Brian Ng
Monash University
I am in the middle of my PhD degree at Monash University. Originally from Hong Kong, I did my undergraduate degree in the UK in aeronautics and am now researching in the department of electrical and computer systems engineering. Previously, I have worked in the Environmental Protection Department in Hong Kong to develop solutions that facilitate acoustic modelling and noise pollution related law enforcement.
1. Can you give me a quick rundown about the type of mathematics you are studying and its potential impacts for the broader community (think how you would explain your work and studies to others who don’t study maths)
I am interested in convex optimisation, polynomial optimisation in particular. As you might notice from my biography, I studied aeronautical engineering before working on polynomial optimisation in the department of Electrical Engineering. While the three areas seem disjoint at first glance, they employ the same technique in Mathematics to solve questions from different applications.
Aeronautics is about aircrafts, there is structural engineering, material engineering, and perhaps most importantly, fluid mechanics. The behaviour of fluid is postulated to be described by the Navier-Stokes equations. While we have successful numerical approximations to the equation leading to the advancement of computational fluid dynamics and much better designed aircrafts, it remains an open problem to prove the existence and uniqueness of its solution. While there has been substantial progress made in understanding the stability of fluid flow analytically, mathematicians are still experimenting with different methods. Recent attempts involve the use of polynomial optimisation to obtain bounds to fluid problems (Chernyshenko, et al., 2014). This introduced me to the field of polynomial optimisation. Optimisation is a popular word in society these days, but it has a very precise mathematical meaning. In simple terms, we have an objective expression that we wish to minimise (or maximise), followed by some constraints expressed in Mathematical formulations. This forms an optimisation problem.
Polynomial optimisation entails the use of polynomials in the expression of the optimisation problem. It was first popularized in control engineering, but subsequently found itself in various fields including operational research, biomedical engineering, signal processing, material science, speech recognition etc. The functioning of mechanisms or design is not always optimal, the development of polynomial optimisation allows the operator to extract the maximum benefit from tweaking designs.
My research interest is about nonnegative polynomial certificates. They are the primary toolset that enables polynomial optimisation. However, despite its trivial name, there remains a lot of unanswered questions about the set of nonnegative polynomials. How do we certify any multivariate polynomials as nonnegative? Hilbert’s seventeenth problem says “Given a multivariate polynomial that takes only non-negative values over the reals, can it be represented as a sum of squares of rational functions?” The problem was answered. But it leads to more: “If they are not sum of squares, what are they?” “How do we design a certificate that includes all (or more than sum of squares) of the nonnegative polynomials?” Chernyshenko, S. I., Goulart, P., Huang, D. & Papachristodoulou, A., 2014. Polynomial sum of squares in fluid dynamics: a review with a look ahead. Phil. Trans. R. Soc. A., 372(2020).
2. How did you get into mathematics/statistics/data science? Was there someone or something that inspired you to this field?
I was fortunate enough to have a great pool of talented mathematicians teaching in the Aeronautics department in my undergraduate years, the mathematical rigour and precision drew my attention and interest two years into my degree. Traditionally, engineering involves a lot of hand-wavy approximations, compromising mathematical rigour for practicality and efficiency. This leads to appealing graphs and results that can be difficult to verify rigorously. On the other hand, while Mathematics might appear inaccessibly abstract and convoluted, it provides surety in the theorems established, ultimately guaranteeing performance following strict adherence to all technical assumptions.
The abstraction of Mathematics also means less work to a certain extent: the idea developed from one application, once conceptualised and turned into abstraction, can be applied to other fields, much like the example of optimisation being applied to various engineering fields aforementioned. An inspiring figure I have in mind is Andrey Kolmogorov. Originally an expert in probability, under the assignment of the government, he made substantial contributions in fluid mechanics, bringing about the idea of ‘scaling laws’. Who would have thought statistical homogeneity would advance our understanding of turbulence? This showcases the flexibility of Mathematics owing to its abstraction.
While uncompromisingly inflexible in discerning true from false, Mathematics demonstrates its power in its adaptability once proofs are established. There are a lot more mathematicians that I aspire to, it never ceases to amaze me when I learn of the breadth of their research expertise, often spanning so many specialties in Mathematics that are difficult to imagine.
3. You received a Travel Grant to attend AMSI Summer School 2024. How important was this in terms of your ability to attend, fully participate in the program and meet others studying in similar fields? Do you think it was an advantage to attend the program in-person?
Definitely. Friendship and networking takes time to develop. In-person participation, especially when people stay in the same accommodation, maximises time for connections. It is also substantially easier to participate in tutorials and exchange ideas/ ask questions (both to the lecturer and fellow students) in-person. While it may be rare to come across participants who share the same research field, it is always nice to learn of what happens and what intrigues people in the greater Mathematics community. Inspiration rarely follows from a predictable routine.
4. What was the most valuable part of the program for you?
The concentration of nerds. The courses are also superb as they don’t usually run even in institutions where the lecturers are from. The courses offer a much wider breadth yet much deeper content to what I suppose a more enthusiastic audience, on steroids.
5. In the long-term, what do you think are the benefits of having attended Summer School?
I think the long-term benefit is in the network with like-minded students. The material and lectures were great, but the opportunity to learn alongside students from across Australia was amazing. On top of this, it was a great chance to improve my ability to communicate mathematics I find interesting with students in different fields.
6. Summer School included a special Careers Day program which aims to help give students an idea of the kinds of career paths available to maths graduates in industry and private sector research areas. Do you feel better equipped to explore career options in the mathematical sciences after attending AMSI Summer School?
It is interesting to listen to various talks given by representatives of different companies, some of which like ASD and Optiver do not have as much information accessible to the general public so the chance to be able to speak to the representatives is valuable.
7. What advice would you give to someone who is considering applying for Summer School in 2025? Should they apply and why?
I would recommend applying. It brings together like-minded people of a similar age range which is not an occasion worth missing out on. The contents of the courses are also very interesting and insightful, sometimes integrating various fields in Mathematics into just 4 weeks of intense study. I did hyperbolic knot theory and convex optimisation in the Summer School. The former nicely integrated hyperbolic geometry, algebraic topology, and of course knot theory into a single course. The latter explored, very systematically and packed with examples and insights, important topics in convex optimisation, or some might put it as convex algebraic geometry.
8. What are your current career ambitions in the mathematical sciences sector?
To be an academic.
9. How did connecting with the community at AMSI Summer School support your experience?
Learning is often more enjoyable when done with pals. The interaction does not stop at intellectual exchange, but also exploring new places and new cuisine etc. The community stimulates some intense but inspiring discussions about topics in Mathematics, sometimes lasting well into midnight. Solving problem sheets together also helps as participants often have different mathematical backgrounds so the assistance from peers would aid in filling gaps in knowledge. It also introduces different approaches to questions and life in general.