AMSI grant recipient profile: Steven Buchanan

 

Give me a quick overview of the type of mathematics you are studying, and/or the aims of your research and its potential applications/outcomes (how you would explain your work and studies to friends who don’t study maths?)

I study geometric flows in 2 dimensions, where a surface changes shape according to its curvature. Perhaps the most concrete example in this direction is heat diffusion. If you imagine an object with a small region of high temperature, this heat will diffuse quickly to the surrounding areas. Thinking about the graph of temperature, the small area with high temperature corresponds to a sharp spike on a graph (think of the graph as the object that is changing shape over time). The heat diffusing quickly reflects the high curvature of this spike (and the evolution equation describing heat diffusion).

One part of my work looks at flows in two dimensions where the rate of evolution is determined by a power of the scalar curvature. I am interested in understanding how these surfaces behave over long periods of time, how singularities can form, and what kinds of special solutions are possible.

How did you get into the mathematical sciences? Was there someone or something that inspired you to this field?

I was impressed by the idea of a mathematical proof, by which one can arrive at what seem to be fundamental truths about the universe, or about some particular object of study, simply by sitting and thinking.

Winter School is designed to give students a deeper understanding of their area of research and expose them to others working in different fields/industries. Tell me about your Winter School experience. What was the most valuable part of the program for you?

The most valuable part of the program for me was the excellent lectures! I learned a lot in these two weeks about some unfamiliar areas of geometric analysis.

What was your main take away/s from AMSI Winter School? Something you learnt? A connection you made? Do you have new ideas for your work/research or see it in a new light?

I found the questions about constructing cohomogeneity-one Einstein metrics and the questions relating to proving/disproving concavity of solutions to certain classes of equations quite fascinating; if I have an opportunity in the future, I may like to dedicate some time to studying these questions. Additionally, I hope to explore the Łojasiewicz–Simon methods that we learned about, as these have some connection to my current research.

What surprised you most about Winter School?

What surprised me most was the breadth of mathematics covered at Winter School. Even within a relatively focused area, there were many different perspectives, techniques, and connections to other parts of mathematics.

What is the importance of in-person communication at an event like Winter School?

For me, attending lectures in person is much more engaging than online. Additionally, being present with other participants allows many opportunities for discussion about the material, as well as learning about new perspectives, research problems, and other areas. I think that these sorts of meetings form the basis for the mathematical community.

You received a grant to attend AMSI Winter School. How important was this in terms of your ability to attend, fully participate in the program and meet others studying in similar fields?

This grant was very important, as it made the difference between going or not!

What advice would you give to someone who is considering applying for Winter School? How would you describe the conference to them?

I would highly recommend the Winter School for anyone studying in a relevant field. The conference was stimulating, enjoyable, and productive!

Tell us about a moment that changed a preconceived idea about your field?

Throughout the process of my latest project, I have been surprised by how much analysis is needed to solve certain geometric problems.

Where do you want the mathematical sciences to take you? Where do you see yourself in five, ten years time?

I love mathematical research and would like mathematics to remain a central part of my career, but I am also interested in applying the skills I have developed through academic research to practical problems outside academia. Over the next five to ten years, I hope to find a role where I can combine rigorous mathematical thinking, computation, and problem-solving with problems that have tangible real-world outcomes.

Any other feedback/comments you would like to provide on the travel grant or AMSI Winter School?

Just want to say thanks to everyone who made the grant and the school possible!