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 …
Topology Puzzles | The n-Category Café - University of Texas at …
Nov 11, 2017 · Okay, I didn’t know whether these puzzles were stepping stones to something else you wanted to talk about, or what. (Actually, it feels like calling them “puzzles” is a little strange, or at least so for Puzzle 1, since AFAIK the characterization is a pretty difficult theorem that I wouldn’t expect someone to work out to completion without considerable effort, if they didn’t know it ...
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.
E8
There are 70 nilpotent orbits in e8 \mathfrak {e}_8. Excluding the regular orbit (which corresponds to the trivial defect), that gives 69 codimension-2 defect operators (“regular punctures”) in the E8 E_8 (2,0) theory. We label them by their Bala-Carter labels, where 0 0 denotes the full puncture and E8(a1) E_8 (a_1) (the subregular orbit ...
Dimensional Analysis in Algebra and Geometry
Aug 8, 2024 · Another name for the study of categories of line objects is “dimensional analysis”. In dimensional analysis, a physical theory is described by specifying an abelian group of “dimensions” (these are the line objects) together with a commutative algebra of “quantities” (these are the sections of the line objects) which is graded by ...
Entropy Modulo a Prime | The n-Category Café - University of …
Dec 14, 2017 · In 1995, the German geometer Friedrich Hirzebruch retired, and a private booklet was put together to mark the occasion. That booklet included a short note by Maxim Kontsevich entitled “The 1 1 2 1\tfrac{1}{2}-logarithm”.. Kontsevich’s note didn’t become publicly available until five years later, when it was included as an appendix to a paper on polylogarithms by Philippe Elbaz-Vincent ...
Three Phases of Continued Fraction Theory | The n-Category Café
Sep 6, 2020 · But in all these cases we actually get much more: we get continued fraction expansions for the functions ln z \ln z, arctan z \arctan z and e z e^z — and making the right choice of z z we get our formulas for ln 2 \ln 2, 4 / π 4/\pi and e e, respectively. Indeed Euler’s formula gives nice continued fractions for all functions with simple ...
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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 ...
Instiki - University of Texas at Austin
Feb 3, 2018 · Instiki is free software, licensed under the same license as Ruby. The current release version is 0.30.2. The current development version (which is what this site is running on) is available on Github, from the BZR repository or as a tarball. The BZR Repository has an Atom feed, with which you can stay abreast of the development of this software.