KPP road-field system with spatially periodic exchange terms, Nonlinear Anal. 128 (2015), 273-302.
7. Y.Du, B.Lou and M.Zhou, Nonlinear diffusion problems with free boundaryes: convergence, transition speed and zero number arguments, SIAM J. Math. Anal. 47 (2015), no. 5, 3555-3584.
8. Y.Du, H.Matsuzawa and M.Zhou, Spreading speed and profile for nonlinear Stefan problems in high space dimensions, J. Math. Pures Appl. (9) 103 (2015), no. 3, 741-787.
9. J.Cai, B.Lou and M.Zhou, Asymptotic behavior of solutions of a reaction diffusion equation with free boundary conditions, J. Dynam. Differential Equations 26 (2014), no. 4, 1007-1028.
10. Y.Du, H.Matsuzawa and M.Zhou, Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal. 46 (2014), no. 1, 375-396.
Noninear Partial Differential Equations: Bubbles, Layers and Stability

Project title: Nonlinear Partial Differential Equations: Bubbles, Layers and Stability

Principal investigators: Prof. Yihong Du (UNE), Prof. Norm Dancer (Univ of Sydney), Prof. Shusen Yan (Central China Normal Univ.) and Prof. Chang-Shou Lin (National Taiwan Univ.)

Postdoctoral Fellow: Dr Benniao Li (UNE)

Funding body:   Australian Research Council (2017-2019)

Project description:

In this project we investigate several nonlinear elliptic partial differential equations that are of great concern in the recent study of the elliptic problems. They arise from well-established models in various applied fields, and the treatment of them poses great challenges to the current mathematical theory. We want to answer a number of important questions on these equations through deep analysis of the properties of their solutions, therefore enriching and expanding the existing mathematical theory in this area. One of the properties of the solutions under our investigation is known as bubbling, which describes the situation that as a key parameter in the equation approaches a certain critical value, the solution \mass" concentrates more and more at some points in the underlying space, like a Dirac delta function (in the limit). Such solutions are called "bubbling solutions", or "blowup solutions", and they frequently arise in various applied sciences, such as condensed matter physics and fluid mechanics.  Many important phenomena in the natural sciences are also described by solutions with sharp layers, representing sharp transitions of phases in physics, chemistry or biology. In the limit, a solution with a sharp layer converges to a discontinuous function. Since bubbling and sharp layered solutions do not converge to a continuous function in the limit, they pose great difficulties in partial differential equations, and have been a central topic of research of many groups of first rate mathematicians around the world. Another important property of the solutions that we will study in this project is characterized by the level of stability, measured by the Morse indices of the solutions. The bigger the Morse index, the less stable the solution, with Morse index 0 meaning the solution is stable under small perturbations (or is a stable equilibrium solution of the corresponding parabolic problem). We want to show that the behaviour of solutions with finite Morse index is usually not beyond reach and can sometimes be completely classified.

Related publications:

