Exploring Mathematics 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.

Lately, I’ve also been obsessed with a related question: where does mathematical insight actually come from — and what happens when we try to make that insight explicit in formal proof systems and AI?

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

If you want a post that captures what I’ve been thinking about lately — contest problems, structure, and what formal proofs reveal — start here:

Putnam 2016 A1: Where Insight Lives (and Why Formal Proofs Expose It)

From problems to systems

I’m fascinated by how reasoning works — not just how to finish a proof once the right idea is known, but how that idea is found in the first place.

That’s why I like Putnam-style problems: they force you to notice structure before you can “compute” anything. And it’s also why I’m interested in formal proof assistants (like Lean) and math-focused AI: they’re excellent at verification, but they expose exactly where the human part of math still lives.

If that kind of thinking resonates, you’ll probably enjoy exploring Path and the essays on the Blog.

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.

I love it because it’s a theorem about decomposition: when symmetry looks complicated, it can often be broken into clean, understandable pieces — the kind of structural clarity I care about in both pure math and modern computational settings.

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