In the following I give a brief summary about what has been done. I concentrate on M^{8}H duality since the most significant results are achieved here.
It is fair to say that the new view answers the following a long list of open questions.
 When M^{8}H correspondence is true (to be honest, this question emerged during this work!)? What are the explicit formulas expressing associativity of the tangent space or normal space of the 4surface?
The key element is the formulation in terms of complexified M^{8} identified in terms of octonions and restriction to M^{8}. One loses the number field property but for polynomials ring property is enough. The level surfaces for real and imaginary parts of octonionic polynomials with real coefficients define 4D surfaces in the generic case.
Associativity condition is an additional condition reducing the dimension of the spacetime surface unless some components of RE(P) or IM(P) are critical meaning that also their gradients vanish. This conforms with the quantum criticality of TGD and provides a concrete first principle realization for it.
 How this picture corresponds to twistor lift? The twistor lift of Kähler action (dimensionally reduced Kähler action in twistor space of spacetime surface) one obtains two kinds of spacetime regions. The regions, which are minimal surfaces and obey dynamics having no dependence on coupling constants, correspond naturally to the critical regions in M^{8} and H.
There are also regions in which one does not have extremal property for both Kähler action and volume term and the dynamics depends on coupling constant at the level of H. These regions are associative only at their 3D ends at boundaries of CD and at partonic orbits, and the associativity conditions at these 3surfaces force the initial values to satisfy the conditions guaranteeing preferred extremal property. The nonassociative spacetime regions are assigned with the interiors of CDs. . The particle orbit like spacetime surfaces entering to CD are critical and correspond to external particles.
 The surprise was that M^{4}⊂ M^{8} is naturally coassociative. If associativity holds true also at the level of H, M^{4} ⊂ H must be associative. This is possible if M^{8}H duality maps tangent space in M^{8} to normal space in H and vice versa.
 The connection to the realization of the preferred extremal property in terms of gauge conditions of subalgebra of SSA is highly suggestive. Octonionic polynomials critical at the boundaries of spacetime surfaces would determine by M^{8}H correspondence the solution to the gauge conditions and thus initial values and by holography the spacetime surfaces in H.
 A beautiful connection between algebraic geometry and particle physics emerges. Free manyparticle states as disjoint critical 4surfaces can be described by products of corresponding polynomials satisfying criticality conditions.
These particles enter into CD , and the nonassociative and noncritical portions of the spacetime surface inside CD describe the interactions. One can define the notion of interaction polynomial as a term added to the product
of polynomials. It can vanish at the boundary of CD and forces the 4surface to be connected inside CD. It also spoils associativity: interactions are switched on. For bound states the coefficients of interaction polynomial are such that one obtains a bound state as associative spacetime surface.
 This picture generalizes to the level of quaternions. One can speak about 2surfaces of spacetime surface
with commutative or cocommutative tangent space. Also these 2surfaces would be critical. In the generic case commutativity/cocommutativity allows only 1D curves.
At partonic orbits defining boundaries between Minkowskian and Euclidian spacetime regions inside CD the string world sheets degenerate to the 1D orbits of point like particles at their boundaries. This conforms with the twistorial description of scattering amplitudes in terms of point like fermions.
For critical spacetime surfaces representing incoming states string world sheets are possible as commutative/cocommutative surfaces (as also partonic 2surfaces) and serve as correlates for (long range) entaglement) assignable also to macroscopically quantum coherent system (h_{eff}/h=n hierarchy implied by adelic physics).
 The octonionic polynomials with real coefficients form a commutative and associative algebra allowing besides algebraic operations function composition. Spacetime surfaces therefore form an algebra and WCW has algebra structure. This could be true for the entire hierarchy of CayleyDickson algebras, and one would have a highly nontrivial generalization of the conformal invariance and CauchyRiemann conditions to their nlinear counterparts at the n:th level of hierarchy with n=1,2,3,.. for complex numbers, quaternions, octonions,... One can even wonder whether TGD generalizes to this entire hierarchy!
What mathematical challenges one must meet?
 One should prove more rigorously that criticality is possible without the reduction of dimension of the spacetime surface.
 One must demonstrate that SSA conditions can be true for the images of the associative regions (with 3D or 4D). This would obviously pose strong conditions on the values of coupling constants at the level of H.
What questions should be answered?
 Does associativity hold true in H for minimal surface extremals obeying universal critical dynamics? As found, the study of the known extremals supports this view.
 Could one construct the scattering amplitudes at the level of M^{8}? Here the possible problems are caused by the exponents of action (Kähler action and volume term) at H side. Twistorial construction however leads to a proposal that the exponents actually cancel. This happens if the scattering amplitude can be thought as an analog of Gaussian path integral around single extremum of action and conforms with the integrability of the theory. In fact, nothing prevents from defining zero energy states in this manner! If this holds true then it might be possible to construct scattering amplitudes at the level of M^{8}.
