Research Projects
Some math research I've done

[Back to Main Page]

The Number Rotation Puzzle
Imagine a rectangular grid of numbers in scrambled order. You can pick any square of numbers (of some set side length) and rotate it. The puzzle is solved when the numbers have been rearranged into increasing order via such moves. What is the strategy for solving this puzzle, provided that it can be solved?



To summarize my results: It turns out that, except for some small boards, the only solvability criteria are based in parity restrictions.

My solving algorithm involves obtaining a 3-cycle algorithm using a commutator and then

A bit of background for what motivated this project: Back in high school, I downloaded an app on my phone called Simon Tatham's Puzzles, and there was one puzzle on there called Twiddle. It didn't take too long to figure out the $3 \times 3$ board with $2 \times 2$ rotating blocks, but some of the other preset-modes taunted me, demanding to be solved.

The timeline for the evolution of the mathematical theory for this game proceeded as follows.

Being a Regeneron Finalist was kinda surreal, and I'm still surprised that I was selected. I wrote an essay about it for a Chinese newspaper, if you're interested.

Singularly-Perturbed Phase Transition Models with Boundary Conditions
The "proper physical context" for motivating this is a bit different in $N$-dimensions, but here's a one-dimensional sort of motivation. Let's say that we have this potential energy system that has two wells, i.e. locations of low energy that we want to be at. We also call these wells phases. Going from one phase to the other is a phase transition, and it is natural to ask what is the "best" way to execute a phase transition, in the sense that the least possible amount of energy is spent.

One could imagine, for instance, travelling between two planets. It's a bit unrealistic to just "teleport" to get from one planet to the other, or even "accelerate really fast and appear at the other planet". This is because speeding up is hard. Thus, it is natural to penalize high velocity when we build a model for this sort of phase transition.

This gives rise to one possible model for the energy spent in a phase transition: $$F_{\varepsilon}(u) := \int_a^b \varepsilon^{-1}(u(x)^2-1)^2 + \varepsilon|u'(x)|^2\,dx$$ The $(u(x)^2-1)^2$ term encourages us to stay near the phases $\pm 1$, while the $|u'(x)|^2$ term says that we don't want to go too fast. The $\varepsilon$ is a parameter, and unfortunately explaining why we choose to have an $\varepsilon^{-1}$ on one term and a $\varepsilon$ on the other would be inconvenient to explain here.

Brief Aside: The "correct space" of functions $u$ that we consider is a Sobolev Space, which you can intuitively think of as a really nice space of functions that are "continuous enough to be basically differentiable".

The big point here is that this is a plausible model for phase transitions, and particularly it happens to be a model in the Van der Waals-Cahn-Hilliard theory of phase transitions. The mathematical study of this model is crucial to justifying the model's correctness. Specifically, if it were a correct model, then we would expect that a function $u$ that minimizes the value of $F_\varepsilon(u)$ would tend to "stick to the phases" wherever possible. In particular, as $\varepsilon$ gets small, a minimizer $u$ would spend less and less time between the phases, so that "in the limit" we would expect to see an instantaneous "jump" between the phases. If th math shows that this does not happen, then the model is suspect.

Fortunately, using Gamma convergence, Modica and Sternberg show that this indeed happens. They find something called the Gamma limit of the sequence of funuctionals $F_\varepsilon$ as $\varepsilon \to 0$. Roughly speaking, the formula for this Gamma limit justifies the intuitive notion that the optimal amount of energy spent moving back and forth between two phases is a constant multiple of the number of phase transitions.

Further developments included the following:
At last, we can talk about what I'm studying! I combined the two efforts above by adding boundary condition to the second-order model. I compute the Gamma limit for this, and found that the formula for this Gamma limit consists of two terms as in Owen et al., but also includes different constants as in Fonseca and Mantegazza's work. This work was all done successfully in one dimension, and I am currently working on extending this result to $N$ dimensions using a methodology called slicing.