## The Wente Inequality

By Asama Qureshi, Monash University

I did my project in an area called geometric analysis and our focus was on a well-known result, the Wente inequality. This was my first exposure to geometric analysis, and any kind of mathematical research, so there was a lot to learn about geometric analysis and how to conduct research.

To briefly explain my project, for a specific case of the Poisson equation the Wente inequality states that the norm of the solution and the gradient of the solution is bound above by two expressions in L2 space times a constant. We investigated how, if we changed from L2 space to Lp and Lq, where p and q are Holder conjugates, how this changes the constant term.

One of the first things that stood out to me was how different areas of maths I had studied, multivariable calculus, differential equations, complex analysis and functional analysis, had been seamlessly intermingled. As a student, I had developed the subconscious belief that these subjects, since they had been taught separately from each other, must not have had much common ground. Seeing how the authors of the papers I was reading use this cross-area approach made it look not only useful to blend these areas, but natural, as if all these connections had been here all along and one only had to change their perspective to see them.

The hardest challenge I faced throughout this project was at the start of the project, trying to understand what it is that we were trying to do. Looking at a very abstract problem in, for me, an unknown area, it took me some time before I could even glean any insight into why this is an interesting problem. I was asking myself, “is there some kind of application for this?” and “why is it useful to investigate this?”. On top of that, the literature is so dense and arcane that I had endless frustration trying to interpret what authors were talking about. However, as I read more papers by different authors and worked through them myself, whilst continually asking my supervisors questions, I started to see the elegance of the arguments we were reading. I began to see what made the Wente inequality interesting and our own work related to it, not just because it may have practical applications, but because now were able to discern some new information, which beforehand we would have had no reason to believe that it was true.

On reflection, this project was probably one of the most challenging experiences I’ve had to undertake, because of the mental energy involved in understanding mathematical arguments much more complex than I’m used to, and then creating my own argument from that. But because I was challenged and had to overcome it, I’ve now learned to approach the rest of my work with much more diligence and feel more confident in my abilities as a student. I’m extremely grateful for the opportunity to have worked closely with two professionals in the field and learn from them, and what I’ve learned has been invaluable. The experience has reinvigorated my passion for mathematics and it has excited me for continuing my studies.

Asama Qureshi was one of the recipients of a 2017/18 AMSI Vacation Research Scholarship.