## Research Features

### Can we understand rogue bursts as an effect of broken symmetry?

#### Priya Subramanian

The formation of rogue waves is of interest, from North sea waves [1-3], waves in tanks[4-7], to waves in nonlinear optics[8-11]. Most common models used to investigate rogue bursts have used the nonlinear Schrödinger (NLS) equation and its variants. However, such integrable settings and analytical solutions are rare in higher dimensions. So we use the model of a dissipative system: which describes interaction between standing waves in domains of moderate aspect ratio. When spatial reflection symmetry is broken, the left and right running waves become distinguishable and can interact strongly producing a spatially and temporally localised extremely large amplitude event, i.e., a rogue burst [12].

###### Figure 1: Evolution of amplitude at a location in the system shown in log 10 scale. We see three large amplitude excursions (rogue bursts) with the largest one (in black) reaching value of O(10^{7}). The brown circles are were our analysis can identify a precursor that predicts an impending rogue burst. For more details, see [12].

__References__

- Mori and P. C. Liu, Analysis of freak wave measurements in the Sea of Japan, Ocean Eng. 29, 1399 (2002).
- Haver, A possible freak wave event measured at the Draupner jacket January 1 1995, Rogue Waves 2004: Proceedings of a Workshop, Brest, France (unpublished).
- A. G. Walker, P. H. Taylor, and R. E. Taylor, The shape of large surface waves on the open sea and the Draupner New Year wave, Appl. Ocean Res. 26, 73 (2004).
- Chabchoub, N. P. Hoffmann, and N. Akhmediev, Rogue Wave Observation in a Water Wave Tank, Phys. Rev. Lett. 106, 204502 (2011).
- Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev, Super Rogue Waves: Observation of a Higher-Order Breather in Water Waves, Phys. Rev. X 2, 011015 (2012).
- L. McAllister, S. Draycott, T. A. A. Adcock, P. H. Taylor, and T. S. Van Den Bremer, Laboratory recreation of the Draup- ner wave and the role of breaking in crossing seas, J. Fluid Mech. 860, 767 (2019).
- Xu, A. Chabchoub, D. E. Pelinovsky, and B. Kibler, Ob- servation of modulation instability and rogue breathers on stationary periodic waves, Phys. Rev. Research 2, 033528 (2020).
- R. Solli, C. Ropers, P. Koonath, and B. Jalali, Optical rogue waves, Nature (London) 450, 1054 (2007).
- M. Dudley, F. Dias, M. Erkintalo, and G. Genty, Instabilities, breathers and rogue waves in optics, Nat. Photonics 8, 755 (2014).
- Frisquet, B. Kibler, P. Morin, F. Baronio, M. Conforti, G. Millot, and S. Wabnitz, Optical dark rogue wave, Sci. Rep. 6, 20785 (2016).
- Tikan, C. Billet, G. El, A. Tovbis, M. Bertola, T. Sylvestre, F. Gustave, S. Randoux, G. Genty, P. Suret, and J. M. Dudley, Universality of the Peregrine Soliton in the Focusing Dynamics of the Cubic Nonlinear Schrödinger Equation, Phys. Rev. Lett. 119, 033901 (2017).
- Subramanian, E. Knobloch and P. G. Kevrekidis, Forced symmetry breaking as a mechanism for rogue bursts in a dissipative nonlinear dynamical lattice, Phys. Rev. E 106, 014212 (2022).

### Mathematical recipes for never-repeating quasicrystals

#### Priya Subramanian

The dynamics of many physical systems often evolve to asymptotic states that exhibit spatial and temporal variations in their properties such as density, temperature, etc. Regular patterns such as graph paper and honeycombs look the same when moved by a basic unit and/or rotated by certain special angles. They possess both translational and rotational symmetries giving rise to discrete spatial Fourier transforms. Among all possible arrangements, such regular arrangements are preferred in nature because they are associated with the least amount of energy required to assemble them. However an aperiodic crystal was discovered by Schectman [1] in 1982 which displayed long range order but no periodicity. Such quasicrystals lack the lattice symmetries of regular crystals, yet they still have discrete Fourier spectra.