1. Wei, Lei; Du, Yihong Positive solutions of elliptic equations with a strong singular potential. Bull. Lond. Math. Soc. 51 (2019), no. 2, 251–266.
2. Dancer, E. Norman On a degree associated with the Gross-Pitaevskii system with a large parameter. Discrete Contin. Dyn. Syst. Ser. S 12 (2019), no. 7, 1835–1839.
3. Dancer, E. N.; Yang, Hui; Zou, Wenming Liouville-type results for a class of quasilinear elliptic systems and applications. J. Lond. Math. Soc. (2) 99 (2019), no. 2, 273–294.
4. Cao, Daomin; Guo, Yuxia; Peng, Shuangjie; Yan, Shusen Local uniqueness for vortex patch problem in incompressible planar steady flow. J. Math. Pures Appl. (9) 131 (2019), 251–289.
5. Deng, Yinbin; Guo, Yuxia; Yan, Shusen Multiple solutions for critical quasilinear elliptic equations. Calc. Var. Partial Differential Equations 58 (2019), no. 1, Art. 2, 26 pp.
6. Lin, Chang-Shou; Yan, Shusen On the mean field type bubbling solutions for Chern-Simons-Higgs equation. Adv. Math. 338 (2018), 1141–1188.
7. Guo, Yuxia; Li, Benniao; Pistoia, Angela; Yan, Shusen Infinitely many non-radial solutions to a critical equation on annulus. J. Differential Equations 265 (2018), no. 9, 4076–4100.
8. Peng, Shuangjie; Wang, Chunhua; Yan, Shusen Construction of solutions via local Pohozaev identities. J. Funct. Anal. 274 (2018), no. 9, 2606–2633.
9. Zhang, Xuemei; Du, Yihong Sharp conditions for the existence of boundary blow-up solutions to the Monge-Ampère equation. Calc. Var. Partial Differential Equations 57 (2018), no. 2, Art. 30, 24 pp.
10. Du, Yihong; Efendiev, Messoud Existence and exact multiplicity for quasilinear elliptic equations in quarter-spaces. Patterns of dynamics, 128–137, Springer Proc. Math. Stat., 205, Springer, Cham, 2017.
11. Wei, Lei; Du, Yihong Exact singular behavior of positive solutions to nonlinear elliptic equations with a Hardy potential. J. Differential Equations 262 (2017), no. 7, 3864–3886.
12. Dancer, E. N.; Gladiali, F.; Grossi, M. On the Hardy-Sobolev equation. Proc. Roy. Soc. Edinburgh Sect. A 147 (2017), no. 2, 299–336.
13. Lin, Chang-Shou; Yan, Shusen On condensate of solutions for the Chern-Simons-Higgs equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 5, 1329–1354.
14. Guo, Yuxia; Peng, Shuangjie; Yan, Shusen Local uniqueness and periodicity induced by concentration. Proc. Lond. Math. Soc. (3) 114 (2017), no. 6, 1005–1043.
15. Musso, Monica; Wei, Juncheng; Yan, Shusen; Infinitely many positive solutions for a nonlinear field equation with super-critical growth. Proc. Lond. Math. Soc. (3) 112 (2016), no. 1, 1-26.
16. Deng, Yinbin; Peng, Shuangjie; Yan, Shusen; Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations. J. Differential Equations 260 (2016), no. 2, 1228-1262.
17. Cao, Daomin; Peng, Shuangjie; Yan, Shusen; Planar vortex patch problem in incompressible steady flow. Adv. Math. 270 (2015), 263-301.
18. Deng, Yinbin; Lin, Chang-Shou; Yan, Shusen; On the prescribed scalar curvature problem in R^N, local uniqueness and periodicity. J. Math. Pures Appl. (9) 104 (2015), no. 6, 1013-1044.
19. Du, Yihong; Guo, Zongming; Finite Morse index solutions of weighted elliptic equations and the critical exponents. Calc. Var. Partial Differential Equations 54 (2015), no. 3, 3161-3181.
20. Du, Yihong; Wei, Lei; Boundary behavior of positive solutions to nonlinear elliptic equations with Hardy potential. J. Lond. Math. Soc. (2) 91 (2015), no. 3, 731-749.
21. Dancer, E. N.; A remark on stable solutions of nonlinear elliptic equations on R^3 or R^4. Differential Integral Equations 27 (2014), no. 5-6, 483-488.
22. Du, Yihong; Guo, Zongming; Wang, Kelei; Monotonicity formula and e-regularity of stable solutions to supercritical problems and applications to finite Morse index solutions. Calc. Var. Partial Differential Equations 50 (2014), no. 3-4, 615-638.
23. Lin, Chang-Shou; Yan, Shusen; Bubbling solutions for the SU(3) Chern-Simons model on a torus. Comm. Pure Appl. Math. 66 (2013), no. 7, 991-1027.
24. Lin, Chang-Shou; Yan, Shusen; Existence of bubbling solutions for Chern-Simons model on a torus. Arch. Ration. Mech. Anal. 207 (2013), no. 2, 353-392.
25. Du, Yihong; Guo, Zongming; Finite Morse-index solutions and asymptotics of weighted nonlinear elliptic equations. Adv. Differential Equations 18 (2013), no. 7-8, 737-768.
26. Dancer, E. N.; Du, Yihong; Efendiev, Messoud; Quasilinear elliptic equations on half- and quarter-spaces. Adv. Nonlinear Stud. 13 (2013), no. 1, 115-136.
Analysis and Modelling of Collective Animal Motion

Project title: Analysis and Modelling of Collective Animal Motion

Project participants: Dr Timothy Schaerf (University of New England), Professor Ashley Ward (University of Sydney)

Project description:
The spectacular patterns of collective movement have been, and remain, a long standing and major interest in many branches of science, including biology, mathematics, physics and computational science. From the early 1980s until the late 2000s computational self-propelled particle models were the dominant methods for examining how, and what type of, individual-level interactions could result in the amazing emergent patterns formed by groups such as flocking birds, shoaling fish and even crowds of humans. The influence of these models on the study of collective motion has been immense, with multiple model-based studies led by contemporary researchers cited thousands of times each. It is only in the last decade that advances in automated tracking methods have led to the exciting development of techniques for estimating the local rules of interaction used by real animals to coordinate collective motion directly from observational data (as opposed to inferring possible rules using models). In spite of these advances, many of the techniques for analysis are still in their infancy. This project will focus on refining existing techniques and developing new techniques for quantifying individual-interactions from automated tracking data. The more accurate analysis of experimental data will then be used to help refine and build on simulation models for collective animal movement.

