Bounded Gaps Between Primes | The n-Category Café
WEBMay 15, 2013 · We know of course that this list is countably infinite. A prime gap is an integer pn+1 − pn p_ {n+1}-p_n. The Prime Number Theorem tells us that pn+1 − pn p_ {n+1}-p_n is approximately log(pn) \log (p_n) as n n approaches infinity. The twin primes conjecture, on the other hand asserts that.
Timing, Span (Graph) and Cospan (Graph) | The n-Category Café
WEBAug 1, 2022 · This is a category whose objects are graphs, morphisms are cospans X → Z ← Y of graphs and composition is given by pushout, which exist for Graph. Cospan(Graph) is a symmetric monoidal category due to the cocartesian monoidal structure on Graph. For a set A, define the graph 𝒱(A) ≡ (A, ∅) to contain no edges.
Bernoulli Numbers and the Harmonic Oscillator
WEBAug 16, 2024 · Re: Bernoulli Numbers and the Harmonic Oscillator. The definition of the Bernoulli numbers that sticks in my head is the one in Pierre Cartier’s paper Mathemagics: they are defined by the equation. B n = (B + 1) n B^n = (B + 1)^n. for n ≥ 2 n \geq 2, and by B 0 = 1 B^0 = 1. Then you change superscripts to subscripts.
Dynamical Systems and Their Steady States | The n-Category Café
WEBBaez–Pollard’s approach The category Dynam \mathbf{Dynam} of open dynamical systems. Let’s start by introducing the main players. A dynamical system is usually defined as a manifold M M whose points are “states”, together with a smooth vector field on M M saying how these states evolve in time. Since the motivation in this paper comes from chemistry, our manifolds will be euclidean ...
4d QFT for Khovanov Homology | The n-Category Café
WEBFeb 4, 2011 · 4d QFT for Khovanov Homology. Khovanov homology is a knot invariant that is a categorification of the Jones polynomial. Khovanov homology has long been expected to appear as the observables in a 4-dimensional TQFT in higher analogy of how the Jones polynomial arises as an observable in 3-dimensional Chern-Simons theory.
Topos Theory and Measurability | The n-Category Café
WEBJul 28, 2021 · There was an interesting talk that took place at the Topos Institute recently – Topos theory and measurability – by Asgar Jamneshan, bringing category theory to bear on measure theory. Jamneshan has been working with Terry Tao on this: Asgar Jamneshan, Terence Tao, Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical ...
Parametrized Mates and Multivariable Adjunctions
WEBOf course, there’s an obvious “profunctorial” simplification. Encode a multivariable adjunction as a “spider” - a circle with legs that are undirected this time and labeled in the following manner. Looking at “the left adjoint” for the adjunction, there is a leg labeled by each of the input objects and also a leg labeled by the opposite of the output object.
The Algebraic K-Theory of the Integers | The n-Category Café
WEBDec 9, 2020 · The algebraic K K-groups of the integers, and other rings, are a big deal.A lot of fairly recent progress on understanding them is due to Voevodsky, Rost, Rognes and Weibel, whose work culminated in a proof of the Bloch–Kato Conjecture connecting K K-theory to Galois cohomology.But lots of other mathematicians deserve credit too.
Galois Theory | The n-Category Café - University of Texas at Austin
WEBAug 15, 2024 · About 40 short explanatory videos. A large collection of problems. Nearly 500 multiple choice questions. I’m a little bemused by the popularity of the Galois theory notes. I’ve made quite a few sets of course notes public before, e.g.: Fourier analysis. General topology. Linear algebra. Category theory.
Weirdness in the Primes | The n-Category Café - University of …
WEBMar 15, 2016 · Another pretty intuitive explanation by Ryan Alweiss ():You are on the first rock. In one case you advance to the next rock or stay put with probability .5 and in the other the same is true if you are on the first rock, but if you’re on the second rock and flip you have a one half chance of finishing… and a one half chance of moving backward.