I wrote a brief summary about basic ideas of enumerative algebraic geometry and proposals for applications to TGD.
Here is a short abstract of the article summarizing the new results.
String models and M-theory involve both algebraic and symplectic enumerative geometry. Also in adelic TGD enumerative algebraic geometry emerges. This article gives a brief summary about the basic ideas involved and suggests some applications to TGD.
In the sequel I will summarize the understanding of novice about enumerative algebraic geometry and discuss possible TGD applications. This material can be also found in earlier articles but it seemed appropriate to collect the material about enumerative algebraic geometry to a separate article.
- One might want to know the number of points of sub-variety belonging to the number field defining the coefficients of the polynomials. This problem is very relevant in M8 formulation of TGD, where these points are carriers of sparticles. In TGD based vision about cognition they define cognitive representations as points of space-time surface, whose M8 coordinates can be thought of as belonging to both real number field and to extensions of various p-adic number fields induced by the extension of rationals. If these cognitive representations define the vertices of analogs of twistor Grassmann diagrams in which sparticle lines meet, one would have number theoretically universal adelic formulation of scattering amplitudes and a deep connection between fundamental physics and cognition.
- Second kind of problem involves a set algebraic surfaces represented as zero loci for polynomials - lines and circles in the simplest situations. One must find the number of algebraic surfaces intersecting or touching the surfaces in this set. Here the notion of incidence is central. Point can be incident on line or two lines (being their intersection), line on plane, etc.. This leads to the notion of Grassmannians and flag-manifolds. In twistor Grassmannian approach algebraic geometry of Grassmannians play key role. Also in twistor Grassmannian approach to TGD algebraic geometry of Grassmannians play a key role and some aspects of this approach are discussed.
- In string models the notion of brane leads to what might be called quantum variant of algebraic geometry in which the usual rules of algebraic geometry do not apply as such. Gromow-Witten invariants provide an example of quantum invariants allowing sharper classification of algebraic and symplectic geometries. In TGD framework M8-H duality suggests that the construction of scattering amplitudes at level of M8 reduces to a super-space analog of algebraic geometry for complexified octonions. Candidates for TGD analogs of branes emerge naturally and G-W invariants could have applications also in TGD.
See the chapter Does M8-H duality reduce classical TGD to octonionic algebraic geometry? or the article About enumerative algebraic geometry in TGD framework.