Related publications:

1. M J Hansen, T M Schaerf and A J W Ward, The influence of nutritional state on individual and group movement behaviour in shoals of crimson-spotted rainbowfish (Melanotaenia duboulayi), Behavioral Ecology and Sociobiology, 69(10):1713-1722, (2015).
2. M J Hansen, T M Schaerf and A J W Ward, The effect of hunger on the exploratory behaviour of shoals of mosquitofish Gambusia holbrooki, Behavior, 152:12-13, (2015).
3. J E Herbert-Read, S Krause, L J Morrell, T M Schaerf, J Krause and A J W Ward, The role of individuality in collective group movement, Proceedings of the Royal Society B, 280:20122564, (2013).
4. A Strandburg-Peshkin et al., Visual sensory networks and effective information transfer in animal groups, Current Biology, 23:R709-R711, (2013).
5. Y Katz, K Tunstrøm, C C Ioannou, C Huepe and I D Couzin, Inferring the structure and dynamics of interactions in schooling fish, Proceedings of the National Academy of Sciences of the United States of America, 108:18720-18725, (2011).
6. J E Herbert-Read, A Perna, R P Mann, T M Schaerf, D J T Sumpter and A J W Ward, Inferring the rules of interaction of shoaling fish, Proceedings of the National Academy of Sciences of the United States of America, 108:18726-18731, (2011).
7. M Nagy, Z Àkos, D Biro and T Vicsek, Hierarchical group dynamics in pigeon flocks, Nature, 464:890-893, (2010).
8. R Lukeman, Y X Li and L Edlestein-Keshet, Inferring individual rules from collective behaviour, Proceedings of the National Academy of Sciences of the United States of America, 107(28):12576-12580, (2010).
9. M Ballerini et al., Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study, Proceedings of the National Academy of Sciences of the United States of America, 105(4):1232-1237, (2008).
10. I D Couzin, J Krause, N R Franks and S A Levin, Effective leadership and decision-making in animal groups on the move, Nature, 433:513-516, (2005).
11. I D Couzin, J Krause, R James, G D Ruxton and N R Franks, Collective memory and spatial sorting in animal groups, Journal of Theoretical Biology, 218:1-11, (2002).
12. D Helbing, I Farkas and T Vicsek, Simulating dynamical features of escape panic, Nature, 407:487-490, (2000).
13. W L Romey, Individual differences make a difference in the trajectories of simulated fish schools, Ecological Modelling, 92:65-77, (1996).
14. T Vicsek, A Czirók, E Ben-Jacob, I Cohen and O Shochet, Novel type of phase transition in a system of self-driven particles, Physical Review E, 75(6):1226-1229, (1995).
15. D Helbing and P Molnar, Social force model for pedestrian dynamics, Physical Review E, 51(5):4282-4286, (1995).
16. A Huth and C Wissel, The simulation of fish schools in comparison with experimental data, Ecological Modelling, 75:135-146, (1994).
17. A Huth and C Wissel, The simulation of the movement of fish schools, Journal of Theoretical Biology, 156(3):365-385, (1992).
18. D Helbing, A mathematical model for the behavior of pedestrians, Behavioral Science, 36(4):298-310, (1991).
19. C W Reynolds, Flocks, herds, and schools: a distributed behavioral model, Computer Graphics, 21:25-34, (1987).
20. I Aoki, A simulation study on the schooling mechanism in fish, Bulletin of the Japanese Society of Scientific Fisheries, 48(8):1081-1088, (1982).
Removable Singularities for Monopoles on a Sasakian 3-manifold (PhD project)

Project title: Removable Singularities for Monopoles on a Sasakian 3-manifold (PhD project)

PhD student: Mr Kumbu Dorji

Project description:
A classic problem of higher-dimensional complex analysis concerns the extension of complex-analytic structures, such as holomorphic functions or vector bundles, across a gap locus, which in the first instance may simply be a point. Theorems of this kind for functions are more than a century old, whereas "removable singularities" theorems for vector bundles have a more recent origin. A famous theorem of Uhlenbeck shows that point singularities of unitary vector bundles over a punctured ball in real four-dimensional space are removable if certain conditions are imposed on the curvature form associated with a choice of hermitian metric. Such conditions are typically representable as differential equations governing gauge-fields which arise in theoretical physics.  One such is the so-called anti self-dual form of the Yang-Mills equation, in terms of which a unitary bundle can be endowed with a natural holomorphic structure. Techniques available for the study of removable singularities in this setting are applied to this project by means of a two-fold reduction from four dimensions to three. On one hand, a particular form of the magnetic monopole equation, first studied by Dirac, can be realised as a time-independent reduction of the anti self-dual equation. On the other, every Sasakian three-manifold is the canonical section of a certain complex Kahler surface.

