Hi, I'm Ramzy.
I build software.

Coursework

Reading

• Algebra: Chapter 0 Paolo Aluffi
• Algebra Thomas Hungerford
• Linear Representations of Finite Groups Jean-Pierre Serre
• Introduction to Commutative Algebra Atiyah & Macdonald
• Real and Functional Analysis Serge Lang
• Fundamentals of Differential Geometry Serge Lang

Favourite theorems

• First Isomorphism Theorem

  Every group homomorphism $\varphi: G \to H$ induces an isomorphism:

  $$G/\ker\varphi \;\cong\; \mathrm{im}\,\varphi.$$

  This shows up all over the place in math. This construction takes a map and boils it down to the information that it preserves; any homomorphism factors as a projection followed by an embedding. The same statement holds verbatim for rings, modules, and vector spaces.

• Absolute Galois group is a Cantor set

  The absolute Galois group $G_\mathbb{Q} = \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, equipped with its Krull topology, is compact, totally disconnected, perfect, and uncountable, hence homeomorphic (as a topological space) to the Cantor set.

  The “symmetry of all algebraic numbers” is fractal at its core: no isolated points, no smooth structure. Motivates profinite groups and Galois representations.

• Peter–Weyl Theorem

  For a compact group $G$, matrix coefficients of irreducible unitary representations are dense in $C(G)$ and orthonormal in $L^2(G)$:

  $$L^2(G) \cong \bigoplus_{\pi \in \hat G} V_\pi \otimes V_\pi^*.$$

  Fourier analysis is just a special case of this theorem. Fourier analysis works because the circle is a compact group; Peter-Weyl says that we can do something similar for all other compact groups. In physics, Peter–Weyl explains why angular momentum and spherical harmonics work.

• Profinite completion of the integers

  The profinite completion of the integers decomposes as

  $$\widehat{\mathbb{Z}} = \varprojlim_n\,\, \mathbb{Z}/n\mathbb{Z}\cong \prod_p \mathbb{Z}_p,$$

  a product over all primes of the $p$-adic integers.

  This is just super cool; integers are determined by all their modular shadows.

• Probability that an integer is squarefree

  The natural density of positive integers not divisible by any $p^2$ is

  $$\frac{6}{\pi^2} = \frac{1}{\zeta(2)} = \prod_p \left(1 - \frac{1}{p^2}\right) \approx 0.6079.$$

  About 61% of integers have no repeated prime factor — and the answer involves $\pi$. The Euler product is the punchline: independence of primes makes the density multiplicative.