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About Me

Me pretending to play the guitar, 2025
Hello! I'm Manan Jain, a second-year MS-PhD student in mathematics at the Indian Institute of Science Education and Research, Mohali. My mathematical interests lie at the interface of (Riemannian) geometry and analysis (of PDEs). I like reading, listening to music, trekking, and taking long walks. Recently, I have started learning how to play the guitar.
Contact Email: mananifold [AT] gmail [DOT] com

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Maths Teachers Orientation Camp: Some Reflections on Maths Education in India

Last week, IISER-Mohali hosted the Mathematics Teachers Orientation Camp (MTOC) sponsored by Homi Bhabha Centre for Science Education (HBCSE), TIFR. The objective of this camp (and similar camps conducted by HBCSE elsewhere in the country) is to introduce and expose high school mathematics teachers and educators to fresh perspectives on their subject through interactive talks and problem-solving sessions. This is not strictly bound to the school curriculum, but the topics centre around things that can be assumed to be reasonably accessible for someone out of touch with (under)graduate level mathematics for years. As a volunteer for this camp (primarily engaged in hospitality for the educators arriving on the campus), I got to interact with many of these educators who hailed from Uttarakhand and Punjab (teachers from other regions, especially J&K, Himachal, and Haryana, had to back out due to the approaching final exams). While I missed out on the opportunity to hear about the exper...
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The Poincaré Recurrence Theorem

Recurrence in phenomena, physical or otherwise, has piqued the curiosity of humans for a long time. Without going into a historical detour on the study of periodic phenomena (of which the author is painfully ignorant), we cut to the chase and introduce one of the foundational results of ergodic theory: the Poincaré Recurrence Theorem. The theorem essentially states that under certain transformations of a space (to be made precise shortly), the system will almost return to its initial state under repeated iterations of the transformation. Preliminaries In order to state the theorem, we recall the definition of a probability space and a measure-preserving transformation here. Definition. [Probability Space] A probability space $(X,\mathcal{B},\mu)$ is a nonempty set $X$, a $\sigma$-algebra $\mathcal{B}$ on $X$, and a measure $\mu$ on $(X,\mathcal{B})$ with $\mu(X)=1$. Assume further that the space $(X,\mathcal{B},\mu)$ is complete, in the sense that all subsets of measurable subs...
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