Related publications:

1. I. Biswas and J. Hurtubise, Monopoles on Sasakian three-folds, arXiv:1412.4050v1 (2014)
2. C.P. Boyer and K. Galicki, Sasakian Geometry, Oxford Math. Monographs, OUP (2008)
3. N.P. Buchdahl and A. Harris , Holomorphic connections and extension of complex vector bundles. Math. Nachr. 204 (1999) 29-39
4. A. Harris and Y. Tonegawa, Analytic continuation of vector bundles with Lp-curvature. Int. J. Math. 11, No. 1 (2000) 29-40
Symmetries and Mapping in Real and Complex Geometry

Project title: Symmetries and Mappings in Real and Complex Geometry

Principal investigator: A/Prof. Gerd Schmalz (UNE)

Other participants: A/Prof. Vladimir Ejov (Flinders), Prof. Dmitri Alekseevsky (Moscow), A/Prof. Martin Kolar (Brno), Dr Alessandro Ottazzi (UNSW)

Funding body: Australian Research Council (2013-2016)

Project description:
This project is focused on a wide range of interrelated areas of Differential and Cauchy-Riemann geometry and Geometric Analysis.  Cauchy-Riemann-geometry has its roots in the study of the geometric properties of boundaries of domains in complex space. In recent time it has received much attention in Differential and Riemannian geometry. On the one hand, Cauchy-Riemann manifolds serve as test objects for the general machinery of parabolic geometry, on the other hand, there is a deep relation between CR geometry and Riemannian, pseudo-Riemannian and conformal geometry, which was established by Fields prize winner (and 2016 Wolf prize winner) Charles Fefferman.  Particular aims of the project are: studying mappings and embeddings of CR-manifolds into other CR-manifolds and complex space, exploring the relation between CR-manifolds and its generalisations with shear-free Lorentzian geometry, finding new classes of homogeneous CR-manifolds and classifying flat Sasakian manifolds.

Related publications:

1. OTTAZZI, A., SCHMALZ, G., Singular multicontact structures. J. Math. Anal. Appl. 443:2, 1220-1231, 2016.
2. EJOV, V., SCHMALZ, G., The zero curvature equation for rigid CR-manifolds, Complex Variables and Elliptic equations, 61:4, 443-447, 2016.
3. JOO, Ch.-J., KIM, K.-T., SCHMALZ, G.,  On the generalization of Forelli's theorem, Mathematische Annalen, 365:3-4, 1187-1200, 2016.
4. EJOV, V., SCHMALZ, G., Explicit description of spherical rigid hypersurfaces in C^2,  Complex Analysis and its Synergies,  1.1,  Paper no. 2, 10 pp, 2015.
5. DE HOOG, F., SCHMALZ, G., GUREYEV, T.E., An uncertainty inequality, Appl. Math. Lett., 38, 84-86, 2014.
6. EJOV, V., SCHMALZ, G., Spherical rigid hypersurfaces C^2, Differential Geom. Appl., 33, suppl., 267-271, 2014.
7. JOO, Ch.-J., KIM, K.-T., SCHMALZ, G., A generalization of Forelli's theorem, Mathematische Annalen, 355, no. 3, 1171-1176, 2013.
8. EJOV, V., Kolar, M., SCHMALZ, G., Normal forms and symmetries of real hypersurfaces of finite type in C^2, Indiana Univ. Math. J., 62, no. 1, 1-32, 2013.
9. SCHMALZ, G., SLOVAK, J., Free CR distributions, Cent. Eur. J. Math., 10,  1896-1913, 2012.
Models of the honeybee nest-site selection process

Project title: Models of the honeybee nest-site selection process

Project participants: Dr Timothy Schaerf (University of New England), Professor Mary Myerscough (University of Sydney), Professor Madeleine Beekman (University of Sydney)

Project description:

As part of their reproductive cycle, colonies of honeybees issue swarms. These swarms must then find a site to establish a new colony quickly, as a swarm is unable to store food without established combs, and is often vulnerable to the elements. Aspects of the complex process used by the cavity nesting honeybee Apis mellifera to select a new nest-site are relatively well understood due to a century of scientific research founded on the works of Karl von Frisch and Martin Lindauer (A. mellifera is the species of honeybee used by the honey industry). It is only in the last decade that detailed scientific investigations of the nest-site selection process of other species, of which there are at least six, have been undertaken. Common to each of the species studied with some detail (the cavity nesting A. mellifera, the red dwarf honeybee, A. florea, and the giant honeybee, A. dorsata) swarms release scouts to investigate the surrounding landscape for potential nest-sites. Upon their return to the swarm, scouts that have found a suitable target then perform a waggle dance, a stylised behaviour first recognised by Karl von Frisch as encoding the distance and direction to a resource of interest in the environment (in the context of nest-site selection, this resource is a potential new home). Other bees on the swarm follow such waggle dances, and then use the encoded information to help them find and assess the advertised site themselves. If these recruited bees also think that the advertised site is suitable, then they will return to the swarm, and advertise the location of the site to other bees with their own waggle dances. Over time, if enough bees are recruited to a particular specific site (in the case of A. mellifera) or a general direction (as seems to be the case for A. florea and perhaps A. dorsata), then the swarm chooses that site as a collective, and then flies to the new site to start building their new colony.

