Exploring Mathematics, Physics, and Patterns

Hey, I'm Jorge — a student of mathematics with a constant curiosity for the deep structures that connect abstract theory and the physical world.

This is my digital lab — a space where I explore how ideas from pure math, physics, and data intertwine, from symmetry in tensor spaces to the hidden patterns behind real-world systems.

Whether you're a fellow math enthusiast, a student navigating science, or someone intrigued by how abstract reasoning explains reality, this blog offers a mix of rigorous analysis, creative inquiry, and reflections on the evolving landscape of mathematical and scientific discovery.

Where to start

If you're new here, here's the easiest way to explore:

• Path — my problem-solving hub. This is where I collect interesting problems I've worked on (Putnam, GRE, and others), together with the tactics, visual ideas, and rabbit holes that came out of them.
• Blog — essays and explanations on topics from algebra and representation theory to physics and dynamics, plus anything I find mathematically beautiful or surprising.

Featured post

A good place to begin is with a bit of circle geometry inspired by Stranger Things:

How a Circle Trick Revealed the Upside Down

Problem-solving toolbox

I'm fascinated by problem-solving itself — how a clever trick or a small shift in perspective can make a tough problem unfold. Over time I've been building a toolbox of tactics I use on contests and challenging problems: modular arithmetic shortcuts, symmetry arguments, geometric transformations, and more.

You can explore these ideas, along with full write-ups of the problems that motivated them, in Path.

One theorem on my mind

Currently, one of my favorite theorems is Maschke’s Theorem — a fundamental result in representation theory which states:

Every finite-dimensional representation of a finite group over a field of characteristic not dividing the order of the group is completely reducible.

It’s a beautifully precise insight into how symmetry can be decomposed — and it reflects the kind of structure and elegance I hope to uncover in both mathematics and physics. I write more about results like this, and how they connect to other areas, on the Blog.

Thanks for visiting — I hope you find something here that makes you think a little differently about mathematics.