My personal notes

 From now on I am goin to share what I have learned in another format: html files. They are easier to write. They will be allocated at


https://notesmath.vercel.app/

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About the definition of angle, from Geometric Algebra

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In GA, bivectors represent  2-dimensional directions. Any simple bivector can be written as \[ \theta e_1e_2 \] with \(\theta \in \mathbb{R}\) and \(\{e_i\}\) unitary and orthogonal vectors. The unitary vectors specify the 2- direction itself, and \(\theta\) is the size (analogous to the length of a vector, that is a 1-dimensional direction). Given two vectors \(a\) and \(b\) (that we take to be unitary, wlog) we say that the simple bivector \(\theta e_1e_2\) is the angle formed by them if \[ ab=e^{\theta e_1 e_2} \] where we take as a definition \[ e^A=1+A+A^2 /2+\cdots=\lim_N (1+\frac{A}{N})^N \] Real analysis let us assure that this expression is well defined when \(A\) is a 2-blade. In fact, since \(e_1e_2e_1e_2=-1\), we can write \[ e^{\theta e_1 e_2}=cos(\theta)+e_1 e_2sin(\theta) \] where \(cos\) and \(sin\) are defined by their power series. Oserve that \(b=a e^{\theta e_1 e_2}\) and then \[ b=a(cos(\theta)+e_1 e_2sin(\theta))=a\cdot cos(\theta)+ ae_1e_2 \cdot sin(\theta...
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Spherical, hyperbolic and parabolic geometry

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Let's consider the space \(\mathbb{M}^4\), that is, the manifold \(\mathbb{R}^4\) with coordinates \((x,y,z,t)\) but with the pseudo-Riemannian metric given in each tangent space by \[ ds^2=dx^2+dy^2+dz^2-dt^2. \] This metric is left invariant by the group \(O(3,1)\) when acting on \(\mathbb{M}^4\). It also leaves invariant the cone \[ t^2-x^2-y^2-z^2=0 \] so is logical that we pay attention to it. Now, following TRTR from Penrose page 423, we can consider the family of hyperplanes inside \(\mathbb{M}^4\) given by \[ z+t+\lambda(t-z)=2 \] intersecting our cone in different 2-dimensional submanifolds \(P_{\lambda}\) of \(\mathbb{M}^4\). Here we have a picture from that book, where we consider only \((x,z,t)\) for purposes of drawability. If we analyse the case \(\lambda=0\), we observe that we obtain the submanifold \(P_0\), \(E\) in the picture with a shape of a parabola, given by the embedding \begin{align*}   \phi \colon \mathbb{R}^2 &\longrightarrow \mathbb{M}^4\\   (u_...
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PhD Thesis Acknowledgement

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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...
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