Mathematics {Content}

Foundations and metamathematics

• FOM - Foundations of Mathematics mailing list with its archives
• Logicomix
• Wikipedia:Metamathematics
• New Trends in Logic, Conference honoring the Winners of the Kurt Gödel Research Prize Fellowships 2008 and 2011, Austrian Academy of Sciences April 2011
  • http://videolectures.net/godelfellowship2011_vienna/
• consider On proof and progress in mathematics by William P. Thurston, 1994
  • recommended by franki
• Wikipedia:Constructivism (mathematics)
• The QED Project single, distributed, computerized repository that rigorously represents all important, established mathematical knowledge.
• Wikipedia:Proof complexity

Complexity theory

• Center for Computational Intractability
• Wikipedia:Computational complexity theory
• ECCC discovered via WithoutNotesAugust11#ScottAaronson
• http://qwiki.stanford.edu/index.php/Complexity_Zoo discovered via WithoutNotesJune11#RussellImpagliazzoIAS
• LePouvoirDesMathematiques#LArbreDeLaComplexite

Also important for Foundations and metamathematics

Graph theory

• Wikipedia:Hypergraph
• Wikipedia:Topology, Wikipedia:Homology (mathematics), Wikipedia:Homotopy
• important link with combinatorics thus complexity

In particular motivated by Programming#Graph, Internet routing and Programming#HardwareStack for circuit design, in particular because of the democratization of Electronics#FPGA but also to clarify Mind (according to Cognition#ExocognitionAsNetworkTraversal) understanding how a graph via solely its activation can produce meaning, update its topology, etc... and this via software models e.g. WithoutNotesJuly11#GraphsBrainsAndGremlin.

Statistics

• Wikipedia:Ronald Fisher
• "I keep saying that the sexy job in the next ten years will be statisticians" by Hal Varian, co-author of Information Rules
• Stack Exchange Statistical Analysis Q&A for statisticians, data analysts, data miners and data visualization experts
• The strength of evidence versus the power of belief: Are we all Bayesians? by Jessica Utts, ICOTS 2010
• R free software environment for statistical computing and graphics.
• to explore
  • manifold learning, e.g. Manifold Learning and its Applications, AAAI 2010 Fall Symposium
  • techniques used in Urban Scaling and Its Deviations: Revealing the Structure of Wealth, Innovation and Crime across Cities, PLoS ONE 2010
    • read in WithoutNotesJanuary11

Probability distributions

• Wikipedia:List of probability distributions
  • Custom Distribution Plushie by NausicaaDistribution on Etsy
• Wikipedia:Kullback–Leibler divergence (|| notation)
  • specific Wikipedia:f-divergence but Wikipedia:Metric (mathematics)#Premetrics
  • discovered in WithoutNotesMay11#IITManifesto and required to estimate Φ
  • check if used in WithoutNotesSeptember10#TDlearning
  • note that it mentions Renyi entropy, related to low Rényi entropy required in the end of Leonid A. Levin's paper WithoutNotesMay11#LeonidLevin
  • recently added to Cognition#Thinking
  • is it also related to simulated annealing used in quantum computing?

Applications

• finance
  • FinancialTools#API_statistical_analysis
• machine learning
  • mloss machine learning open source software
  • MLcomp free website for objectively comparing machine learning programs across various datasets for multiple problem domains.
  • Machine Learning Demos (aka MLDemos) by Basilio Noris, Learning Algorithms and Systems Laboratory at Ecole Polytechnique Fédérale de Lausanne (EPFL)
  • Optimization Algorithm Toolkit (OAT)
  • http://videolectures.net/nips2010_meng_mlhi/
  • Waffles command-line tools for researchers in machine learning, data mining, and related fields.
  • cheat sheets
    • Theoretical CS http://www.tug.org/texshowcase/cheat.pdf
    • probability and statistics http://www.cs.berkeley.edu/~mavam/dl/probstat.pdf
    • introductory machine learning http://homepages.inf.ed.ac.uk/imurray2/pub/cribsheet.pdf
    • Mathematical Foundations for the Information Age http://www.cs.cornell.edu/courses/cs4850/2010sp/Useful_Formulas.pdf
    • HMM http://www.bcl.hamilton.ie/~barak/misc/hmm.pdf