The vast majority of the quasicrystals discovered so far are metallic alloys with at least two components (e.g., Al/Mn or Cd/Ca). However, quasicrystals have recently been found in nanoparticles [2], mesoporous silica [3], and in soft-matter [4, 5] systems. Metallic quasicrystals (3D quasicrystals in bulk) have low friction and high corrosion resistance which is advantageous in prosthesis manufacture and are excellent heat insulators while polymeric quasicrystals (2D quasicrystals on surfaces) can be used in photonic devices to manipulate light. Considering the difference in scale between metallic and polymeric quasicrystals, there is a need for mathematical models that explore the unifying mechanisms that generate quasicrystals independent of the microscopic structure.

Previous theoretical investigations have focused on understanding the formation of 2D quasicrystals on surfaces [6, 7, 8, 9, 10]. I develop mathematical recipes, consisting of both models to understand the formation of quasicrystals both on surfaces and in bulk along with tools to characterize their properties, analyse their growth and stability within the framework of the following four focus questions. Below we see examples of dodecagonal quasicrystal in 2D (panels (*a*) and (*b*)) and icosahedral quasicrystals in 3D (panels (*c*) and (*d*)) from [11].

__References __

- Shechtman, I. Blech, D. Gratias and J.W. Cahn, Phys. Rev. Lett 53, 1951 (1984).
- V. Talapin, E. V. Shevchenko, M. I. Bodnarchuk, X. Ye, J. Chen, and C. B. Murray, Nature (London) 461, 964 (2009).
- Xiao, N. Fujita, K. Miyasaka, Y. Sakamoto, and O. Terasaki, Nature (London) 487, 349 (2012).
- Dotera, Isr. J. Chem. 51, 1197 (2011).
- Fischer, A. Exner, K. Zielske, J. Perlich, S. Deloudi, W. Steurer, P. Lindner, and S. Frster, Proc. Natl. Acad. Sci. U.S.A. 108, 1810 (2011).
- Barkan, H. Diamant, and R. Lifshitz, Phys. Rev. B 83, 172201 (2011).
- J. Archer, A. M. Rucklidge, and E. Knobloch, Phys. Rev. Lett. 111, 165501 (2013).
- Dotera, T. Oshiro, and P. Ziherl, Nature (London) 506, 208 (2014).
- Barkan, M. Engel, and R. Lifshitz, Phys. Rev. Lett. 113, 098304 (2014).
- J. Archer, A. M. Rucklidge, and E. Knobloch, Phys. Rev. E 92, 012324 (2015).
- Subramanian, A. J. Archer, E. Knobloch and A. M. Rucklidge, Phys. Rev. Lett. 117, 075501 (2016).

### The mathematics of public key cryptography

#### Professor Steven Galbraith

Professor Steven Galbraith works on the mathematics of public key cryptography, which is an interdisciplinary subject that bridges foundational research in pure mathematics and applied areas of computer science. Cryptography is one of the tools that secures the internet.

The security systems being used today will not be secure in the future when powerful quantum computers are built. So it is natural to design new systems that will be secure in a post-quantum future. Since 2010 Professor Galbraith has been working on developing such cryptographic systems. He has made a number of high-impact and foundational contributions to the understanding of the field.

Here are two case studies:

**1.**

One of the hot topics in post-quantum cryptography is lattice-based cryptography, which exploits a mathematical object called a lattice which is used in many areas of mathematics. Together with a post-doctoral researcher Dr Shi Bai at the University of Auckland, Professor Galbraith published the paper:

Shi Bai and Steven D. Galbraith, An Improved Compression Technique for Signatures Based on Learning with Errors, in J. Benaloh (Ed.), CT-RSA 2014, LNCS 8366 (2014) 28-47.

This paper introduces a general tool for making lattice-based digital signature schemes more efficient. This tool was later used by the designers of the CRYSTALS-Dilithium digital signature, which has been selected by the US National Institute of Standards and Technology for standardization. In a few years time, the CRYSTALS-Dilithium signature will be widely used in practice.

Press release on this work from last year:

https://www.auckland.ac.nz/en/news/2022/07/08/quantum-hackers.html

**2. **

Isogeny-based post-quantum cryptography is based on mathematical objects that arise in algebraic geometry and number theory. Professor Galbraith, together with a PhD student, conducted foundational research to describe and explain certain structures that appear in isogeny graphs of supersingular elliptic curves. This work was published in the paper:

Christina Delfs and Steven D. Galbraith, Computing isogenies between supersingular elliptic curves over F_p, Designs, Codes and Cryptography, Volume 78, Issue 2 (2016) 425-440.

Subsequently, these ideas were used by others to invent a new cryptosystem called CSIDH. It is not clear whether CSIDH will ever be used in practice, but it has created an entirely new and very active field of research.