Day 4 of ICTS-SSWP '26: Exploring Hidden Connections
Day 4 of the ICTS-SSWP 20261 began with a talk by Dr. Jayanth Vyasanakere from APU on one of the most famous unsolved problems in mathematics: the Collatz Conjecture. The rules are simple:
- start with any positive integer $n$
- if $n$ is even, divide it by 2; $n →$ ($\frac n 2$)
- if $n$ is odd, multiply it by 3 and add 1; $n →$ $(3n+1)$
- repeat this process to generate a sequence of numbers
The conjecture states that regardless of the starting number, the sequence will eventually reach $1$, entering the familiar loop of $4 → 2 → 1$.
Despite its simple formulation, the conjecture remains unproven. To gain some intuition about its behavior, we explored variations of the rule, such as replacing $(3n+1)$ with $(n+1)$ or $(5n+1)$, and examined how these changes affect the resulting sequences. Some variations still led numbers back to 1, while others produced dramatically different behavior, offering a glimpse into the rich complexity hidden within seemingly simple iterative processes. You can see some of my observations on Collatz Conjecture here.
Our discussion then shifted to unit-digit circles in base 10, where the mathematics became even more intriguing. By mapping the possible transformations of the units digit under the Collatz operation, a network of connections emerged. Extending the same idea to base 100 produced a much denser structure, revealing beautiful and unexpected geometric patterns. Distinct mathematical curves such as cardioids (heart-shaped curves) and nephroids (kidney-shaped curves) appeared in the resulting diagrams. Even more surprisingly, these same patterns could be recreated using just an ordinary bangle and a light source, demonstrating how elegant mathematical structures can emerge in the most unexpected places. We discussed several paradoxes in physics - many of which revolved around questions of energy conservation.
In the evening, we attended a talk by Dr. Sumathi Rao from ICTS titled “Why You Shouldn’t stand on a Topological Insulator and Change Bulbs.” The talk introduced us to the fascinating world of topological insulators, materials that act as insulators inside the surface but have metallic, highly conductive states on their surfaces. Such materials have been successfully synthesized in laboratories and are the subject of research worldwide. Their unique properties could help develop more efficient electronic devices (reducing overheating problems) and may play a role in the development of future quantum technologies.
We delved deeper into the physics underlying these materials. We learned about quasiparticles - which aren’t fundamental particles like a single electron, but rather a collection of electrons behaving like a single particle (called quasi-electrons). In two-dimensional systems, quasiparticles can exhibit even more exotic behavior, giving rise to entities known as anyons. Unlike ordinary particles, anyons possess a kind of “memory” of how they move around one another, a property encoded in their quantum state.
This remarkable feature makes anyons particularly attractive for quantum computing. Researchers hope to exploit their robustness against external disturbances to build topological quantum computers, devices that could potentially perform computations while being far less susceptible to errors than conventional quantum computers.
Apart from the core physics lectures, discussions, and laboratory sessions, we also attended an insightful panel discussion on women in physics. The panel, comprising researchers and academicians with decades of experience, shared their perspectives on the current state of the field, the challenges one often encounters, and the progress that has been made over the years. The discussion also explored different paths for research.
Overall, Day 4 showed how ideas from different areas of science and mathematics can be connected in surprising ways. From the Collatz conjecture and the geometric patterns hidden within simple numerical transformations to the physics of topological insulators and anyons, the sessions revealed how seemingly different topics can share common underlying principles.