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 …
Galois' Fatal Duel | The n-Category Café - University of Texas at …
May 30, 2023 · Galois’ Fatal Duel. On this day in 1832, Evariste Galois died in a duel. The night before, he summarized his ideas in a letter to his friend Auguste Chevalier. Hermann Weyl later wrote “This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of ...
2-Rigs and the Splitting Principle | The n-Category Café
Oct 10, 2024 · The 2-rig map f is characterized by the fact that it sends the generating object x ∈ ¯ kS to the infinite coproduct s1 ⊕ s2 ⊕ ⋯. Here is our new splitting principle: Theorem. The 2-rig map f: ¯ kS → A ⊠ ∞ has these properties: f(r) splits as a countably infinite coproduct of subline objects. K(f) is injective.
Axiomatic Set Theory 4: Subsets | The n-Category Café
Oct 11, 2024 · One of the axioms (loosely stated) is that there exists a set 2 \mathbf {2} such that for all sets X X, functions X → 2 X \to \mathbf {2} correspond to injections into X X (taken up to isomorphism over X X). We didn’t assume that 2 \mathbf {2} has two elements. But we prove here that this does in fact follow from the axioms.
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The n-Category Café - University of Texas at Austin
Nov 8, 2024 · Previously: Part 7. By this point in the course, we’ve finished the delicate work of assembling all the customary set-theoretic apparatus from the axioms, and we’ve started proving major theorems.This week, we met well ordered sets and developed all the theory we’ll need. The main results were: every family of well ordered sets has a …
Operads and Trees | The n-Category Café - University of Texas at …
Mar 31, 2014 · Thanks, James! I’m not sure “application” is the best term for what we’re doing with the operad structure on the space of phylogenetic trees — and I’m quite sure “practical application” is the wrong way of describing it. You can see an overview of what we’ve done here.I hope there will be applications to reconstructing phylogenetic trees, but our work so far is more ...
Inner Automorphisms of the Octonions | The n-Category Café
Nov 22, 2022 · Does he describe the other conjugacy class of order 3 automorphisms? (There are exactly two, and only one class of order 2s.) At the far end of the Weyl alcove, i.e. the conjugate points of the exponential map from the Lie algebra to the compact group, there’s an 11-dim family of automorphisms that includes the C-based and H-based ones we were discussing, at …
Bernoulli Numbers and the Harmonic Oscillator
Aug 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.
The Quintic, the Icosahedron, and Elliptic Curves
Apr 18, 2024 · There is a remarkable relationship between the roots of a quintic polynomial, the icosahedron, and elliptic curves. This discovery is principally due to Felix Klein (1878), but Klein’s marvellous book misses a trick or two, and doesn’t tell the whole story. The purpose of this article is to present this relationship in a fresh, engaging ...