Interview with Dr. Bakul Sathaye

Bakul Sathaye is a postdoc at Mathematics Münster. She specialises in geometric group theory and geometric topology

What is your research area?
My research is in areas of mathematics called geometric group theory and geometric topology. These lie at the interface of three broad areas of geometry, topology, and algebra.

Topology is the field that concerns itself with properties of spaces that are preserved under deformations like stretching, bending, and twisting. Geometry is the study of properties of spaces related with distances, shapes, and sizes. In school we study what is called the Euclidean geometry, the geometry on a flat surface like a sheet of paper or a chalkboard. In Euclidean geometry, the distance between two points is determined by drawing a straight line between them and measuring the length of this line. Based on this we can construct other shapes like triangles, polygons, and circles. Now, one can ask what the distance is between two points if we were on a sphere. This is a very relevant question for us, because we live on one! If you have ever flown internationally you might have noticed that the flight route on the map is a curve instead of a straight line. Even though it looks like a long curve on the map, it is in fact the shortest path on the sphere. The most natural way to define distance on a sphere is by using shortest paths. One can ask the same question for other surfaces, like a torus or a pretzel. The distance on these surfaces is defined using something called a Riemannian metric, which uses the idea of looking at paths of shortest length.

Once you have such a metric you can do things that you do on a plane, like defining paths, computing their speed, acceleration etc. One can also ask questions such as what do triangles or polygons on the surface look like and use them to understand more about the geometry of the surface.
Using this theory one can also define what is called the curvature of a surface. This is a fairly intuitive idea. You can easily see that a sphere is "curved" while a plane is not. Similarly, a sphere of radius 1 cm is "more curved" than a sphere of radius 6370 km - we don't even notice the curvature of earth. A sphere is said to have positive curvature while a saddle shaped surface has negative curvature and a flat plane has zero curvature. One can generalize the idea of metric spaces and curvatures for higher dimensional spaces.

Which scientific question are you currently working on?
In my research, I study certain spaces of nonpositive curvature. These are spaces that, in fact, cannot be described using a Riemannian metric, but are constructed using cubes. Such spaces are called cubical complexes and they can be understood by using group actions. Here we enter the field of geometric group theory, where we aim to understand the geometry of the space through the action of a group on it.

Such actions can reveal a lot about the geometry of the space. For example, one can detect when there are subspaces of zero curvature sitting inside a space of nonpositive curvature through group actions. I am particularly interested in exploring the boundary at infinity of spaces of nonpositive curvature. These are similar to the idea of infinity, positive and negative, that mark the "ends" of the real line. Studying what happens at the "ends" or "boundary" of an infinite space often reveals a lot about the space itself. The study of spaces of nonpositive curvature is very important and has several applications in theoretical physics and understanding the nature of our universe.

Have you always wanted to become a mathematician and researcher?
My interest in mathematics was sparked by my school teacher who taught us geometry. The visual aspects of Euclidean geometry and coordinate geometry interested me. The mathematics that I learned in college was very different from what I had studied in school and that was a turning point for me. The ideas of limits, infinity and how they are dealt with using mathematical rigour were eye opening. I learned how to express my ideas in a logical and coherent manner. This is often what makes mathematics difficult, but it also lies at the heart of mathematics. Most of the ideas are very intuitive. However, our intuition can fail us and it is only our logical reasoning that can show us the right direction.

Learning about non-Euclidean geometry was another turning point in my path towards choosing to do research in geometry. Euclidean geometry is based on five postulates given by Euclid, of which the fifth postulate, also called the parallel postulate, consistently appears more complicated than the first four. In fact, for over a thousand years, mathematicians believed that it could be proven using the other axioms and tried to do so. Finally, at the beginning of the 19th century, it was discovered that one obtains different geometries by considering the negation of the fifth postulates and thus the non-Euclidean geometries were discovered. Of these, the most studied are spherical and hyperbolic geometries with curvatures positive and negative respectively.

Since then I have been fascinated by spaces of nonpositive curvature. This idea of curvature can be generalized to spaces of dimension bigger than two. This is what I focus on in my research. I try to understand the behaviour of spaces of nonpositive curvature by using the tools of groups and their actions on these spaces.

At which institutions did you do your studies and further research stays?
I went on to do my master’s at Indian Institute of Technology (IIT) Bombay, India and PhD in Mathematics at the Ohio State University in Columbus, USA. My interest in theoretical mathematics was furthered by my time at IIT. Professors at both the institutes have had a big impact on my continued interest and research in mathematics. I was fortunate to have a conducive atmosphere at Ohio State where I could discuss math with other graduate students and postdocs. I was also able to participate in and speak at several conferences about my research.

Following my PhD, I did a postdoc at Wake Forest University, NC, USA, for a year. After that I spent a year at Ben Gurion University in Be'er Sheva, Isreal, before coming to Munster. I enjoyed my time at these institutions, both as a researcher and an educator.

What are the most beautiful aspects about being a mathematician and a researcher for you?
Mathematics feels like a very rigid field, because of the mathematical formalism. One needs to use rigour and discipline when applying logic and writing mathematics in order to avoid the pitfalls of stating something untrue. However, that makes it harder for new students to understand and appreciate mathematics. I was fortunate to have had inspiring and encouraging teachers who helped me crack open the rigid exterior of mathematics and see the beauty that lies within.

As a researcher and educator in mathematics, I hope that I can do the same for young girls and boys and help them see the beauty in mathematics. I thoroughly enjoyed teaching mathematics at Ohio State and at Wake Forest. I hope to continue to teach and engage students at Münster as well.

Which were the reasons for you to come to the Cluster of Excellence Mathematics Münster?
The Cluster of Excellence Mathematics Münster is an excellent place for young researchers. There are many mathematicians whose work is close to my research interests, so it was a no brainer for me to come here to work. My time here will certainly help me grow as a mathematician. I also hope to develop some strong collaborations with the researchers and institute at Münster.

You have started at the University of Münster during the Corona pandemic: Which challenges did you have to overcome? How does the pandemic affect your current work?
Coming to Münster during the pandemic certainly has had its own challenges. However, I had a lot of help from the Welcome Center and the Cluster’s Buddy Programme through the whole process. All the information, which kept changing at all times during to the uncertainty of the situation, was communicated to me in a timely manner.

In mathematics, a lot of meaningful ideas are shared and collaborations initiated in coffee lounges and the informal gatherings around the formal seminars and lectures. The pandemic has certainly put a damper on mathematical activities. Nonetheless, I have been communicating with fellow researchers as best as I can. I am hopeful that soon I will be able to continue to meet my colleagues in person and truly experience working at Mathematics Munster.

Have you already settled in in Münster?
I am yet to truly experience the city. The weekly market is certainly a wonderful place to be on Saturday mornings. I am extremely happy to be in a bike and pedestrian friendly city. People in Münster are very friendly. Even with the language barrier, people are usually willing to help.