What I have learned today

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 the sum of the amplitudes of all the paths that end at that point. $$ \mathcal{A}_\text{total} = \int_{\mathcal{P}(a \to b)} \mathcal{A}[\gamma] \, \mathcal{D}\gamma, $$ where \( \mathcal{D}\gamma \) is some sort of mysterious "Lebesgue measure" on the space \( \mathcal{P}(a \to b) \). The probability amplitude of an individual path is given by the exponential of th...

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}...

In April 2024 I defended my PhD Thesis, entitled: New methods for the integration of distributions of vector fields .  I want to share here the acknowledgement section: In reflecting upon the journey that led to the completion of this thesis, a famous phrase attributed to Isaac Newton comes to mind: "If I have seen further, it is by standing on the shoulders of giants". However, in my own judgement, this statement might be more accurately extended to include not only giants: "If I have seen further, it is by standing on the shoulders of giants - and ants ". Let me explain. The lesser-acknowledged, yet equally vital, contributions of numerous individuals have been the foundation upon which the academic progress has been built. These ants , often unnamed and unrecognized, are akin to the countless mathematicians whose quiet yet essential contributions have paved the way for larger discoveries. We can think of this using the analogy of the evaporation of a liquid, whe...

My collaborators and I have recently published two papers ( this one and this other ) in which  we develop a method to obtain the integral manifolds of involutive distributions. Exploring the integral manifolds of involutive distributions contributes to a broader understanding within differential geometry, an area with ties to many other mathematical branches. Furthermore, the study of these distributions and manifolds holds relevance in physics, especially within classical mechanics and field theory. A deeper grasp of these mathematical constructs can be beneficial for ongoing research in both mathematics and physics.  Given a distribution, for example \(\mathcal{Z}=\{Z_1,Z_2\}\) in \(\mathbb R^n\), the idea of our work is to complete it with a sequence of \(n-2\) vector fields \(Y_1, Y_2,Y_3,\ldots\) in such a way that - \(Y_1\) is a \(\mathcal{C}^{\infty}\)-symmetry of \(\mathcal{Z}\). - \(Y_2\) is a \(\mathcal{C}^{\infty}\)-symmetry of \(\mathcal{Z}\oplus \{X_1\}\)....