## Meet the speaker: Professor Jessica Purcell

Summer School 2018 lecturer Professor Jessica Purcell (Monash University) talks to us about the importance of mentorship and communication for budding mathematicians.

*Tell me about your research field, what drew you to this area and its impacts on discovery – its real-world applications?*

I work in topology, which can be described as the mathematics of shape and space. Topologists try to answer questions about shapes. For example, they may study knots. If someone hands you an extension cord with the ends fused together, should you bother trying to unknot it? How would you know whether or not it could be unknotted? If not, can you at least remove a lot of crossings? How many crossings can you remove? How long will it take you? How many knots are there with a fixed number of crossings, and how do you know? How can you tell them apart? Sometimes such questions can be answered by looking at more complicated objects, such as the space around the knot, or higher dimensional objects whose boundaries are related to the knot. Topologists try to analyse all these spaces and shapes and others.

*You are a researcher at Monash University. What are you working on currently? Can you tell me about some recent achievements?*

I am working on the interplay of hyperbolic geometry and 3-dimensional spaces called manifolds. Hyperbolic geometry is a negatively curved geometry that arises when spaces have saddles or flared ends. To go back to the knot example, the space around a knot is 3-dimensional. It often admits a unique hyperbolic structure, which gives a very natural way of measuring distances and volumes. One aspect of my research involves trying to determine hyperbolic metric properties of knots. A problem I have been investigating recently is to determine, out of all knots with a fixed crossing number, the knot with the largest hyperbolic volume. While we are still a long way from answering this question, we have some good candidates. Some recent work of mine investigates the geometric properties of these candidates, and shows that at least in a limiting sense, they have as much volume per crossing as possible. I have also been looking at other limiting behaviour of other knots, and showing that these limits can have surprising geometric properties.

*What are the biggest challenges in this area and more broadly facing the global mathematics community?*

I believe one of the biggest challenges in my area is relating and applying new knowledge to old problems, or new instances of similar problems. In the last 15 years, my specific area of mathematics has seen a phenomenal amount of progress. Many old questions and conjectures have been answered, beginning with the century old Poincare conjecture that was proved in 2002-3 and confirms a suspected property of 3-dimensional spheres. This progress and success has led to great tools to study 3-dimensional manifolds, and a better understanding of their structure. However, it is still very difficult to apply these tools to examples that arise in practice, both in low-dimensional topology and in other fields. To give an example, physicists have been working with quantum invariants associated to knots and 3-dimensional manifolds, and experimentally some of these invariants seem intricately related to hyperbolic geometry. But we cannot prove this! Moreover, we really don’t have a good understanding of why this happens in a lot of cases. Many remaining challenges in my subfield involve relating results on geometry to other invariants, and giving geometric reasons why the relations hold.

More broadly, I think the biggest challenges facing the global mathematics community involve communication. Mathematics thrives when researchers are able to work together and communicate. It requires global societies that are open and value the process of seeking answers. It also requires work on the part of the mathematicians. We need to be better at sharing our ideas, and trying to communicate the value of what we do. We need to have the courage to look at problems from different points of view, and to try to understand the work that is done in other fields and to contribute.

*You are lecturing on *Low Dimensional Topology* at AMSI Summer School 2018, can you give us the elevator pitch for your session?*

Low-dimensional topology is the mathematics of spaces in dimensions 2, 3, and 4. We will encounter spaces that can often be described by sketching pictures or diagrams encoding their forms. The mathematical tools we develop will help us to analyse the pictures and diagrams, and confirm or contradict our intuition, and help us understand spaces when diagrams are not available.

*We will start by considering surfaces. We’ll look at maps from a surface back to itself, and determine a set of generating maps. We will use these to build up 3-manifolds, and discuss related topics along the way such as knots and applications to 4-dimensional spaces.*

Questions from low-dimensional topology are becoming more common and more broadly interesting in our world. Triangulations and properties of surfaces and 3-manifolds are important in computer vision and visualisation. As mentioned above, 3-dimensional manifolds appear in quantum physics. The knots you find in your extension cords and shoe laces are also appearing in biochemistry, as knotted and folded proteins and DNA strands, and the knotting seems to affect function. There is much current research in these directions. However, our session will focus on the developing the theory and not much on the applications.

*How important are opportunities such as AMSI Summer School as we seek to strengthen national and international engagement within the mathematical sciences and prepare emerging research talent to drive innovation?*

The AMSI Summer School is a great way for students to learn about aspects of mathematics that are deeper or broader than what they see in a standard curriculum. The extra knowledge should help them prepare for a future in which knowledge is important. Another huge benefit of the AMSI Summer School that I see is the opportunity for mathematically inclined students to meet each other, and to interact with other students who do well in mathematics and who love it. My research career has been shaped by interactions with people I met as a student, especially fellow students. These are my mathematical friends, now residing all over the world. I go to them to bounce off ideas or share intriguing problems and projects. I hope the students take the opportunity to form their own mathematical friendships at the AMSI Summer School.

*What do you see as the biggest barriers to driving innovation? How important are initiatives to provide industry experience and knowledge to graduates and address issues such as participation of women and indigenous Australians?*

In mathematics a lot of innovation comes from the sharing of ideas. Putting restrictions on the people who are encouraged to participate in this sharing of ideas is a big barrier to innovation. Another barrier to innovation comes from the attitudes we sometimes communicate as a society. I have found that students sometimes believe that they will be happier if they avoid challenges. In fact the opposite is true. Working on challenging problems can lead to deep satisfaction. Providing experience with industry and programs like the AMSI Summer School can help graduates develop the skills they will need to tackle challenging problems, and to enjoy the challenge!

As part of Choose Maths, we are in the process of establishing a mentoring program particularly in relation to encouraging the participation of women. Who are your biggest maths influences or mentors, how have they impacted your maths journey and career?

As an undergraduate, I had a mentor in mathematics who helped me immensely. She was able to point out which classes would be most important to help me reach my goals, and suggest opportunities for further development and exploration, such as summer schools. I would have had no idea these options existed without her mentoring. Since then, I have found mentors among colleagues at different institutions at different stages of my career. They have all helped me to achieve my goals, and navigate changes.

*Did you grow up mathematical or did maths find you along the way? Was it always a career dream?*

I liked mathematics in school, but I didn’t understand that mathematics could be a career. I remember telling someone early in my undergraduate days that I had already learned nearly everything there was to learn about mathematics — I couldn’t imagine maths beyond multivariable calculus. With the support of strong mentors and the encouragement of my professors, I continued taking more and more mathematics courses, and discovering that in fact there was more and more to learn.

A few years later, I did have a hard time deciding whether or not to get a PhD in mathematics. While I was doing well in my maths classes, I imagined that a mathematician had to be clever and quick – the kind of person who did really well on maths contests and competitions. I was not that kind of person. Again with some encouragement, I built up enough confidence to try a PhD. And again I found that I liked it, and continued to enjoy mathematics most of the time, and I realised that the day to day job of a mathematician was not much like a maths contest. In any case, I am happy to be a mathematician now.