 What about coupling constants? Coupling constants make themselves visible at H side both via the vanishing conditions for Noether charges in subalgebra of SSA and via the values of the nonvanishing Noether charges. M^{8}H correspondence determining the 3D boundaries of interaction regions within CDs suggests that these couplings must emerge from the level M^{8} via the criticality conditions posing conditions on the coefficients of the octonionic polynomials coding for interactions.
Could all coupling constant emerge from the criticality conditions at the level of M^{8}? The ratio of R^{2}/l_{P}^{2} of CP_{2} scale and Planck length appears at H level. Also this parameter should emerge from M^{8}H correspondence and thus from criticality at M^{8} level. Physics would reduce to a generalization of the catastrophe theory of Rene Thom!
 There are questions related to ZEO. Is the notion of CD general enough or should one generalize it to the analog of the polygonal diagram with lightlike geodesic lines as its edges appearing in the twistor Grassmannian approach to scattering diagrams? Octonionic approach gives naturally the lightlike boundaries assignable to CDs but leaves open the question whether more complex structures with lightlike boundaries are possible. How the spacetime surfaces associated with different quaternionic structures of M^{8} and with different positions of tips of CD interact?
 Real analyticity requires that the octonionic polynomials have real coefficients. This forces the origin of octonionic coordinates to be at real line (time axis) in the octonionic sense. All CDs cannot be located along this line. How do the varieties associated with octonionic polynomials with different origins interact? The polynomials with different origins neither commute nor are associative. How could one avoid losing the extremely beautiful associative and commutative algebra? It seems that one cannot form their products and sums and must form the Cartesian product of M^{8}:s with different origins and formulate the interaction in this framework. Could CayleyDickson hierarchy be necessary to describe the interactions between different CDs (note however that the dimension are powers of 2) than multiples of 8?
Is the interaction nr welldefined only at the level of H inside CD to which these 4D varieties arrive through the boundary of this CD? All CDs, whose tips are along lightlike ray of CD boundary, share this ray. There is a common M^{2}_{0} shared by these CDs. Could M^{2}_{0} make possible the interaction. The CDs able to interact with given CD would have tips at the 3D boundary of this CD and share common M^{2}_{0}. These M^{2}_{0}:s are labelled by the points of twistor sphere so that twistoriality seems to enter into the game in nontrivial manner also at the level of M^{8}!
This however allows interactions only between varieties with same M^{4}_{0}. What about a more general picture in which unit octonions define 6D sphere S^{6} of directions of 8D lightrays and parameterize different quaternionic structures with fixed M^{2}_{0}. Could the condition for interaction be that S^{6} coordinates are same so that M^{2}_{0} is shared. In this case light rays in 7D lightcone would parameterize the allowed origins for octonionic polynomials for which the interaction of zero loci is possible. The images of these lightrays in H would be more complex. This could allow varieties with different M^{4}_{0} but common M^{2}_{0} to interact via the common light ray/M^{2}_{0}. Somewhat similar picture involving preferred M^{2}_{0} for given connected part of twistor graph emerges from the construction of twistor amplitudes.
What would be the interaction at the intersection? Lightlike ray naturally defines a string like object with fermions at its ends. Could this fermionic string be transferred between spacetime surfaces in the intersecting CDs. Also a branching of a string like object between spacetime varieties in different CDs intersecting along this ray would be possible. This would describe stringy reaction vertex with incoming strings in different CDs. It must be admitted that this unavoidably brings in mind branes and all the nasty things that I have said about them during years!
Or could the strange singularity behavior in Minkowski signature play a role in the interactions? In the generic situation the intersection consists of discrete points but as the study of o^{2} shows the surface RE(o^{2})=0 and IM(o^{2})=0 can have dimensions 5 and 6 and their intersection naively expected to consist of discrete points can be interior or exterior of lightcone. Could the zero loci of singular polynomials play a key role in the interaction and allow 4varieties in M^{8} to interact by being glued to this higher than 4D objects. 4D spacetime variety could have 2D intersection with 6D variety and the 6D variety could allow the interactions to between two 4D varieties by correlating them. Also this strongly brings in mind branes but looks less elegant that above proposal.
 What is the connection with Yangian symmetry, whose generalization in TGD framework is highly suggestive?
See the chapter Does M^{8}H duality reduce classical TGD to octonionic algebraic geometry? or the articles Does M^{8}H duality reduce classical TGD to octonionic algebraic geometry?: part I and Does M^{8}H duality reduce classical TGD to octonionic algebraic geometry?: part II.
