Bounded Gaps Between Primes | The n-Category Café
WEBMay 14, 2013 — A prime gap is an integer pn + 1 − pn. The Prime Number Theorem tells us that pn + 1 − pn is approximately log(pn) as n approaches infinity. The twin primes conjecture, on the other hand asserts that. liminf n → ∞(pn + 1 − pn) = 2. i.e., that there are infinitely many pairs of twin primes for which the prime gap is just two.
Octonions and the Standard Model (Part 2) | The n-Category Café
WEBJul 22, 2020 — The payoff for all these contortions is a very pretty formula for octonion multiplication. We’ll take elements of ℂ ⊕ ℂ3 and write them like a + →a, where the ‘scalar part’ a lives in ℂ and the ‘vector part’ →a lives in ℂ3. And these will be our octonions! Theorem 2. If we define multiplication on ℂ ⊕ ℂ3 by.
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 ...
Structure and Pseudorandomness | The n-Category Café
WEBTerence Tao has written three delightful posts, starting here, detailing his views delivered at the Simons’ lectures at MIT on the relationship between structure and pseudorandomness in mathematics. We read. Structured objects are best studied using the tools of algebra and geometry. Pseudorandom objects are best studied …
Bernoulli Numbers and the Harmonic Oscillator
WEBAug 16, 2024 — The Bernoulli numbers can be defined like this: x ex − 1 = B0 + B1x + B2x2 2! + B3x3 3! + ⋯. and if you grind them out, you get. B0 = 1 B1 = −1 2 B2 = 1 6 B3 = 0 B4 = − 1 30. and so on. The pattern is quite strange. Bernoulli numbers are connected to hundreds of interesting things.
You Could Have Invented De Bruijn Indices | The n-Category Café
WEBAug 30, 2021 — One of the most curious things about formal treatments of type theory is the attention paid to dummy variables. The point is simple enough to explain to a calculus student: just as the integrals ∫ 0 2 x 3 dx \int_0^2 x^3 \, dx and ∫ 0 2 t 3 dt \int_0^2 t^3 \, dt are equal, so are the functions λ x. x 3 \lambda x. x^3 and λ t. t 3 \lambda t. t^3 — the name of the variable doesn’t matter.
Discreteness, Concreteness, Fibrations, and Scones
WEBNov 23, 2011 — Theorem 1: Suppose C has a terminal object preserved by U. If U is a fibration, then C has codiscrete objects. Proof: Define G: S → C by G(a) = the result of pulling back the terminal object 1 ∈ C along the unique map a → 1 in S. It is easy to verify that U ⊣ G and that G is fully faithful. .
Prime Numbers and the Riemann Hypothesis | The n-Category Café
WEBMar 21, 2016 — The Riemann Hypothesis is one of the great unsolved problems of mathematics, and the reward of $1,000,000 of Clay Mathematics Institute prize money awaits the person who solves it. But — with or without money — its resolution is crucial for our understanding of the nature of numbers. There …
Holomorphic Gerbes (Part 1) | The n-Category Café
WEBMar 27, 2022 — Thanks for all those references! I’d never guessed you were studying holomorphic gerbes until you came out and told me the other day. If you’re studying Deligne cohomology and n n-gerbes, it sounds like you’ll inevitably need to learn what Urs Schreiber is doing with those things.. As for me, I’m studying algebraic geometry as a kind of retirement hobby, the way other people take up ...
Globular | The n-Category Café - University of Texas at Austin
WEBDec 5, 2015 — Mathematical foundations. Globular is based on the theory of finitely-presented semistrict n n-categories; at the moment, it works up to the level of 3-categories, with an extension to 4-categories actively in development.(You can build cells of any dimension, but from 4-cells and up, some structures are missing.) Definitions of n n-category vary in how strict they are; a definition is ...