The graduate-school days are zooming away quickly from my life. It seems that the space of human memory is hyperbolic in nature. Things get thin and small at an exponential rate. I defended my thesis in July 2020 and reached India in August of the same year. The pandemic was in full swing. It was a conscious choice to return to my aging family who needed support.
Almost all of 2021 was spent on the paper that Chris and I were working on. It is an extension of the results in my doctoral thesis. We showed that connected boundary of a relatively hyperbolic group is locally connected. This removed some of the tameness restrictions that Bowditch’s theorem has on the peripheral subgroups. At the end of 2021, my advisor suggested that I should work on some projects alone.
For a few days, I felt like a radarless ship in the ocean of mathematics. Since I was not associated with any university at the time, research had to be a solo adventure. I decided to build a small research group at Cheenta. It is the organisation that I developed from scratch since 2010.
Cheenta was conceived as a training school for math olympiads for school students. Subsequently we have also accepted college students for university level programs. We already have a strong alumni and student base spread all around the world. I could easily get a few people who became curious about geometric group theory.
We started meeting weekly. In order to keep a psychological leverage, I put the meeting time on Tuesdays at 10:30 PM IST or 11 AM CST. In graduate school, that was the time when I met my advisor weekly. My brain-clock responded to this procedure and a group of 7 students was assembled for weekly adventures in geometric group theory.
2022 was also productive. I managed to prove a small theorem related to Dehn fillings and connectedness of Bowditch boundary. The entire team participated in a translation project of the famous green book by Ghys and Harpe from French to English. I also started collaborating with Arka Banerjee on another problem related to embedding of hyperbolic plane in relatively hyperbolic groups.
I hope 2023 will be productive. I want to understand how small cancellation theory, and splittings of relatively hyperbolic groups interact. It could be a powerful source of examples in group theory. Another area that interests me is the theory of hierarchically hyperbolic groups and spaces.
There are a few obstacles for my research activities. Books and journals are not easily available outside the university system. Access to conferences is hard. I was invited to speak at a conference in Ohio (to be held in April). However due to VISA and funding issues I was forced to decline the offer. There are few positive ends as well. My work at Cheenta allows me to have flexible work-hours and financial security. It also helps me to stay with my family at home.
Lets hope 2023 will be productive with what I have.