A number of mathematical models have been developed to better understand the honeybee nest-site selection process, including systems of ordinary and stochastic differential equations, matrix models and individual-based simulation models. Most of these models have been developed specifically to examine nest-site selection by A. mellifera, and take into account a number of known species specific behaviours. Participants in this project will build on existing models, and develop new models, to best reflect current knowledge of the overall nest-site selection processes of A. mellifera, A. florea and A. dorsata. These models will then be used to examine effects of swarm size (and hence the number of active participants in nest-site selection), noisy communication, and the number and relative positions of viable nesting sites on the decision-making ability of swarms of each species. Further work could then include developing a multi-generational nest-site selection model to examine how quickly species can adjust their nest-site selection process, if at all, when changing environmental conditions force a change in the type of nest-site sought.

Related publications:

1. T D Seeley and S C Buhrman, Group decision making in swarms of honey bees, Behavioral Ecology and Sociobiology, 45:19-31, (1999).

2. T D Seeley and S C Buhrman, Nest-site selection in honey bees: how well do swarms implement the “best-of-N” rule?, Behavioral Ecology and Sociobiology, 49:416-427, (2001).

3. N F Britton, N R Franks, S C Pratt and T D Seeley, Deciding on a new home: how do honeybees agree?, Proceedings of the Royal Society of London B, 269:1383-1388, (2002).

4. M R Myerscough, Dancing for a decision: a matrix model for nest-site choice by honeybees, Proceedings of the Royal Society of London B, 270:577-582, (2003).

5. K M Passino and T D Seeley, Modeling and analysis of nest-site selection by honeybee swarms: the speed and accuracy trade-off, 59:427-442, (2006).

6. B P Oldroyd, R S Gloag, N Even, W Wattanachaiyingcharoen and M Beekman, Nest site selection in the open-nesting honeybee Apis florea, Behavioral Ecology and Sociobiology, 62:1643-1653, (2008).

7. C List, C Elshotlz and T D Seeley, Independence and interdependence in collective decision making: an agent-based model of nest-site choice by honeybee swarms, Philosophical Transactions of the Royal Society Bm 364:755-762, (2009).

8. J C Makinson, B P Oldroyd, T M Schaerf, W Wattanachaiyingcharoen and M Beekman, Moving home: nest-site selection in the Red Dwarf honeybee (Apis florea), Behavioral Ecology and Sociobiology, 65:945-958, (2011).

9. K Diwold, T M Schaerf, M R Myerscough, M Middendorf and M Beekman, Deciding on the wing: in-flight decision making and search space sampling in the red dwarf honeybee Apis florea, Swarm Intelligence, 5:121-141, (2011).

10. T M Schaerf, M R Myerscough, J C Makinson and M Beekman, Inaccurate and unverified information in decision making: a model for the nest site selection process of Apis florea, Animal Behaviour, 82:995-1013, (2011).

12. T D Seeley, P K Visscher, T Schlegel, P M Hogan, N R Franks and J A R Marshall, Stop signals provide cross inhibition in collective decision-making by honeybee swarms, Science, 335:108-111, (2012).

13. T M Schaerf, J C Makinson, M R Myerscough and M Beekman, Do small swarms have an advantage when house hunting? The effect of swarm size on nest-site selection by Apis mellifera, Journal of the Royal Society Interface, 10:20130533, (2013).

14. J C Makinson, T M Schaerf, A Rattanawannee, B P Oldroyd and M Beekman, Consensus building in giant Asian honeybee, Apis dorsata, swarms on the move, Animal Behaviour, 93:191-199, (2014).

15. M Beekman, J C Makinson, M J Couvillon, K Preece and T M Schaerf, Honeybee linguistics – a comparative analysis of the waggle dance among species of Apis, Frontiers in Ecology and Evolution, 3:11, (2015).

16. J C Makinson, T M Schaerf, A Rattanawannee, B P Oldroyd and M Beekman, How does a swarm of the giant Asian honeybee Apis dorsata reach consensus? A study of the individual behaviour of scout bees, Insectes Sociaux, 63:395-406, (2016).

