This is a contributed topic on the mathematical structures associated with physical theories. Hopefully, this topic will be of interest to both physicists and mathematicians (as well as logicians and philosopy of science researchers).

1 Mathematical and Logical Foundations of Physics

1.1 Physical Theories

• Classical, or Newtonian, mechanics: Kinematics and Dynamics, Netwon’s Laws of motion
• Classical molecular dynamics
• Classical Thermodynamics and Statistics
• Classical Electromagnetism: Maxwell’s theory and Maxwell’s equations
• Optics and Spectroscopic Theories
• General and special relativity
• quantum mechanics
• quantum field theories
• Mathematical Foundations of Quantum Mechanics
• Axiomatic Quantum Field Theories
• Local Quantum Physics theories
• Statistical Quantum Mechanics
• Standard Model (SUSY)
• Electroweak Theory
• Quantum Chromodynamics
• Quantum Gravity theories
• Strings and Brane theories
• Astrophysics theories
• Chaotic Dynamics and Complex systems theories
• XYZ

1.2 Mathematical structures and fields underlying physical theories

• Calculus and Analytical Geometry
• non-Newtonian calculus
• vector Calculus
• Functional Analysis
• Differential Geometry
• Algebra, matrices and tensors
• category theory
• algebraic topology and non-Abelian algebraic Theory
• noncommutative geometry
• Algebraic Geometry
• Harmonic and anharmonic Analysis
• Numerical Analysis
• Statistics and Probability Theories
• Model Theory
• Superstructures: supergroups, superalgebras, algebraic supervarieties, super-categories
• XYZ

1.3 Logical structure of physical theories

• Axiomatics and Physical Postulates
• quantum logics
• automata theory and Boolean logic designs
• quantum automata and quantum computers
• many-valued logic Algebra