1. Introduction

    1. Various approaches to classical TGD

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

    3. Topics to be discussed

  2. Some basic notions, ideas, results, and conjectures of algebraic geometry

    1. Algebraic varieties, curves and surfaces

    2. About algebraic curves and surfaces

    3. The notion of rational point and its generalization

  3. About enumerative algebraic geometry

    1. Some examples about enumerative algebraic geometry

    2. About methods of algebraic enumerative geometry

    3. Gromow-Witten invariants

    4. Riemann-Roch theorem

  4. Does M8-H duality allow to use the machinery of algebraic geometry?

    1. What does one really mean with M8-H duality?

    2. Is the associativity of tangent-/normal spaces really achieved?

    3. M8-H duality: objections and challenges

  5. 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

  6. 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

  7. 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

  8. From amplituhedron to associahedron

    1. Associahedrons and scattering amplitudes

    2. Associations and permutations in TGD framework

    3. Questions inspired by quantum associations

  9. 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?

  10. Could the precursors of perfectoids emerge in TGD?

    1. About motivations of Scholze

    2. Attempt to understand the notion of perfectoid

    3. Second attempt to understand the notions of perfectoid and its tilt

    4. TGD view about p-adic geometries

  11. Cognitive representations and algebraic geometry

    1. Cognitive representations as sets of generalized rational points

    2. Cognitive representations assuming M8-H duality

    3. Are the known extremals in H easily cognitively representable?

    4. Twistor lift and cognitive representations

    5. What does cognitive representability really mean?

  12. Galois groups and genes

    1. Could DNA sequence define an inclusion hierarchy of Galois extensions?

    2. Could one say anything about the Galois groups of DNA letters?

  13. 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

  14. Secret Link Uncovered Between Pure Math and Physics

    1. Connection with TGD and physics of cognition

    2. Connection with Kim's work

    3. Can one make Kim's idea about the role of symmetries more concrete in TGD framework?

  15. Are fundamental entities discrete or continuous and what discretization at fundamental level could mean?

    1. Is discretization fundamental or not?

    2. Can one make discretizations unique?

    3. Can discretization be performed without lattices?

    4. Simple extensions of rationals as codons of space-time genetic code

    5. Are octonionic polynomials enough or are also analytic functions needed?

  16. Summary and future prospects

  17. Appendix: o2 as a simple test case

    1. Option I: M4 is quaternionic

    2. Option II: M4 is co-quaternionic