Himani's mathematics PhD experience

Publication date
Monday, 19 Dec 2022
Two graduands in front of a building


What degree did you graduate from?

I did PhD in mathematics at MSI under the supervision of Associate Professor Pierre Portal.

What was your research about? 

My research was about studying the functional calculus of abstract differential operators. They include the well-known Schrödinger operators, but my results were a bit more general. By functional calculus, we mean the functions of operators defined using an appropriate formula where the functions are in a suitable class. This method uses properties of functions to model the properties of operators. In fact, this method allows us to pass easily and flexibly from estimates on known operators to estimates on these functions of operators. I worked on obtaining their boundedness on specific function spaces. Functional calculus is quite useful in spectral analysis and has applications in mathematical physics. My thesis is available on the ANU thesis repository with the title: “Spectral Multiplier Theorems for Abstract Differential Operators”. 

What made you choose that degree and why did you decide to come to ANU to pursue those interests?

I wanted to do a PhD in mathematics with a main interest in Operator Theory. This interest started developing in me when I was in my undergrads. At that time I did not know about Operator Theory and that it is one of the major topics in pure maths. I was more into Linear Algebra then, but I realised that it is somewhat an extension of Linear algebra in higher dimensions and in different spaces, while doing my MSc. I started liking it, and in fact, went on to do an MPhil in this area. I did my MPhil under the supervision of Prof. Sachi Srivastava where my thesis was on “Semigroups on Composition Operators”. It was she who suggested I apply for PhDs outside India.

I contacted many mathematicians working in Operator Theory and Functional Analysis all around the world for a PhD position. ANU being the top university in Australia was one of my prime choices. I contacted Pierre after going through his personal webpage where he had nicely summarised his research in Fourier and Harmonic Analysis as well as his interest in operator theory. I felt that it will be a good opportunity to connect my experience from MPhil with his expertise and broaden my field of research. Eventually the results in my PhD thesis come from the combination of harmonic analysis and operator theoretic techniques.

How was your experience studying at MSI?

Overall, my experience was pretty good. It was my first time studying at a university outside of India and the admin staff at MSI made this transition quite smooth for me. I am very impressed with the academic culture here, and I found everyone at MSI quite approachable and helpful.

Tell us a few highlights of your experience studying at MSI.

There were many things that I really liked while studying at MSI apart from our beautiful building (which many people are jealous of!), but I am going to mention only a few.

  • Graduate seminars. It started with the objective to prepare PhD students to give talks at conferences and to interact with students from different fields of maths and know their research. I also got the opportunity to give a talk in it. I found it quite helpful and also made some good friends through this.
  • The weekly Analysis and PDE seminars in which we used to have mathematicians coming from within and outside Australia to give talks. This was a wonderful opportunity to connect with researchers outside the home university.
  • I found Thursday’s afternoon tea breaks as one of the best times to interact with all the people of MSI. I still miss it!

Unfortunately, I enjoyed all this only in the first two years of my PhD, and then COVID happened. Glad to see that it is almost back to normal now, and all these are happening in person.


Did you experience any challenges while studying? How did you overcome those challenges? 

Yes, of course! There were quite a few actually, but the major one happened within the first six months of my PhD. After 3-4 months in PhD, I was able to get a result, and I remember Pierre and I were very excited about it. I went to India for Diwali that year and a month after I came back, we realised that there was a mistake in my result which could not be rectified. It was heartbreaking and challenging since I had to look for another problem to work on. 

I overcame those challenges thanks to Pierre, who was very supportive then and throughout my PhD. He suggested me a couple of problems and gave me time to understand their background to decide which area I would be more interested in working in. I am glad that he gave me the freedom to choose from multiple options. This built my confidence, and I was able to restart, learnt more about harmonic analysis and was able to use my experience from operator theory in my work. I realised that such ups and downs are a part of the PhD journey, and one must not feel demotivated. We must learn from our mistakes and put those lessons into practice.

What is next for you? Any special plans for the future?

After submitting my PhD thesis in February this year, I started working as a Postdoctoral researcher at Macquarie University in Sydney. This position is now coming to an end, and I am moving to Sweden in January, where my husband is working. Currently, I have applied for a few postdoc positions in Europe and waiting for their results. My plan is to stay in academia, so I am working towards bettering my research and teaching skills with the aim to get a permanent position in a good university.

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