Introduction
Could one identify space-time surfaces as zero loci for octonionic polynomials with real coefficients?
Topics to be discussed
Some challenges of octonionic algebraic geometry
Could free many-particle states as zero loci for real or imaginary parts for products of octonionic polynomials
Questions related to ZEO and CDs
About singularities of octonionic algebraic varieties
The decomposition of space-time surface to Euclidian and Minkowskian regions in octonionic description
About rational points of space-time surface
About h_{eff}/h=n as the number of sheets of Galois covering
Connection with infinite primes
Super variant of octonionic algebraic geometry and space-time surfaces as correlates for fermionic states
About emergence
Does physics emerge from the notion of number field?
About physical interpretation
Could scattering amplitudes be computed in the octonionic framework?
Could scattering amplitudes be computed at the level of M^{8}-H?
Interaction vertices for space-time surfaces with the same CD
How could the space-time varieties associated with different CDs interact?
Twistor Grassmannians and algebraic geometry
About the concrete construction of twistor amplitudes
From amplituhedron to associahedron
Associahedrons and scattering amplitudes
Associations and permutations in TGD framework
Questions inspired by quantum associations
Gromov-Witten invariants, Riemann-Roch theorem, and Atyiah-Singer index theorem from TGD point of view
About the analogs of Gromow-Witten invariants and branes in TGD
Does Riemann-Roch theorem have applications to TGD?
Could the TGD variant of Atyiah-Singer index theorem be useful in TGD?
A possible connection with family replication phenomenon?
How the homology charge and genus correlate?
Euler characteristic and genus for the covering of partonic 2-surface
All genera are not representable as non-singular algebraic curves
Summary and future prospects