1. Modeling of interactions between physics and mathematics
Arash Rastegar
Department of Mathematical Sciences
Sharif University of Technology

2. Moving from mathematics to physics
When mathematicians do physics.
Classical Mechanics
Relativity
Quantum mechanics
String theory
Supper symmetry

3. Moving from physics to mathematics
When physicists do mathematics.
Differential equations
Infinite dimensional group representations
Operator theory
Algebraic topology
Algebras and operads
Deformation of metric
Category theory

4. Mathematics imitates physics
When mathematics is developed similar to physical theories
Energy integral
Mathematical expectation
Classical mechanics over fiber bundles
Deformation quantization
Geometric quantization

5. Physics imitates mathematics
When physics is developed similar to mathematical theories
Quantum theory imitates operator theory
Relativity imitates deformation of metric
Gauge theory imitates fiber bundles
Quantum field theory imitates cobordism theory
Conformal field theory imitates algebras over operads

6. Physics invades mathematics
When parts of mathematics are considered as parts of physics
Parts of the theory of differential equations
Deformation quantization
Quantum groups
Geometric quantization
Lorenz spaces and space-time in general
Super symmetry

7. Mathematics invades physics
When parts of physics are considered as parts of mathematics
Classical Mechanics
Quantum theory
Solution of Einstein equations
Solution of wave equations
Super mathematics

8. Mathematical paradigms in physics
Differentiation
Integration
Operators
Algebras over operads
Spaces
Singularity

9. Physical paradigms in mathematics
Length
Area
Volume
Center of gravity
Energy
Stability
Harmonicity

10. Making assumptions in physics
What are important assumptions in physics?
What is the role of mathematics in them?
Assumptions on space
Assumptions on space-time
Assumptions on observers
Assumptions on observables
Assumptions on measurement

11. Making assumptions in mathematics
What are important assumptions in mathematics?
What is the role of physics in them?
Definition of spaces
Assumptions on spaces
Assumptions on Hilbert spaces
Structures on algebras
Assumptions on group representations

12. Theorization in physics
What is a good theory in physics?
Compatibility with related paradigms
Compatibility with experiments
Compatibility with atlas of concepts
Computability
Disprovability
Relations with measurements
Simplicity

13. Theorization in mathematics
What is a good theory in mathematics?
Computability
Illumination
Relations with other theories
Insightfulness
Generalizability
Existence of a toy theory
Simplicity

14. Problem solving in physics
What are problem solving habits in physics?
Patience
Divergent thinking
Criticizing conjectures
Looking for equivalent formulations
Fluency in working with ideas and concepts
Looking for simpler models

15. Problem solving in mathematics
What are problem solving habits in mathematics?
Tasting the problem
Gaining personal view towards the problem
Considering relations with similar theories
Considering generalizations
Checking special cases
Performing a few steps computationally
Thinking simple

16. Teaching habits of physicists
Drawing formulas
Geometric imagination
Recognizing simple from difficult
Decomposition and reduction to simpler problems
Jumps of the mind
Estimating how much progress has been made
Finding the trivial propositions quickly
Formulating good conjectures
Being creative and directed in constructions
Understanding an idea independent of the context
Imagination and intuition come before arguments & computations

17. Teaching habits of mathematicians
Deciding where to start
Listing different strategies to attack the problem
Mathematical modeling in different frameworks
Deciding about using symbols or avoiding symbols
Deciding about what not to think about
Organizing the process of coming to a solution
How to put down the proof
Clarifying the logical structure
Writing down side results
Putting down the full proof after finishing the arguments
Notifying important steps in form of lemmas
Considering the mind of reader

18. Learning habits of physicists
Considering concept relations
Comparing similar assumptions for theorization
Comparing the related paradigms
Criticizing conjectures
Looking for equivalent formulations
Fluency in working with ideas and concepts
Looking for simpler models

19. Learning habits of mathematicians
What are logical implications
Considering relations with similar theories
Considering generalizations
Checking special cases
Performing a few steps computationally
Thinking simple
Solving problems

20. What do physicists care about?
Atlas of concepts
Paradigms of physical theories
Computability
Disprovability
Measurements
Intuition

21. What do mathematicians care about?
Understanding simple cases completely
Understanding the ways one finds generalizations
Understanding the logical structure
Finding the exact assumptions needed
How to solve similar problems
Flexibility of techniques

22. Contributions of physicists to mathematics
Introduction of new mathematical concepts
Motivating computations
Motivating theorizations
Motivating assumptions
Motivating conjectures in mathematics
Fields moving from physics to mathematics
Introducing important special cases of theories

23. Contributions of mathematicians to physics
Providing mathematical formulations
Mathematical rigor
Categorizing exceptions
Mathematical computations
Expansion of the realm of physics
Study of the generalizations
Criticism
Guidance

24. Contributions of mathematics to physicists
Rigor
Postulation
Logical structure
Problem solving
Critical thinking
Abstraction
Categorization

25. Contributions of physics to mathematicians
Applications
Philosophical thought
Intuition
Computation
Divergent thinking
Searching for the truth
Relations with nature

26. Physics and mathematics form a pair
Seems that mathematics is mother and physics is father!!!!!! It should have been vice versa
What are pairs inside them?
Quantum theory and Hilbert spaces
Relativity and space-time
Classical mechanics and manifolds
Dynamical systems and differential equations

27. Mathematics is the father
What is the role of father?
Provides ideas and intuitions.
Management of relations with other theories.
Provides the global structure.
Determines how to generalize.
Furnishes the soul.

28. Physics is the mother What is the role of mother?
Provides appropriate formulation and language.
Management of internal relations between sub-theories.
Provides the local structures.
Determines how to solve problems.
Furnishes the body.

29. Mathematical physics is the fruit of this marriage
What are similarities with mathematics?
Based on geometric ideas and intuitions.
Mathematics provides the global structure and determines relations with other theories.
What are similarities with physics?
Based on physical formulation and language.
Management of internal relations between sub-theories according to physical intuition.
Physics determines methods of solving problems.

30. A word on personality of mathematical-physicists
Mathematical-physicists have double standards:
Mathematical standards criticize geometric ideas and intuitions.
Management of relations with other theories is according to mathematical standards of theorization.
Mathematical intuition provides the global structure and determines how to generalize.
Physical standards provide appropriate formulation and language and manages internal relations between sub-theories.
Physical standards provides the local structure and determines how to solve problems.