• Unicity of Primary Decomposition

Let's recall the fundamental theorem of arithmetics: every integer decomposes in a unique way in a product of prime factors, and a unit $u \in \{+1,-1\}$. Phrased in terms of ideals, this statement becomes: any non-zero ideal can be written in a unique way as the intersection of ideals of the form $\langle p_i^{e_i}\rangle$ where the $p_i$ are prime numbers and the $e_i$ are positive integers. In this post, I will examine how this theorem generalizes to ideals in polynomial rings, with a special emphasis on the unicity property.

• Une remarque sur les unités SI

La semaine dernière, j'ai posé le petit problème suivant sur discord : calculer la quantité $$5000000 \frac{h \alpha}{c e^2}$$ où $h$ est la constante de Planck, $c$ la vitesse de la lumière dans le vide, $e$ la charge de l'électron, et $\alpha$ la constante de structure fine. On veut le résultat dans le système international d'unités (SI), et on demande de commenter la valeur obtenue.

• Definition of Quiver varieties

This is the third post of a series (see here and there) where I explore the mathematical theory of quivers, following [1]. Today, we will focus on the geometry of the isomorphism classes of quiver representations, and proceed to the definition of the quiver varieties.

• Geometric Invariant Theory

Today I continue my reading of Kirillov's textbook on quiver representations. Before coming to quiver varieties, we need to review the basics of Geometric Invariant Theory (GIT), some symplectic geometry, and see how these are combined to provide symplectic resolution of singular spaces. Again I will follow very closely Kirillov's book [1].

• Embedding the Hyperbolic Plane

I define the hyperbolic plane as the space $$\mathbb{H} = \left\{ (x,y) \in \mathbb{R}^2 | y>0 \right\}$$ endowed with the metric $$\mathrm{d}s^2_{\mathbb{H}} = \frac{\mathrm{d} x^2 + \mathrm{d} y^2}{y^2} \, .$$ I want to study here various "models" of this hyperbolic plane. Most of what I will say has an analogous un higher dimension, but I will keep the discussion as simple and as visual as possible, so I will stick to two dimensions.

• Introduction to Quiver Representations

A quiver $\vec{Q}$ is a directed graph, defined by a set of vertices $I$ and a set of edges $\Omega$, with two maps $s,t : \Omega \rightarrow I$ that indicate respectively the source and the target of the oriented edges. Here I will focus only on finite and connected quivers (the number of edges and vertices are finite, and the underlying graph $Q$ is connected). My main concern today will be quiver representations. A representation $V$ of $\vec{Q}$ is comprised of a finite-dimensional vector space $V_i$ associated to each vertex $i \in I$, together with linear operators $x_h : V_i \rightarrow V_j$ for each edge $h \in \Omega$ with source $i$ and target $j$. There is a natural notion of morphisms between two representations, and I will denote by $\mathrm{Rep}(\vec{Q})$ the category of representations of $\vec{Q}$. The main goal is to classify these representations.

• Rank 1 four-dimensional SCFTs

In this post I will review a series of papers by Argyres, Lotito, Lü and Martone published in recent years. The main problem is to classify some four-dimensional $\mathcal{N}=2$ SCFTs, based on the possible geometries that can appear on the moduli space of vacua. Finding a geometry does not mean that a corresponding theory exists, but the complete classification provides constraints on the possible theories that can exist.

• Rational maps and Connected formulas

Some time ago, I began to talk about Amplitudes and the scattering equations. Today a new paper by Cachazo, Guevara, Heydeman, Mizera, Schwarz and Wen appeared, and I thought this was an opportunity to understand a bit more this topic. Section 2 of this paper contains a very enlightening review of the rational maps, scattering equations and CHY formulas in arbitrary dimension.

• Inégalités Isopérimétriques et Problèmes associés

Quelle est la plus grande aire que l'on peut délimiter dans le plan en utilisant une courbe de longueur 1 ? La réponse est "naturellement" le cercle. Mais comment le démontrer ? Voici une preuve, fortement inspirée de ce document.

• Méthode pour résoudre un problème d'échecs

Pour s'entraîner à la tactique aux échecs, il est bon de faire quelques exercices, par exemple ici sur Lichess. Quand on ne voit pas directement la solution, il est bon de procéder méthodiquement; voici une liste d'idées à explorer.