1. Introduction

    1. Could one identify space-time surfaces as zero loci for octonionic polynomials with real coefficients?

    2. Topics to be discussed

  2. Some challenges of octonionic algebraic geometry

    1. Could free many-particle states as zero loci for real or imaginary parts for products of octonionic polynomials

    2. Questions related to ZEO and CDs

    3. About singularities of octonionic algebraic varieties

    4. The decomposition of space-time surface to Euclidian and Minkowskian regions in octonionic description

    5. About rational points of space-time surface

    6. About heff/h=n as the number of sheets of Galois covering

    7. Connection with infinite primes

  3. Super variant of octonionic algebraic geometry and space-time surfaces as correlates for fermionic states

    1. About emergence

    2. Does physics emerge from the notion of number field?

    3. About physical interpretation

  4. Could scattering amplitudes be computed in the octonionic framework?

    1. Could scattering amplitudes be computed at the level of M8-H?

    2. Interaction vertices for space-time surfaces with the same CD

    3. How could the space-time varieties associated with different CDs interact?

    4. Twistor Grassmannians and algebraic geometry

    5. About the concrete construction of twistor amplitudes

  5. From amplituhedron to associahedron

    1. Associahedrons and scattering amplitudes

    2. Associations and permutations in TGD framework

    3. Questions inspired by quantum associations

  6. Gromov-Witten invariants, Riemann-Roch theorem, and Atyiah-Singer index theorem from TGD point of view

    1. About the analogs of Gromow-Witten invariants and branes in TGD

    2. Does Riemann-Roch theorem have applications to TGD?

    3. Could the TGD variant of Atyiah-Singer index theorem be useful in TGD?

  7. Intersection form for 4-manifolds, knots and 2-knots, smooth exotics, and TGD

    1. Basic ideas

    2. Intersection form in the case of 4-surfaces

    3. About ordinary knots

    4. What about 2-knots and their cobordisms?

    5. Could the existence of exotic smooth structures pose problems for TGD?

  8. A possible connection with family replication phenomenon?

    1. How the homology charge and genus correlate?

    2. Euler characteristic and genus for the covering of partonic 2-surface

    3. All genera are not representable as non-singular algebraic curves

  9. Summary and future prospects