The following is a comment from FB discussion. Since the answer developed to a summary of the mathematics needed in TGD, I decided to make it blog post.
I think that TGD is a "problem" for anyone in the sense that it is very difficult to get graps of what it really is. The reason is that TGD have been silenced for 4 decades - censorship in archives started around 93 or so and has had fatal consequences. The idea about the hegemony of M-theory estabilished by censorship was terribly silly.
TGD is an outcome of concentrated effort lasted more than 4 decades and involves 24 books. I guess that the minimum time to get some perspective to TGD is one year. For instance, I have worked with twistors for about 10 years and gradually begin to really understand the twistorialization of TGD. I am skeptic about communication of TGD without a long series of lectures and personal face to face discussions. This has not been possible, and now also my age poses strong limitations.
Concerning math, I am not interested in technical things such as producing a new p-adic variant of self-interacting phi4 scalar field theory. Usually mathematically oriented people do this kind of things in lack of wider physics perspective. I have a big developing a vision binding physics, consciousness, and biology, and I want to identify and even develop the mathematics needed if needed.
- The math involves sub-manifold geometry intended to geometrize field theory by replacing field patterns with 4-surfaces using generalization of the induction procedure for metric, spinor connection, and spinors (also twistors). Kaehler geometry is in central role. Induction is something new for physics, but well-known for mathematicians.
- The great vision involves geometrization of quantum theory in terms of infinite-D Kaehler geometry for the world of "classical world". Wheeler with his superspace was pioneer in general relativity and loop spaces are predecessors in string theory. Infinite-D symmetries as generalization of superstringy conformal symmetries are in pivotal role: they guarantee the existence of infinite-D Kaehler geomery.
- Number theoretic vision is complementary to geometric vision and number theory including extensions of rationals, p-adic number fields and their extensions induced by extensions of rationals and classical number fields. Extending the physics to a description of also cognition is the great vision and brings in p-adic number fields, adeles, and predicts hierarchy of Planck constants characterizing dark matter in TGD sense. Cognitive representations are fundamental notion and provide a unique number theoretical discretization of classical and quantum physics. One powerful implication is discretization of coupling constant evolution forced solely by number theoretical universality.
- The generalization of twistors to 8-D context to solve basic problem of standard twistor approach (only massless particles are allowed) is part of the vision. A new element is the replacement of space-time surfaces with their twistor spaces represented as 6-surfaces in 12-D twistor space of imbedding space. One powerful prediction is that M4×CP_2 implied by standard model is the only possible choice for the imbedding space besides M8 which is equivalent dual choice: the reason is that only the spaces M4, E4 and CP2 allow twistor space with Kaehler structure. TGD is unique from its existence.
- I have considered also categories occasionally. I tend to see them as tools to organize and excellent for bureaucratic challenges: they do not seem to code for core ideas. As a physicist I see interpretations of quantum theory as as unsuccessful attempts to get rid of the basic problem of the measurement theory: here zero energy ontology (ZEO) provides the great vision and leads to a theory of consciousness and quantum biology.
The great idea is terribly simple: transform initial value problem to boundary value problem by replacing initial values at t= constant hyperplane with boundary values characterizing deterministic time evolution t1→>t2 and by replacing quantum states with superpositions of deterministic time evolutions.
Those interested in material about TGD can find links from Material about TGD.
For a summary of earlier postings see Latest progress in TGD.