17. J C Makinson, T M Schaerf, N Wagner, B P Oldroyd and M Beekman, Collective decision making in the red dwarf honeybee Apis florea: do the bees simply follow the flowers?, Insectes Sociaux, 64:557-566, (2017).

Modelling the effects of climate change on dung beetle populations

Project title: Modelling the effects of climate change on dung beetle populations

Project participants: Associate Professor Nigel Andrew (University of New England), Dr Timothy Schaerf (University of New England)

Project description: As part of their life-cycle, dung beetles actively remove dung from the environment, either for use as food or breeding chambers for their offspring. The removal of dung by these beetles is also an important ecosystem service, with impact that extends to human agriculture. For example, in Australia dung beetles remove pasture dung, and as a consequence: enhance soil nutrient cycling, water penetration and soil aeration; destroy pest-fly breeding sites; and reduce the impacts of livestock gastro-intestinal parasites.

Available evidence suggests that climate change will have a substantial impact on dung beetles. At the individual level, warming advances egg laying and hatching rates, decreases egg and larval size, and decreases dung beetle survival. At population levels, warmer conditions have been linked to altitudinal shifts in the ranges of dung beetle species, and to changes in habitat selection by dung beetles. There is also evidence that warming substantially reduces dung breakdown by beetles, with an associated reduction in growth of plants surrounding dung patches. Logically, the impacts on beetles are likely to effect the services that they provide to their ecosystems.

This project will involve the development of individual based models that model dung beetle growth, development and reproductive output using dynamic energy budget theory (systems of ordinary differential equations that model how organisms allocate available energy resources in order to grow and reproduce). These models will be informed by experimental and observational work performed by Associate Professor Nigel Andrew and members of the Insect Ecology Laboratory at the University of New England. Observational data from multiple species present in Australia will include the effects of variable climate on beetle growth, maturation and reproductive output, competition within and between species for available dung resources, movements of beetles through their environment, and the general availability of dung. The models can be developed to examine possible climate effects on dung beetle populations at varying spatial scales, and to examine possible adaptation of beetles across multiple generations subject to changing climatic conditions.

Related publications:

1. X Wu and S Sun, Artificial warming advances egg-laying and decreases larval size in the dung beetle Aphodius erraticus (Coleoptera: Scarabaeidae) in a Tibetan alpine meadow, Annales Zoologci Fennici, 49:174-180, (2012).

2. T H Larsen, Upslope range shifts of Andean dung beetles in response to deforestation: compounding and confounding effects of microclimatic change, Biotropica, 44:82-89, (2012).

3. R Menéndez and D Gutiérrez, Shifts in habitat associations of dung beetles in northern Spain: climate change implications, Écoscience, 11:329-337, (2004).

4. X Wu, J E Duffy, P B Reich and S Sun, A brown-world cascade in the dung decomposer food web of an alpine meadow: effects of predator interactions and warming, Ecological Monographs, 81: 313-328, (2011).

5. S A L M Kooijman, Dynamic Energy Budget Theory for Metabolic Organisation, Cambridge University Press, (2010).

6. B T Martin, E I Zimmer, V Grimm and T Jager, Dynamic energy budget theory meets individual-based modelling: a generic and accessible implementation, Methods in Ecology and Evolution, 3:445-449, (2012).

7. B T Martin, T Jager, R M Nisbet, T G Preuss and V Grimm, Predicting population dynamics from the properties of individuals: a cross-level test of dynamic energy budget theory, American Naturalist, 181:506-519, (2013).

8. M Renton, Aristotle and adding an evolutionary perspective to models of plant architecture in changing environments, Frontiers in Plant Science, 4:284, (2013).

9. M C Welch, P W Kwan and A S M Sajeev, Applying GIS and high performance agent-based simulation for managing old world screwworm fly invasion of Australia, Acta Tropica, 138S:S82-S93, (2014).

Integration of network design and scheduling

Project Title: Integration of network design and scheduling

Project Participants: Natashia Boland (Georgia Tech), Konrad Engel (University of Rostock), Thomas Kalinowski (UNE), Dmytro Matsypura (University of Sydney), Martin Savelsbergh (Georgia Tech), Hamish Waterer (University of Newcastle)

Project Description:
We study optimisation problems in which a network is used over a certain time horizon, and at the same time the network is modified using limited resources. Applications arise in the optimal reconstruction planning for infrastructure networks (transportation, water, electricity) after natural disasters like earthquakes or hurricanes, and in planning maintenance or capacity expansions in transportation networks. The class of incremental network design problems has been introduced to study essential features of these complex optimisation problems in a simplified context. Every classical network optimisation problem (shortest path, maximum flow,...) yields an incremental variant. In general, these problems are NP-complete, and this motivates the following questions: Can the problem be solved in polynomial time under additional assumptions which might be satisfied in some application contexts? Can we find approximation algorithms, that is, algorithms with polynomial run-time and a guarantee for the quality of the solution? Or is the problem even hard to approximate? How can we understand the combinatorial and polyhedral structure of the problem better, and then use this understanding for the development of practical algorithms?

