1. Introduction

    1. What could be the deeper mathematics behind dualities?

    2. Correspondence along common rationals and canonical identification: two manners to relate real and p-adic physics

    3. Brief summary of the general vision

  2. Quantum arithmetics and the notion of commutative quantum group

    1. Quantum arithmetics

    2. Canonical identification for quantum rationals and symmetries

    3. More about the non-uniquencess of the correspondence between p-adic integers and their quantum counterparts

    4. The three options for quantum p-adics

  3. Do commutative quantum counterparts of Lie groups exist?

    1. Quantum counterparts of special linear groups

    2. Do classical Lie groups allow quantum counterparts?

    3. Questions

    4. Quantum p-adic deformations of space-time surfaces as a representation of finite measurement resolution?

  4. Could one understand p-adic length scale hypothesis number theoretically?

    1. Number theoretical evolution as a selector of the fittest p-adic primes?

    2. Only Option I is considered

    3. Orthogonality conditions for SO(3)

    4. Orthogonality conditions for SO(3)

    5. Number theoretic functions rk(n) for k=2,4,6

    6. What can one say about the behavior of r3(n)?

  5. How quantum arithmetics affects basic TGD and TGD inspired view about life and consciousness?

    1. What happens to p-adic mass calculations and quantum TGD?

    2. What happens to TGD inspired theory of consciousness and quantum biology?

  6. Appendix: Some number theoretical functions

    1. Characters

    2. Divisor functions

    3. Class number function and Dirichlet L-function