Mathematics Research Projects

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.
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.
Nonlinear 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.
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.
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
Removable Singularities for Monopoles on a Sasakian 3-manifold (PhD project)

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

Supervisor: Dr Adam Harris (UNE)

PhD student: Mr Kumbu Dorji

Funding: UNE International Postgraduate Scholarship  

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 Mappings 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.