Meet the speaker: Dr Vanessa Robins
Dr Vanessa Robins from ANU fills us in on topological data analysis, her lecture topic for AMSI Summer School 2018.
Tell me about your research field, what drew you to this area and its impacts on discovery – its real-world applications?
Computational topology and topological data analysis have developed rapidly over the past 20 years or so. The goal is to establish methods for quantifying the shape of data. It is quite simple for a person to look at a 2D pattern of points and “see” the shape those points trace out, but it requires a lot of mathematics and good algorithms to get a computer to do the same. And if the points live in a higher-dimensional space, it is difficult even for a person to identify the shapes.
I’ve always loved geometry and patterns and the way they manifest in nature. And I’ve always loved seeing how different areas of mathematics work together to help describe real-world phenomena. My first research project as an honours student investigated patterns that arise in models of plasma fusion experiments.
The shape of the magnetic fields controls how much of the space the plasma particles can move through. If the fields fill out a smooth surface the plasma remains contained, while if certain field lines break up and develop large holes, the particles will escape and fusion will fail. This was well quantified in two-dimensional models, but much more difficult to study in the more realistic higher-dimensional models. The question of how to detect holes in higher-dimensional point data then inspired my doctoral research on computational topology. For example, holes in the trajectories traced out by a particle signal the presence of non-linear periodicities, or the detection of small-scale local holes allows us to discriminate between different classes of random point processes. This helps scientists understand and test the validity of their models for processes that underlie the data generated. My thesis explored the mathematical foundations for extracting topological quantities from point data. Simultaneous studies by others around the same time led to efficient algorithms and a field of research now called topological data analysis.
You are a researcher at ANU. What are you working on currently? Can you tell me about some recent achievements?
I have two main areas of research at ANU. One is working with the ANU x-ray CT group who developed world-leading lab-based x-ray imaging techniques. We have adapted the primary tool of TDA (persistent homology of a filtration) to analyse 3D voxel data rather than point-cloud data. Our work established the first good definition of critical saddle point for a function on a digital grid and has also led to efficient algorithms for the simultaneous skeletonisation and partitioning of binary images. I’m currently using these tools to analyse x-ray CT images of porous sandstone rock cores to establish correlations between their pore-space geometry and connectivity and their physical properties. For example, the capacity for capillary trapping of CO2 gas for permanent carbon sequestration. Another example of work with this group is our recent Nature Communications paper “Pore configuration landscape of granular crystallization”, where we show how TDA gives clear insights into the local mechanisms behind the order-disorder transition in packings of spherical beads.
My other main area of research (with Stephen Hyde’s group) is on topological crystallography of one and two-dimensional structures embedded in 3D space. One of the biggest areas of research in solid-state physics and chemistry is on framework materials built via the self-assembly of small molecules. A fundamental question here is to describe the range of structures that are possible within some physically relevant classes. Some of the more complicated frameworks consist of multiple inter-threaded components. Ideally we would like to have robust invariants that tell us when two structures are topologically equivalent. Mathematically this translates into a need for computable invariants of linked and knotted graphs in the 3-torus.
What are the biggest challenges in this area and more broadly facing the global mathematics community?
A major challenge for TDA is to extend its methods to more complex situations. One of the main tools of TDA is the persistent homology of a filtration: this tracks the way topology changes as you add elements to the object. So it works only for growing sequences ordered by a single parameter. But many scientific problems have objects that grow and then shrink, or have two or more parameters associated with them. A number of groups around the world are working to develop mathematical theories and computable invariants for this situation.
Another challenge is to make the techniques of TDA easily accessible to scientists who might want to use them. This is something that applies to mathematics more generally. The mathematical community has been improving its expository skills and this is something to be encouraged and expanded. Mathematics is an important part of our culture and global knowledge base, and communicating the key insights and motivations to others is necessary to maintain its vitality. The results don’t need to be immediately applicable, but it is worth taking the time to explain why they are interesting.
You are lecturing on topological data analysis at AMSI Summer School 2018. Can you give us the elevator pitch for your session?
TDA is a rapidly developing field of applied mathematics that uses topology – previously thought of as a very “pure” area of maths – to probe the structure of data from real scientific applications. TDA provides a complete picture of the geometric and topological structure of the data set over a wide range of length scales. This helps scientists to identify and visualise the important structural elements present in their data and provides a means to compare real data with models. What could be more exciting than learning about elegant mathematical quantities and practical algorithms for computing them – tools that can lead to new scientific insights in many different fields.
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?
I’m really excited by the opportunity to pass on my knowledge of applied topology to as many students in Australia as possible. This is a very active area of research in North America and Europe, so the school will certainly help strengthen national and international engagement in this area. The AMSI summer school also provides a great opportunity for mathematics students from around Australia to form friendships and collegial networks that will last throughout their careers.
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?
To me, innovation comes from having the freedom to daydream a little and explore questions that just grab your attention and mean something to you. It is fostered by open access to information and engagement with other like-minded people. Industry experience and knowledge is helpful for “keeping it real” and reminding academics what matters to the wider world. A natural barrier to innovation is the very human tendency to keep doing things the way they have been done in the past – it takes a lot of energy to change. One thing I’ve learnt is that you can combine interests in just about any choice of fields if you think hard enough. So non-Euclidean geometry can be used in fashion design, the physics of nuclear isotopes is vital to dating archaeological artefacts, machine learning is being used to data-mine 18th-century French literature. Perhaps a wider recognition of these interdisciplinary possibilities would help under-represented groups find a way into their fields of interest.
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 a woman in mathematics and physics, I was generally one of only a few in my classes. It has been a struggle at times to overcome the self-doubt instigated by being in the minority and having very few role models. One thing that has kept me going is the encouragement from my academic advisors along the way, and to see other women around me succeed. I’m particularly happy to see the energy and enthusiasm of my younger female colleagues as they develop their careers. The current swell of support for women in disciplines they are under-represented in is also a welcome boost to morale.
Did you grow up mathematical or did maths find you along the way? Was it always a career dream?
I always had an aptitude for mathematics, particularly geometry, and loved learning about all areas of natural science. I started my undergraduate degree in physics, but always felt I needed to know more mathematics to understand the physical models better. So it’s not at all surprising that I’ve ended up doing what I do now.