"Open Questions: May 2026"

2026-05-24

Here's a list of questions that I would love to learn more about. If anyone has any insight or recommendations on how to approach them, I'd love to meet up and learn more!

• What mathematical properties make the Laplace domain so useful? Throughout my undergraduate studies, the Laplace domain has been shown to be useful in solving differential equations and in evaluating the response of a system subject to different inputs. In many cases, when problems are difficult to solve in the time domain, the Laplace domain serves as a more effective alternative to solve these problems. In class, we saw that a Laplace transform can convert a mathematical expression from the time domain to the Laplace domain (which I will henceforth refer to as the s-domain), and an inverse Laplace transform that enables $s \rightarrow t$.

  • What is a concise definition of s? (Sure, I understand that s is an abstraction of the "quantity of osctillations at a given s", though I still don't understand a precise enough definition.)
  • What mathematical properties make the s-domain interesting?
  • Are there domains other than t- and s- that are interesting to study?

• How is the Fourier transform derived? I understand that the frequency domain is related to the s-domain simply by substituting $s=j\omega$. However, what is the intuition that leads us to state that this substitution results in the frequency domain? Furthermore, how can I visually and/or intuitively understand how the Fourier transform works? How is the "original" Fourier transform different from the Fast Fourier Trasform (FFT) and Discrete Fourier Transform (DFT)?

• Where do singular values come from? Singular Value Decomposition! At the surface level, the singular values come from the $\Sigma$ matrix when doing matrix decomposition of $A=U\Sigma V$. However, what is the origin of these singular values? Are they some higher dimensional object whose projection is the initial $A$ matrix that we see on paper?

• Why are singular values so effective at solving problems in optimization and system dynamics? In control systems, we learned that the $H_\infty$ norm is useful at designing controllers with respect to the worst case oscillations of a system, by taking the suprenum (analogous to the maximum) of the singular values of a system. Curiously, singular values are also essential for the Muon optimizer to enable fast convergence for neural network training. Why are singular values so useful in these two distinct domains?

• How does diffusion work (i.e. the DDPM approach), and why do diffusion models work so well? Earlier in undergrad, I got introduced to diffusion models for image generation, and also diffusion models for text generation and even for policies for robots. Why does diffusion work so well? Why hasn't the diffusion architecture replaced its attention-based counterparts?

• For red-black trees, why do a seemingly arbitrary set of rules guarantee self-balancing binary trees? In my algorithms class, we learned that a set of rules involving node colouring and tree rotations guarantees that a binary search tree will always be approximately self-balancing. How did we arrive to this proof? Why does a seemingly arbitrary set of rules arrive to such a guarantee? At a higher level, why can arbitrary rules lead to strong and desirable guarantees?

• How do you learn the art of pre-training and post-training foundational models? I understand that there are some strategies (i.e. scheduled learning rates for pre-trianing, and GRPO and variations for post-training), though what is the best way to amass sufficient intuition and rules that guarantee a desirable pre-trained or post-trained model?

• How many different ways can you play a guitar chord? I previously played mostly open chords and barre chords, though recently I've been trying to learn more ways to play different chords. What are common ways to play chords beyond the basics? How many chord positions should one learn (and know) in order to be proficient at playing guitar?

Thanks for reading! If anyone has any recommendations on how to approach these, no matter how simple the answers may actually be, I'd love to get in touch!