What I have learned today

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 ...

Quantum Fields: a learner’s perspective As someone still learning QFT, I believe there's a special advantage in explaining the ideas while they’re fresh. I still remember what was confusing to me a few months ago—and I can highlight the key insights that helped me move forward, without overwhelming the reader with technical detail. That’s what this paper aims to do .   In this blog post, I’d like to offer you an overview of the first part of my paper, where I lay out the conceptual foundations of QFT—not through abstract postulates, but through familiar terrain: wavefunctions, Hilbert spaces, and harmonic oscillators. If you’ve studied QM before, you’re already halfway there. QFT is Quantum Mechanics—just with a different raw material A common misconception is that QFT somehow replaces quantum mechanics. In truth, it’s the opposite: QFT is quantum mechanics, applied in a particular way. The shift lies not in the rules of quantum theory, but in what we apply them to. ...

Consider, for simplicity, a 2-dimensional manifold \(\Sigma\) embedded into the 3-dimensional Riemannian manifold \((M,g)\), in such a way that the given frame \(\{e_1,e_2,e_3\}\) is adapted to \(\Sigma\), i.e., \(\omega^3|_{T\Sigma}=0\), where \(\omega^1,\omega^2,\omega^3\) is the dual coframe. The surface \(\Sigma\) inherits a Riemannian metric from the ambient manifold, with its corresponding Levi-Civita connection. We will denote by \(\tilde{\omega}^1,\tilde{\omega}^2\) the restrictions of \(\omega^1,\omega^2\) to \(T\Sigma\), and by \(\tilde{\Theta}\) and \(\tilde{\Omega}\) the connection forms and the curvature forms, respectively, of the inherited connection. According to Cartan's first structural equation , \[ d\omega^j=\omega^i \wedge \Theta^j_{\,\,i}. \] By restricting to \(\Sigma\) (and by the uniqueness of \(\tilde{\Theta}\)) we conclude that \[ \tilde{\Theta}^i_{\,\,j}=\Theta^i_{\,\,j} |_{T\Sigma}, \quad i,j=1,2, \] and that \[ 0= \tilde{\omega}^1\wedge\Theta^3_{\,\,1}...