John C. Baez
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View article: Tannaka Reconstruction and the Monoid of Matrices
Tannaka Reconstruction and the Monoid of Matrices Open
Settling a conjecture from an earlier paper, we prove that the monoid $\mathrm{M}(n,k)$ of $n \times n$ matrices in a field $k$ of characteristic zero is the "walking monoid with an $n$-dimensional representation". More precisely, if we tr…
View article: Dirichlet Species and Arithmetic Zeta Functions
Dirichlet Species and Arithmetic Zeta Functions Open
Though Joyal's species are known to categorify generating functions in enumerative combinatorics, they also categorify zeta functions in algebraic geometry. The reason is that any scheme $X$ of finite type over the integers gives a "zeta s…
View article: Groupoid Cardinality and Random Permutations
Groupoid Cardinality and Random Permutations Open
If we treat the symmetric group $S_n$ as a probability measure space where each element has measure $1/n!$, then the number of cycles in a permutation becomes a random variable. The Cycle Length Lemma describes the expected values of produ…
View article: Tradução em séries de televisão
Tradução em séries de televisão Open
The translation field is extensive and has been a focus of studies over time; however, the rise of streaming brought new research opportunities. Considering this, the current study aims to comprehend the translation field, exploring cultur…
View article: The Hexagonal Tiling Honeycomb
The Hexagonal Tiling Honeycomb Open
The hexagonal tiling honeycomb is a beautiful structure in 3-dimensional hyperbolic space. It is called {6,3,3} because each hexagon has 6 edges, 3 hexagons meet at each vertex in a Euclidean plane tiled by regular hexagons, and 3 such pla…
View article: 2-Rig Extensions and the Splitting Principle
2-Rig Extensions and the Splitting Principle Open
Classically, the splitting principle says how to pull back a vector bundle in such a way that it splits into line bundles and the pullback map induces an injection on $K$-theory. Here we categorify the splitting principle and generalize it…
View article: What is Entropy?
What is Entropy? Open
This short book is an elementary course on entropy, leading up to a calculation of the entropy of hydrogen gas at standard temperature and pressure. Topics covered include information, Shannon entropy and Gibbs entropy, the principle of ma…
View article: The Moduli Space of Acute Triangles
The Moduli Space of Acute Triangles Open
As an introduction to the concept of "moduli space" we consider the moduli space of similarity classes of acute and right triangles in the plane. This has a map to the moduli space of elliptic curves which is onto and generically three-to-…
View article: The Beauty of Roots
The Beauty of Roots Open
A "Littlewood polynomial" is a polynomial whose coefficients are all 1 or -1. The set of all complex roots of all Littlewood polynomials exhibits many complicated, beautiful and fascinating patterns. Some fractal regions of this set closel…
View article: The Beauty of Roots
The Beauty of Roots Open
A "Littlewood polynomial" is a polynomial whose coefficients are all 1 or -1. The set of all complex roots of all Littlewood polynomials exhibits many complicated, beautiful and fascinating patterns. Some fractal regions of this set closel…
View article: The Icosidodecahedron
The Icosidodecahedron Open
The icosidodecahedron has 30 vertices, one at the center of each edge of a regular icosahedron -- or equivalently, one at the center of each edge of a regular dodecahedron. It is a beautiful, highly symmetrical shape. But it is just a proj…
View article: Hoàng Xuân Sính's Thesis: Categorifying Group Theory
Hoàng Xuân Sính's Thesis: Categorifying Group Theory Open
During what Vietnamese call the American War, Alexander Grothendieck spent three weeks teaching mathematics in and near Hanoi. Hoàng Xuân Sính took notes on his lectures and later did her thesis work with him by correspondence. In her thes…
View article: Compositional Modeling with Stock and Flow Diagrams
Compositional Modeling with Stock and Flow Diagrams Open
Stock and flow diagrams are widely used in epidemiology to model the dynamics of populations. Although tools already exist for building these diagrams and simulating the systems they describe, we have created a new package called StockFlow…
View article: Motivating Motives
Motivating Motives Open
Underlying the Riemann Hypothesis there is a question whose full answer still eludes us: what do the zeros of the Riemann zeta function really mean? As a step toward answering this, André Weil proposed a series of conjectures that include …
View article: Young Diagrams and Classical Groups
Young Diagrams and Classical Groups Open
Young diagrams are ubiquitous in combinatorics and representation theory. Here we explain these diagrams, focusing on how they are used to classify representations of the symmetric groups $S_n$ and various "classical groups": famous groups…
View article: Compositional thermostatics
Compositional thermostatics Open
We define a thermostatic system to be a convex space of states together with a concave function sending each state to its entropy, which is an extended real number. This definition applies to classical thermodynamics, classical statistical…
View article: Composition and Recursion for Causal Structures
Composition and Recursion for Causal Structures Open
Causality appears in various contexts as a property where present behaviour can only depend on past events, but not on future events. In this paper, we compare three different notions of causality that capture the idea of causality in the …
View article: Isbell Duality
Isbell Duality Open
Mathematicians love dualities. After a brief explanation of dualities, with examples, we turn to one of the purest and most beautiful: Isbell duality. For any category $\mathsf{C}$, this gives an adjunction between the category of presheav…
View article: A Categorical Framework for Modeling with Stock and Flow Diagrams
A Categorical Framework for Modeling with Stock and Flow Diagrams Open
Stock and flow diagrams are already an important tool in epidemiology, but category theory lets us go further and treat these diagrams as mathematical entities in their own right. In this chapter we use communicable disease models created …
View article: The Kuramoto-Sivashinsky Equation
The Kuramoto-Sivashinsky Equation Open
The Kuramoto-Sivashinsky equation was introduced as a simple 1-dimensional model of instabilities in flames, but it turned out to mathematically fascinating in its own right. One reason is that this equation is a simple model of Galilean-i…
View article: Rényi Entropy and Free Energy
Rényi Entropy and Free Energy Open
The Rényi entropy is a generalization of the usual concept of entropy which depends on a parameter q. In fact, Rényi entropy is closely related to free energy. Suppose we start with a system in thermal equilibrium and then suddenly divide …
View article: Compositional Modeling with Stock and Flow Diagrams
Compositional Modeling with Stock and Flow Diagrams Open
Stock and flow diagrams are widely used in epidemiology to model the dynamics of populations. Although tools already exist for building these diagrams and simulating the systems they describe, we have created a new package called StockFlow…
View article: The Gauss-Lucas Theorem
The Gauss-Lucas Theorem Open
The Gauss-Lucas theorem says that for any complex polynomial $P$, the roots of the derivative $P'$ lie in the convex hull of the roots of $P$. In other words, the roots of $P'$ lie inside the smallest convex subset of the complex plane con…
View article: Compositional Thermostatics
Compositional Thermostatics Open
We define a thermostatic system to be a convex space of states together with a concave function sending each state to its entropy, which is an extended real number. This definition applies to classical thermodynamics, classical statistical…