Euclidean, Hyperbolic and Elliptic Geometry | The n-Category Café
Sep 26, 2016 · There are two famous kinds of non-Euclidean geometry: hyperbolic geometry and elliptic geometry (which almost deserves to be called ‘spherical’ geometry, but not quite because we identify antipodal points on the sphere). In fact, these two kinds of geometry, together with Euclidean geometry, fit into a unified framework with a parameter s ...
Overview of the general case 1 The generalized McShane’s identity gives a formula for the length L1 of a pre-specified boundary component 1 of X 2Mg; n(L1;:::;L ) as a sum of terms over all pants containing 1. 2 We integrate this formula over moduli space Mg;n(L1;:::;Ln) using Mirzakhani’s integration formula which gives us an integral in …
Octonions and the Standard Model (Part 3) | The n-Category Café
Jul 24, 2020 · The basic idea is this: if we fix an octonion i ∈ 𝕆 with i2 = −1, the group of automorphisms of the octonions that fix this element is SU(3). This element i also determines a copy of ℂ in 𝕆. This makes 𝕆 into a complex vector space, and SU(3) acts on this vector space. As a representation of SU(3), this vector space is isomorphic to.
Firoozbakht's Conjecture | The n-Category Café - University of …
Oct 30, 2021 · Firoozbakht’s conjecture is a way of saying the gaps between successive primes don’t get too big too fast. As you can see here, it’s stronger than Cramér’s or Granville’s conjectures on prime gaps — and it gets scarily close to being wrong at times, like when the largest prime gap jumps from 924 to 1132 at the prime 1693182318746371 — you can see that big horizontal black line ...
Understanding the Homotopy Coherent Nerve | The n-Category Café
Apr 29, 2010 · I was really happy to understand this construction, both the fact that ℂ \mathbb{C} on categories (and in particular, on Δ \Delta) coincides with the comonad resolution, and the resulting description of the coherent nerve.The definition of the coherent nerve you sometimes read, by first defining the homset of ℂ [n] \mathbb{C}[n] as the nerve of some random poset, always made it seem very ...
Structured Cospans | The n-Category Café
Jul 1, 2019 · The idea is to treat open systems as morphisms in a category of a particular kind: a ‘structured cospan category’. Here is his talk: Kenny Courser, Structured cospans. In July 11th I’m going to talk about structured cospans at the big annual category theory conference, CT2019: John Baez, Structured cospans.
New Normed Division Algebra Found! | The n-Category Café
Sep 23, 2020 · New Normed Division Algebra Found! Posted by John Baez. Hurwitz’s theorem says that there are only 4 normed division algebras over the real numbers, up to isomorphism: the real numbers, the complex numbers, the quaternions, and the octonions. The proof was published in 1923. It’s a famous result, and several other proofs are known.
The Algebraic K-Theory of the Integers | The n-Category Café
Dec 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.
For xed H, this equality follows from 1 x+ i 1 x i = 2i x2 + 2 !0 2ˇi (x): (15) It continues to hold after averaging over H. The thermal partition function Z( ) = Tre H is related to the resolvent R(E) by an integral transform:
Grothendieck--Galois--Brauer Theory (Part 1) | The n-Category Café
Jun 26, 2023 · In 1960 and 1961 Grothendieck developed a new approach to Galois theory in a series of seminars that got written up as SGA1. This is sometimes called Grothendieck’s Galois theory. His approach was very geometrical. He treated the Galois group of a field as the fundamental group of a funny sort of ‘space’ associated to that field, called ...