ReadingNotes

The Princeton Companion to Mathematics

The Princeton Companion to Mathematics edited by Timothy Gowers, June Barrow-Green and Imre Leader - ISBN 9780691118802 - PUP 2008

Motivation

Getting an overview of the mathematics as a whole, not as a fragmented set of fields. Also highly recommended by Wagner. Also, wanting to be able to leverage Shun-ichi Amari's work (including Methods of Information Geometry) and thus requiring a better understanding of several key mathematical notions.

Pre-reading model

Draw a schema (using PmGraphViz or another solution) of the situation of the area in the studied domain before having read the book.

Reading

* article to read
** X read article

I Introduction
- 1 What Is Mathematics About?
  - 1 Algebra, Geometry, and Analysis
  - 2 The Main Branches of Mathematics
- 2 The Language and Grammar of Mathematics
- 3 Some Fundamental Mathematical Definitions
  - 4 Functions between Algebraic Structures
    - 3 Eigenvalues and Eigenvectors
  - 6 What Is Geometry?
    - 9 Manifolds and Differential Geometry
      - 1 Smooth Manifolds
    - 10 Riemannian Metrics
- 4 The General Goals of Mathematical Research
  - 4 Discovering Patterns
    - see also ReadingNotes.ReadingNotes#TheSelf-MadeTapestry
III Mathematical Concepts
- 17 Dimension
- 18 Distributions
- 21 Elliptic Curves
- ~~56 Matroids~~
  - "matroids form a natural home for many optimization problems" (p248)
IV Branches of Mathematics
- 2 Analytic Number Theory
  - 1 Introduction
- 3 Computational Number Theory
  - 4 The Riemann Hypothesis and the Distribution of the Primes
- 7 Differential Topology
VI Mathematicians
- 33 Niels Henrik Abel
- 35 Carl Gustav Jacob Jacobi
- 36 Peter Gustav Lejeune Dirichlet
- 37 William Rowan Hamilton
- 49 Georg Friedrich Bernhard Riemann
- 53 Sophus Lie
- 61 Jules Henri Poincaré
- 63 David Hilbert
- 65 Jacques Hadamard
  - see also ReadingNotes.Hadamard
- 88 Andrei Nikolaevich Kolmogorov
  - (to find) Information Theory and Algorithms (seems inexistant!)
- 89 Alonzo Church
- 91 John von Neumann
  - game theory developped before RAND?
  - Planning and coding of problems for an electronic computing instrument
- 92 Kurt Gödel
  - see also ReadingNotes.TheCodeBook
- 94 Alan Turing
- 96 Nicolas Bourbaki
VII The Influence of Mathematics
- 2 Mathematical Biology
  - 9 Phylogenetics and Graph Theory
- 4 The Mathematics of Traffic in Networks
  - 5 Flow Control in the Internet
- 5 The Mathematics of Algorithm Design
- 7 Mathematics and Cryptography
  - see also ReadingNotes.TheCodeBook
VIII Final Perspectives
- 6 Advice to a Young Mathematician
  - ~~III~~ Alain Connes
    - "Each generation builds a mental picture that reflects their own understanding of this world. They construct mental tools that penetrate more and more deeply into it, so that they can explore aspects of it that were previously hidden." (p1012)
      - see also Cognition.Cognition and Content.CognitiveDrag
      - see also Letters to a Young Mathematician by Ian Stewart, Basic Books 2006

my notes on
Princeton University Press Mathematics Site
The Princeton Companion to Mathematics according to Wikipedia
math resources in France
- IHES
- Mathematiques a l'Ecole Polytechnique
- Alain Connes Professor at the Collège de France, IHES and Vanderbilt University, his youTube account
Récoltes et Semailles by Alexander Grothendieck on "what it is to do math"
- IHES DailyMotion account with the Grothendieck colloque
MathematicsTopics of the Portal:Mathematics Wikipedia
number theory
- Pi trailer
- Number Theory Web online thesis page
  - NTW RSS feed
- Wikipedia Portal:Number theory
Mathematical Texts by the Grothendieck Circle
- previously recommended during a discussion on ReadingNotes.Hadamard
Institute for Advanced Study (IAS) School of Mathematics
http://en.wikipedia.org/wiki/Unifying_theories_in_mathematics
- http://en.wikipedia.org/wiki/Langlands_program
distribution of computation
- http://en.wikipedia.org/wiki/LU_decomposition
The Practice of Mathematics by Robert P. Langlands, Institute for Advanced Study 1999
- 16 lectures serie
Invalid techniques of proof, Faculty of Engineering, Computing and Mathematics at University of Western Australia
Quel statut et nature pour les mathématiques ?, Continent Science, France Culture January 2010
- http://en.wikipedia.org/wiki/Alexander_Grothendieck
- mentioning Hadamard's work at ~min34
- http://www.ihes.fr/~ruelle/Publications.html
Marcus du Sautoy: Symmetry, reality's riddle, TED October 2009
Images des mathématiques La recherche mathématiques en mots et en images
Egres Open open problem forum of the Egerváry Research Group
http://www.khanacademy.org/#Arithmetic

Tools

Mathematics Examples of Wolfram|Alpha
Sage Notebook create, collaborate on, and publish interactive worksheets. In a worksheet, one can write code using Sage, Python, and other software included in Sage.
PLUME, Thème : mathématiques logiciels utilisés dans les domaines de la recherche, de l’enseignement en mathématiques (pures comme appliquées), mais aussi ceux relevant de cette science (calcul scientifique, statistiques …) et pratiqués dans d’autres disciplines (physique, chimie, finance …).
Tricki store of useful mathematical problem-solving techniques.

Overall remarks and questions

main editor Timothy Gowers started the polymath project "using the comment functionality of his blog to produce mathematics collaboratively".

Synthesis

So in the end, it was about X and was based on Y.

Critics

Point A, B and C are debatable because of e, f and j.

Vocabulary

(:new_vocabulary_start:) new_word (:new_vocabulary_end:)

Post-reading model

Draw a schema (using PmGraphViz or another solution) of the situation of the area in the studied domain after having read the book. Link it to the pre-reading model and align the two to help easy comparison.

Fabien Benetou's PIM