Number theory

One-way function

• motivated by Needs#ComplexityOfInverseFunction
  • inspired by MardiInnovation07
  • What's the time complexity for inverse function?, Yahoo! Answers, but only gives an example, not a general theorem
  • Wikipedia:Computational complexity of mathematical operations also mentions that "The complexity of an elementary function is equivalent to that of its inverse, since all elementary functions are analytic and hence invertible by means of Newton's method." but again, is not very general
  • Wikipedia:One-way function (ou "Fonction à sens unique" in french) is related but seems to be an open-conjecture and does not seem to provide a way to automatically qualify a function as such
  • http://www.cs.columbia.edu/~zeph/4261/lectures/one-way-functions.pdf
  • material to explore
    • Complexity-Theoretic One-Way Functions, Cryptography, and Pseudorandom Generators at UR-CS
      • possibly related One-Way Functions in Complexity Theory (1990) (written at Northeastern University, like Barabasi lab), An observation on associative one-way functions in complexity theory (1997) and Computability and Complexity Theory (2001)
    • Luca Trevisan's papers in general but in particular On Hardness Amplification of One-Way Functions (2005)
    • http://www.cs.cornell.edu/courses/cs6830/2009fa/scribes/lecture5.pdf
    • http://cstheory.stackexchange.com/questions/tagged/one-way-function
    • Unsolved Problems in Mathematical Systems and Control Theory edited by V.D. Blondel and A. Megretski, Princeton University Press 2004
    • On a Family of Minimal Candidate One-way Functions and One-way Permutations (2009)
    • Adaptive One-way Functions and Applications (2008)
    • Symmetry of information and one-way functions (1991)
    • Towards Non-Black-Box Lower Bounds in Cryptography (2011)
    • Constant-round Non-malleable Commitments from Any One-way Function (2011)
• Wikipedia:Bremermann's limit
• The On-Line Encyclopedia of Integer Sequences™ (OEIS™)

Applications

• mostly cryptography, also computer science (hash tables)

Geometry

• Wiki.Visualization#hyperbolicmap
• Seedea:Content/Newconcepts#InformationGeometry
  • potential conceptual framework for manifold learning

Visualization

• Gnuplot

Tools

• Programming#R
• Octave high-level language, primarily intended for numerical computations.
  • Web interface to Octave, University of Warsaw
• Numpy, part of SciPy Scientific Computing Tools For Python
• L^AT_EX A document preparation system
• Sage Open Source Mathematics Software
  • The Sage Notebook create, collaborate on, and publish interactive worksheets.
• man:bc
• Wikipedia:Abacus#Tutorials
  • used in Langrolay-sur-Rance
• Programming#FormalMethods

History

• birth of geometry for accounting
• http://www.und.edu/instruct/lgeller/algebra.html
• Wikipedia:Seven Bridges of Königsberg
• Logicomix
• Wikipedia:Hilbert's problems

Remarks

• the notion of "distance" seems to be fundamental in the topics Im most interested in
  • Wikipedia:Computational phylogenetics
    • distances in Inferring Phylogenies Sinauer Associates 2004
  • Wikipedia:Information geometry

See also

• ComplexSystems
• Programming
• read books
  • LePouvoirDesMathematiques
  • Hadamard
  • PCM
  • FromGeometryToTopology
• Computational Information Geometry by Frank Nielsen
• Images des mathématiques, Centre National de la Recherche Scientifique (CNRS)
• http://www.polynomiography.com
• Harvard Mathematics Department
  • Mathematics Courses
• MoMath The Museum of Mathematics
• BetterExplained, Explanations for everyone, by Kalid Azadgi
• Khan Academy
  • e.g. Linear Algebra: Introduction to Eigenvalues and Eigenvectors
• http://www.quora.com/Which-is-the-best-method-for-Mathematics-self-study-up-to-doctorate-level
• We Use Math - Introduction (from http://www.WeUseMath.com )
• CS 4850 Mathematical Foundations for the Information Age at Cornell
  • see also WithoutNotesMay11#JohnHopcroft
• Kill Math by Bret Victor, April 2011
• Patrick JMT (Just Math Tutorials) making FREE and hopefully useful math videos for the world!
  • with Thinkwell
• Newton Institute Seminars Online
• http://www.ProofWiki.org
• SearchOnMath find web pages containing, to various degrees of similarity, a given mathematical formula. It's a search engine for the direct lookup of mathematical content.
• http://www.TouchMathematics.org
• https://github.com/Jam3/math-as-code