Is it really possible to formulate all geometric statements as statements of algebraic geometry?
The TGD view of the geometric Langlands correspondence states that there is a correspondence between the algebraic, essentially linguistic view of physics and the geometric view of physics relying on vision. This leads to a kind of language game. The highly non-trivial challenge is to find whether the geometric picture can be formulated using the language of algebraic geometry involving generalized complex variables of which one is hypercomplex and real.
First of all, one must find out whether the known algebraically universal extremals appearing for practically any conceivable action, deduced by geometric and symmetry arguments, have a simple algebraic description as the roots (P,Q)=(0,0) where P and Q are analytic functions of generalized complex coordinates of H=M4× CP2. This is not at all obvious. One should carefully check whether CP2 type extremals, cosmic strings and monopole flux tubes, and massless extremals allow this kind of formulation.
Inequalities are part of geometric description and involve in an essential manner the notion of distance. The representation of topological boundaries gives rise to inequalities. In TGD a long standing question is whether one should allow boundaries and whether the boundary conditions guaranteeing conservation laws indeed allow space-time boundaries. For instance, could one eliminate CP2 type extremals defining wormhole contacts glued to the Minkowskian background and leaving partonic orbits as boundaries (see this).
- The problem is that well-ordering required by inequalities characterizes only real numbers: the notion of inequality is not algebraically universal. Inequalities have no natural place in pure algebraic geometry involving complex numbers or p-adic numbers. In TGD, the natural variables are generalized complex coordinates and inequalities cannot be represented for the complex numbers using only complex analytic functions.
In TGD, the light-like hypercomplex coordinate u is however an exception. u is real and inequalities make sense for it. For instance, the segment u1≤u≤u2 can be defined in the semialgebraic context and the simplest situation corresponds to a position dependent time interval x-u1≤ t ≤ x+u2 or propagating pulse. The real part Re(w) of the complex coordinate w of the space-time surface defining the analog of the real axis in complex analysis would be a second coordinate of this kind and could be assigned to the partonic 2-surface.
The notion of semi-algebraic geometry makes it possible to represent these observations formally.
- In semi-algebraic geometry inequalities are allowed in the real case but do not make sense for complex and p-adic numbers. In TGD, semialgebraic geometry would make sense for the regions of space-time surface for which the generalized complex coordinates of H or space-time surface are real.
All inequalities should be formulated for the real sub-manifolds, which for ordinary complex 4-manifolds are 2-D. This is the case now. String world sheets parameterized by light-like coordinates u and v, would be naturally 2-D surfaces of this kind but the coordinate v does not appear as the argument of the functions (P,Q). Only the inequalities relating to u seem to make sense.
An interesting question is whether symplectic structure, which is basic element of the WCW geometry and can be seen as a companion of the generalized complex structure, could correspond to the decomposition of the complex space-time coordinate as w= P+iQ and hypercomplex coordinate as (u,v) such that (P,Q) and (u,v) define canonically conjugate coordinate pairs is consistent with the Hamilton-Jacobi structure. Note that the two real coordinates u and x= Re(w) could have interpretation as local choices of light-like direction and polarization direction and inequalities in this sense would be consistent with the notion of semialgebraic geometry. Could one get rid of inequalities altogether by a suitable choices of the real coordinate variants (u,x)? There is indeed a well-known trick allowing to get rid of an inequalities representable in the form t≥ 0 by a change of the coordinate variable as a replacement t → T= t2. Only the points with t≥ 0 are allowed by mere reality conditions. This trick might work to inequalities involving u and x.
See the article About Langlands correspondence in the TGD framework or the chapter with the same title.
For a summary of earlier postings see Latest progress in TGD.
For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.
Source: https://matpitka.blogspot.com/2024/09/is-it-really-possible-to-formulate-all.html
Anyone can join.
Anyone can contribute.
Anyone can become informed about their world.
"United We Stand" Click Here To Create Your Personal Citizen Journalist Account Today, Be Sure To Invite Your Friends.
Before It’s News® is a community of individuals who report on what’s going on around them, from all around the world. Anyone can join. Anyone can contribute. Anyone can become informed about their world. "United We Stand" Click Here To Create Your Personal Citizen Journalist Account Today, Be Sure To Invite Your Friends.
LION'S MANE PRODUCT
Try Our Lion’s Mane WHOLE MIND Nootropic Blend 60 Capsules
Mushrooms are having a moment. One fabulous fungus in particular, lion’s mane, may help improve memory, depression and anxiety symptoms. They are also an excellent source of nutrients that show promise as a therapy for dementia, and other neurodegenerative diseases. If you’re living with anxiety or depression, you may be curious about all the therapy options out there — including the natural ones.Our Lion’s Mane WHOLE MIND Nootropic Blend has been formulated to utilize the potency of Lion’s mane but also include the benefits of four other Highly Beneficial Mushrooms. Synergistically, they work together to Build your health through improving cognitive function and immunity regardless of your age. Our Nootropic not only improves your Cognitive Function and Activates your Immune System, but it benefits growth of Essential Gut Flora, further enhancing your Vitality.
Our Formula includes: Lion’s Mane Mushrooms which Increase Brain Power through nerve growth, lessen anxiety, reduce depression, and improve concentration. Its an excellent adaptogen, promotes sleep and improves immunity. Shiitake Mushrooms which Fight cancer cells and infectious disease, boost the immune system, promotes brain function, and serves as a source of B vitamins. Maitake Mushrooms which regulate blood sugar levels of diabetics, reduce hypertension and boosts the immune system. Reishi Mushrooms which Fight inflammation, liver disease, fatigue, tumor growth and cancer. They Improve skin disorders and soothes digestive problems, stomach ulcers and leaky gut syndrome. Chaga Mushrooms which have anti-aging effects, boost immune function, improve stamina and athletic performance, even act as a natural aphrodisiac, fighting diabetes and improving liver function. Try Our Lion’s Mane WHOLE MIND Nootropic Blend 60 Capsules Today. Be 100% Satisfied or Receive a Full Money Back Guarantee. Order Yours Today by Following This Link.