Related Publications

1. Natashia Boland, Thomas Kalinowski, Hamish Waterer, and Lanbo Zheng. “Mixed integer programming based maintenance scheduling for the Hunter Valley Coal Chain”. Journal of Scheduling 16.6 (2013), pp. 649-659.
2. Natashia Boland, Thomas Kalinowski, Hamish Waterer, and Lanbo Zheng. “Scheduling arc maintenance jobs in a network to maximize total flow over time”. In: Discrete Applied Mathematics 163.1 (2014), pp. 34-52.
3. Matthew Baxter, Tarek Elgindy, Andreas T. Ernst, Thomas Kalinowski, and Martin W. P. Savelsbergh. “Incremental network design with shortest paths”. European Journal of Operational Research 238.3 (2014), pp. 675-684.
4. Thomas Kalinowski, Dmytro Matsypura, and Martin W. P. Savelsbergh. “Incremental network design with maximum flows”. European Journal of Operational Research 242.1 (2015), pp. 51-62.
5. Konrad Engel, Thomas Kalinowski, and Martin W. P. Savelsbergh. “Incremental network design with minimum spanning trees”. Journal of Graph Algorithms and Applications 21 (2017), pp. 417-432.
Convexification of bilinear functions

Project Title: Convexification of bilinear functions

Project Participants: Natashia Boland (Georgia Tech), Akshay Gupte (University of Edinburgh) Thomas Kalinowski (UNE), Hamish Waterer (University of Newcastle)

Project Description:
Optimisation problems with bilinear functions in the constraints or the objective functions are important in applications, for instance in the petrochemical industry, since they appear naturally in the description of blending processes. Due to the non-convexity of these functions and the problem sizes appearing in practice, it is very challenging to solve such problems. An important ingredient in all state-of-the-art solvers is the approximation of the non-convex functions by convex ones, and the quality of this approximation is crucial for the performance of the algorithm. Certain properties of such a function are captured by a graph whose vertices are the decision variables, and two variables are joined by an edge if their product appears in the function. We study the relation between the combinatorics of this graph and the convex hull of the graph of the function. Methods from the theory of random graphs and linear programming duality can be used to characterise the worst case gap between the convex hull and the McCormick hull, which is a standard convexification used in most software packages. In this project we apply geometric methods to characterise the convex hulls for some special classes of functions.

Relevant Publications

1. James Luedtke, Mahdi Namazifar, and Jeff Linderoth. “Some results on the strength of relaxations of multilinear functions”. Mathematical Programming 136.2 (2012), pp. 325-351
2. Mark Zuckerberg. “Geometric proofs for convex hull defining formulations”. Operations Research Letters 44.5 (2016), pp. 625-629.
3. Natashia Boland, Santanu S. Dey, Thomas Kalinowski, Marco Molinaro, and Fabian Rigterink. “Bounding the gap between the McCormick relaxation and the convex hull for bilinear functions”. Mathematical Programming 162 (2017), pp. 523-535
4. Akshay Gupte, Thomas Kalinowski, Fabian Rigterink, and Hamish Waterer. “Extended formulations for convex hulls of some bilinear functions”. Discrete Optimization 36 (2020), paper number 100569.
Propagation in graphs

Project Title: Propagation in graphs

Project participants: Randy Davila (University of Houston), Daniela Ferrero (Texas State University), Thomas Kalinowski (UNE), Joe Ryan (University of Newcastle), Sudeep Stephen (University of Auckland)

Project Desription:
The zero forcing number is a graph theoretical parameter which has been studied under various aspects. It describes a certain propagation process in networks, and is closely related to the power domination number which characterises the minimum number of sensors necessary for monitoring an electrical network. In this project we study the relation between the zero forcing number and other graph parameters, the zero forcing number for special classes of graphs, and for random graphs.

Relevant publications:

1. Randy Davila, Thomas Kalinowski, and Sudeep Stephen. “A lower bound on the zero forcing number”. Discrete Applied Mathematics 250 (2018), pp. 363–367.
2. Daniela Ferrero, Thomas Kalinowski, and Sudeep Stephen. “Zero forcing in iterated line digraphs”. Discrete Applied Mathematics 255 (2019), pp. 198-208.
3. Daniela Ferrero, Cyriac Grigorious, Thomas Kalinowski, Joe Ryan, and Sudeep Stephen. “Minimum rank and zero forcing number for butterfly networks”. Journal of Combinatorial Optimization 37.3 (2019), pp. 970-988.
4. Thomas Kalinowski, Nina Kamčev, and Benny Sudakov. “The zero forcing
Extremal combinatorics of finite sets

Project Title: Extremal combinatorics of finite sets

Project participants: Jerrold Griggs (University of South Carolina), Konrad Engel (Universität Rostock), Thomas Kalinowski (UNE), Uwe Leck (Europa-Universität Flensburg).

