Wednesday, October 15, 2014

Notes on Infinite-Dimensional Projective Algebra

I have no math/phys verifications for following speculations
I have no working knowledge so it might be apart from standard concepts
  • Suppose a huge category Univ, which is category of category of sets, Set
  • and its relation to projections of variations/manifolds, Proj
  • axiom of regularity (foundation) and large cardinal axioms assure limit and inverse limit of Univ in Set
  • they could be related to quantum field theory: regularity assures existence of vacuum (ultraviolet cutoff), large cardinals assures existence of universe (infrared cutoff)
  • Proj might be also found in quantum measurement: lattice of related observations
  • Proj resembles to Univ but generally lacks limits
  • Algebra of integrals forms Proj
  • Univ is a sort of compactification of Proj, but how we compactificate Proj?
  • operator ring theory must have distinction between two ways of compactification, one-point and Stone-Cech
  • section of infinity on circumstance, or hyperbolization of ellipse, make infinity to integrable, by counting leaves of the tree
  • AdS-CFT might be thought as an example of general hyperbolization
  • Can we perform hyperbolization simultaneously in both sides, at UV and IR?
  • regarding any automorphic forms as "ellipses", what does it mean, existence of their (discrete) hyperbolizations?

Saturday, October 11, 2014

Why Stability Matters?

Why Stability Matters: Interrelations between geom/discrete math/HEP/logic

NOTICES
Model Theory and Geometry
  • One of main motivations toward model theory is geometry
    • Source of imagination: Bernhard Riemann's proof on independence of Euclid's fifth postulate (parallel postulate)
    • (added) David Hilbert: the fourth problem and "tables, chairs and beer mugs" axiomatics of geometry
    • Oswald Veblen originated categoricity of axiomatics
    • Alfred Tarski explored decidable theories and their conservative extension
    • Dana Scott: Convergent Sequences of Complete Theories (Prototype of his domain theory)
    Mathematizing Kant's Transcendental Category
    • A disciple of Saunders MacLane - Michael Morley: Categoricity in Power (He later joined Tarski school at UCB, under Vaught)
    • From Loewenheim-Skolem, you cannot determine cardinality of infinite domain, but you can say categoricity under given cardinality
    • Stability of categoricity: Saharon Shelah's efforts
    • In the same period of Morley's above work, MacLane-Eilenberg theory of natural transformations turned into "category theory"
    • William Lawvere, who related Quantifies and Shearves, was a disciple of Samuel Eilenberg, had "workshop" of mathematical logic at UCB Tarski school
    Variational Principle and its relations to disciplines other than analysis
    • PDE, Calculus of Variations in the Large, Harmonic Analysis
    • giving global picture of Any potential distributions: which require infinite dimensions (non-zero energy in infrared), or, so to say, quantum dynamics
    • At least two millennium problems are related: Yang-Mills and Navier-Stokes
    • elliptization of parabolic PDE into infinte system of ODEs
    • soliton characterization by Fermi-Pasta-Ulam and Zabusky-Kruskal
    • Faddeev-Popov ghost
    • In game theory, translation of n-person non-zerosum game to n+1-person zerosum game (extra player, like "natural resource", behave irrationally): In TGEB, von Neumann-Moregenstern gave a suggestion about relation to Morse theory
    • In economics, explaining demands in classical general equilibrium with term structure of finance/dynamic macroeconomics
    Conservation and Stability
    • Stability in Model Theory
    • which means degree of "power division" of time (real time)
    • imaginary time, with conformal invariance, is algebraically determined, but no determinacy with real time: related to validity of path integral
    • residue of energy, in order of log(t) and not absorbable to power structure (orbit), plays main role in non-linear phenomena
    • In engineering, metrics in shorter time make precise control possible and which make systems more stable
    • commensurability of orbits in planetary system
    • ergodic theory
    • basis of fourier analysis
    • lattice structure of natural number >0 represented in product of primes (Dedekind)
    • stability of economic equilibrium
    Kakutani's contribution to understanding of stability
    • I'm asking Shizuo Kakutani's significance to Princeton guys in 1940s
    • origin of Kakutani's FPT and metamathematical counterpart of it
    So What?