Abstract:
This piece explores the role of mathematical collaboration in research. Whether it be seeking feedback, presenting, or informal conversation, the exchange of ideas is an essential aspect of undertaking any form of research. Additionally, my experience conducting research into the geometry of feasible regions in semidefinite programming has revealed that theoretical mathematical questions can address a wide variety of real-world problems.
Thinking Together: My Experience in Researching Semidefinite Programming
From financial portfolio optimisation to supply chains and logistics management, semidefinite programming is an active and exciting area of mathematics with a wide range of applications. This summer, I had the privilege of conducting research into the geometry of feasible regions of semidefinite optimisation. What made the experience especially rewarding was not only the subject itself, but also the opportunity to communicate with other researchers throughout the process.
Despite the stereotype that mathematicians are isolated thinkers who operate in silos, the mathematical research process is incredibly collaborative. Cooperation is important because it exposes you to novel methods and ideas which can help in tackling complex or unfamiliar problems. Further, understanding how another person interprets your work can reveal gaps in explanation and improve the clarity of your mathematical communication.
This was particularly evident through my experience learning the fundamentals of semidefinite programming and its feasible regions. Although the field can be complex, its interdisciplinary nature means that mathematicians from very different backgrounds can still engage with it meaningfully. Researchers may approach it through optimisation, geometry, computation, or applications, and this diversity of perspectives strengthens discussion. While I looked at the faces of shapes, I met researchers who applied semidefinite programming to interesting combinatorial problems.
When engaging with others who are highly accomplished in their field, there is no doubt that people feel a level of imposter syndrome. However, my experience demonstrated that people possessed intellectual humility and maturity, and that these qualities enlivened discussion and the pursuit of mathematical knowledge. I was also struck by the fact that every researcher recognises that their own niche may be unfamiliar to those without the relevant background.
Beyond the collaborative dimension, semidefinite programming taught me about mathematics more broadly. As a generalisation of linear programming, semidefinite programming represents how more abstract mathematical frameworks do not move us away from concrete physical problems but enhance the repertoire of tools we can use to describe them. In fact, this discipline provides a unified language for approaching problems that arise within very different contexts. This represents how mathematics is intrinsically a creative discipline.
I was particularly interested in the way that semidefinite programming lies at the intersection of applied and pure mathematics. On the one hand, academic interest in semidefinite programming is heavily motivated by its real-world applications. Conversely, research into semidefinite programming allows one to grapple with interesting geometric questions.
Emmanuel Travassaros
The University of New South Wales
