What I have learned today

 Isaac Newton is arguably one of the greatest scientist who ever lived. To suggest he made a "mistake" seems almost sacrilegious. But he did make one. And it wasn't a small calculation error. It was a fundamental misinterpretation of reality that held physics back for over two hundred years. The irony of Newton’s legacy is this: he gave us the exact tools required to understand that gravity is an illusion, but he was so deep inside the illusion himself that he couldn't see it.   The ghost forces To understand Newton's mistake, we first need to understand one of his most brilliant insights: the difference between a "true" frame of reference and an accelerating one. Newton established that if you are in a "distinguished frame" (what we now call an inertial frame )—like floating smoothly in deep space—the laws of physics are simple. Objects move in straight lines at a constant speed unless pushed. But if your frame is accelerating, things get weir...

  My latest research has just been published with Springer Nature in Qualitative Theory of Dynamical Systems. Read here . Finding conserved quantities—first integrals—is one of the most powerful ways to tame an ordinary differential equation (ODE). Traditionally, this requires either discovering a symmetry of the equation (à la Lie theory) or digging up an integrating factor. Both approaches are powerful, but they come with heavy baggage: symmetry detection can fail, and solving the partial differential equations for integrating factors is often impossible in practice. In this paper, I propose a different route: a geometric method built on the complete integrability of the Pfaffian system naturally associated with an ODE. By moving the problem into the language of jet bundles and contact forms, the method bypasses both symmetries and integrating factors. Instead, it turns the hunt for a first integral into solving a system of PDEs derived from Frobenius’ theorem. What makes this...

I've just published a small utility: LaTeX Delimiter Swapper This tool allows you to quickly convert between different LaTeX math delimiters: Inline: $...$ ⇄ \(...\) Display: $$...$$ ⇄ \[...\] It's a simple way to clean up or standardize your LaTeX source, depending on your formatting preferences or target rendering environment. Feel free to try it out and let me know if you have suggestions!

 I’m excited to share my latest expository work, titled On theories, symmetries, and gauge , now available on Researchgate! In this work, I provide a foundational exploration of the mathematical definition of physical theories, transformations, and symmetries.    I welcome feedback and discussions—feel free to reach out! 

If you're interested in understanding Principal Components Analysis (PCA) from a linear algebraic and geometric perspective, I've written a detailed note on the topic. You can read it here: Principal components analysis . 

Topos theory is one of the most elegant and far-reaching developments in modern mathematics. It bridges geometry, logic, and category theory , offering a framework where multiple mathematical "universes" can coexist and be rigorously analyzed. In this note , I explore the foundational ideas behind topoi, starting with their relation to the familiar category of sets and expanding toward sheaf theory, internal logic, and the notion of parallel mathematical realities. Whether you're curious about how topoi reinterpret the concept of "truth", this piece aims to provide a clear and structured introduction .

Suppose that we have a quantum-mechanical particle that starts out in the state \(\psi(x)\) at \(t = 0\), that is,  \[ |\Psi(0)\rangle=\int \psi(x)|x\rangle dx. \] And we wish to compute its state \(\phi(x)\) at some other time \(t=T\), i.e., \[ |\Psi(T)\rangle=\int \phi(x)|x\rangle dx. \] Let \[ \mathcal{P}(a \to b) = \{ \gamma: [0, T] \to \mathbb{R}^3 : \gamma(0) = a, \gamma(T) = b \} \] denote the space of all paths that start at the point \(a\) at time 0 and end at the point \(b\) at time \(T\).  The probability of finding the particle at a given point \(b\) at time \(T\) is \(\mathcal{A}_\text{total} \in \mathbb C\), the **sum of the amplitudes**, \(\mathcal{A}[\gamma] \in \mathbb C\), of all the paths that end at that point. That is, \[ \mathcal{A}_\text{total} = \int_{\mathcal{P}(a \to b)} \mathcal{A}[\gamma] \, \mathcal{D}\gamma , \] where \(\mathcal{D}\gamma\) is some sort of mysterious [[measure]] on the space \(\mathcal{P}(a \to b)\). The probability **amplitude of ...