Project Description:
A central topic in extremal set theory is the study of the subset lattice of a finite set. In this project we study the structure of the antichains in this partial order. For instance, we are interested in the possible sizes of maximal antichains under additional assumptions on the cardinalities of the members of the antichain. A long term goal in this area is a simplified proof of the Flat Antichain Theorems, which says that for every antichain we can find a flat antichain (that is, an antichain consisting of k-sets and (k+1)-sets for some k) of the same size and volume.

Relevant publications:

1. Martin Grüttmüller, Sven Hartmann, Thomas Kalinowski, Uwe Leck, and Ian T. Roberts. “Maximal flat antichains of minimum weight”. The Electronic Journal of Combinatorics 16 (2009), R69.
2. Thomas Kalinowski, Uwe Leck, and Ian T. Roberts. “Maximal antichains of minimum size”. The Electronic Journal of Combinatorics 20 (2013), P3.
3. Thomas Kalinowski, Uwe Leck, Christian Reiher, and Ian T. Roberts. “Minimizing the regularity of maximal regular antichains of 2- and 3-sets”. In: Australasian Journal of Combinatorics 64 (2016), pp. 277-288.
Cauchy-Riemann Geometry

Project title: Cauchy-Riemann Geometry

Principal investigator: Prof. Gerd Schmalz (UNE)

Other participants: A/Prof. Vladimir Ejov (Flinders), Prof. Dmitri Alekseevsky (Moscow), A/Prof. Martin Kolar (Brno), Dr Alessandro Ottazzi (UNSW), A/Prof. Andrea Spiro (Camerino), Dr Masoud Ganji

Project description:
This project is focused on a wide range of interrelated areas of Differential Geometry and Geometric Analysis.  Cauchy-Riemann-geometry has its roots in the study of the geometric properties of boundaries of domains in complex space. In recent time it has received much attention in Differential and Riemannian geometry. On the one hand, Cauchy-Riemann manifolds serve as test objects for the general machinery of parabolic geometry, on the other hand, there is a deep relation between CR geometry and Riemannian, pseudo-Riemannian and conformal geometry, which was established by Fields prize winner (and 2016 Wolf prize winner) Charles Fefferman.  Particular aims of the project are: studying mappings and embeddings of CR-manifolds into other CR-manifolds and complex space, exploring the relation between CR-manifolds and its generalisations with shear-free Lorentzian geometry, finding new classes of homogeneous CR-manifolds and classifying flat Sasakian manifolds.

Related publications:

1. MEDVEDEV, A., SCHMALZ, G., EZHOV, V., On the classification of homogeneous affine tube domains with large automorphism groups in arbitrary dimension. Adv. Math. 364 (2020) (online)
2. SCHMALZ, G., GANJI, M., A criterion for local embeddability of three-dimensional CR structures. Ann. Mat. Pura Appl. (4) 198:2, 491-503, 2019.
3. ALEKSEEVSKY, D.V., GANJI, M., SCHMALZ, G., CR-geometry and shearfree Lorentzian geometry. Geometry complex analysis, 11-22, Springer Proc. Math. Stat., 246, Springer, Singapore, 2018
4. EZHOV, V., Kolar, M., SCHMALZ, G., Rigid embeddings of Sasakian hyperquadrics in C^n+1. J. Geom. Anal. 28:3, 2185-2205, 2018.
5. OTTAZZI, A., SCHMALZ, G., Normal forms of para-CR hypersurfaces. Differential Geom. Appl. 52, 78-93, 2017.
6. OTTAZZI, A., SCHMALZ, G., Singular multicontact structures. J. Math. Anal. Appl. 443:2, 1220-1231, 2016.
7. EJOV, V., SCHMALZ, G., The zero curvature equation for rigid CR-manifolds, Complex Variables and Elliptic equations, 61:4, 443-447, 2016.
8. JOO, Ch.-J., KIM, K.-T., SCHMALZ, G., On the generalization of Forelli's theorem, Mathematische Annalen, 365:3-4, 1187-1200, 2016.
9. EJOV, V., SCHMALZ, G., Explicit description of spherical rigid hypersurfaces in C^2, Complex Analysis and its Synergies, 1.1, Paper no. 2, 10 pp, 2015.
10. EJOV, V., SCHMALZ, G., Spherical rigid hypersurfaces C^2, Differential Geom. Appl., 33, suppl., 267-271, 2014.