We learn BEST when we learn by IMMERSION.

It’s NECESSARY to be cognitively overwhelmed, perhaps even so frustrated that we are ready to almost feel like crying!

Without that aggravation of being overwhelmed by some cognitive task, our cognitive faculties will atrophy. When we watch entertainment passively, it’s not just that our faculties are atrophying, but we are being programmed to seek comfort, avoid learning new things and to only IMAGINE that we think critically.

The syllabus for autodidactic study starts with by prompting an AI assistant for a year-long syllabus.

    Develop a 200-module syllabus to study compactification [or de-compactification] and dimensional reduction [[or dimensional expansion] in mathematics and how this changes a theory with respect to one of its spacetime dimensions.

We take the results of from that prompt and then we tear it part critically as we read and research and then we re-factor and re-build a syllabus that is more comprehensive and more detailed … it will typically take at least one hundred additional prompts to the AI assistant to help us put leaves on the branches of the tree.

The following syllabus is an example of developing the result of the above prompt above.

Compactification and Dimensional Reduction in Mathematics and The Effects On Space-Time Dimensions

The following syllabus provides a comprehensive outline of an overview of compactification, de-compactification, dimensional reduction, and dimensional expansion in various areas of mathematics and theoretical physics. It starts with the foundational concepts of topology, compactness, and dimensional reduction in linear algebra and differential geometry.

The course then explores different compactification techniques and their applications, as well as the concepts of de-compactification and localization. It delves into the role of dimensional reduction in field theory, covering topics such as Kaluza-Klein theory, gauge theories, and supergravity.

The syllabus also includes a discussion of dimensional expansion and its applications in string theory and M-theory. It examines the cosmological implications of extra dimensions and the interplay between compactification, dimensional reduction, and cosmology.

The course covers dualities and their relation to dimensional reduction, as well as advanced topics such as flux compactifications, matrix models, and the role of compactification and dimensional reduction in quantum gravity.

Throughout the course, students will gain a deep understanding of how compactification, de-compactification, dimensional reduction, and dimensional expansion can alter the properties of spacetime dimensions and influence the behavior of physical theories. They will be exposed to cutting-edge research in mathematics and theoretical physics and will develop the skills needed to apply these concepts to a wide range of problems.

Introduction to Compactification and Dimensional Reduction (20 modules):

1-4: Topological Spaces and Continuous Functions

5-8: Compact Spaces and Their Properties

9-12: Dimensional Reduction in Linear Algebra and Tensor Analysis

Dimensional reduction is the limit of a compactified theory where the size of the compact dimension goes to zero. In physics, a theory in D spacetime dimensions can be redefined in a lower number of dimensions d, by taking all the fields to be independent of the location in the extra D − d dimensions. Dimensional reduction also refers to a specific cancellation of divergences in Feynman diagrams.

The compact dimension imposes specific boundary conditions on all fields, for example periodic boundary conditions in the case of a periodic dimension, and typically Neumann or Dirichlet boundary conditions in other cases. Now suppose the size of the compact dimension is L; then the possible eigenvalues under gradient along this dimension are integer or half-integer multiples of 1/L (depending on the precise boundary conditions). In quantum mechanics this eigenvalue is the momentum of the field, and is therefore related to its energy. As L → 0 all eigenvalues except zero go to infinity, and so does the energy. Therefore, at this limit, with finite energy, zero is the only possible eigenvalue under gradient along the compact dimension, meaning that nothing depends on this dimension. This is the essence of dimensional reduction. The compact dimension is often called the compactified dimension, and the non-compact dimensions are called the uncompactified dimensions. The process of dimensional reduction is often called compactification, although this term is also used in a different sense in algebraic geometry. The process of dimensional reduction is often used in string theory and M-theory, where it is used to relate different theories in different dimensions. For example, the original 11-dimensional M-theory is related to the 10-dimensional type IIA string theory by compactifying one of the dimensions of M-theory on a circle. The resulting theory is then a 10-dimensional theory, and the extra dimension is the compactified dimension. The process of dimensional reduction is also used in the Kaluza-Klein theory, where it is used to unify gravity and electromagnetism in a higher-dimensional theory. In this case, the extra dimensions are compactified on a circle or a torus, and the resulting theory is a 4-dimensional theory with additional scalar fields that can be interpreted as the electromagnetic potential. The process of dimensional reduction is also used in the AdS/CFT correspondence, where it is used to relate a higher-dimensional theory of gravity in an anti-de Sitter space to a lower-dimensional theory of conformal field theory. In this case, the extra dimensions are compactified on an anti-de Sitter space, and the resulting theory is a lower-dimensional theory that describes the boundary of the anti-de Sitter space. The process of dimensional reduction is a powerful tool in theoretical physics, and has many applications in string theory, M-theory, and other areas of theoretical physics.

13-16: Dimensional Reduction in Differential Geometry

17-20: Applications of Dimensional Reduction in Physics

The discovery of the Higgs boson has heralded the era of precision in hadron collider physics. Disentangling potential new physics effects from the wealth of data requires a very high level of control over theoretical predictions for Standard Model cross sections which is very often limited by our ability to compute complicated Feynman diagrams. Feynman integrals are a rapidly developing field and there are many competing methods which each have their own merits and limitations and state-of-the art problems often require a combinations of various tools and basic concepts of dimensional regularization and Feynman parametrization, which provide the foundation for more advanced topics including sector [decomposition], Mellin Barnes representations, reduction to master integrals using integration-by-parts identities, solving master integrals by differential equations and the expansion by regions.

Compactification Techniques (30 modules):

21-24: One-Point Compactification (Alexandroff Compactification) The Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Alexandroff (1896-1982). More precisely, let X be a topological space. Then the Alexandroff extension of X is a certain compact space X* together with an open embedding c : X → X* such that the complement of X in X* consists of a single point, typically denoted ∞. The map c is a Hausdorff compactification if and only if X is a locally compact, noncompact Hausdorff space. For such spaces the Alexandroff extension is called the one-point compactification or Alexandroff compactification. The advantages of the Alexandroff compactification lie in its simple, often geometrically meaningful structure and the fact that it is in a precise sense minimal among all compactifications; the disadvantage lies in the fact that it only gives a Hausdorff compactification on the class of locally compact, noncompact Hausdorff spaces, unlike the Stone–Čech compactification which exists for any topological space (but provides an embedding exactly for Tychonoff spaces).

25-28: Stone-Čech Compactification

29-32: Compactification by Ends

33-36: Martin’s Axiom and the Continuum Hypothesis

37-40: Applications of Compactification in Analysis and Topology

41-44: Compactification in Functional Analysis (Banach and Hilbert Spaces)

45-50: Compactification in Algebraic Geometry (Projective Varieties, Stein Spaces)

De-compactification and Localization (20 modules):

51-54: De-compactification and the Inverse Limit

55-58: Localization and Sheaf Theory

59-62: Grothendieck Topologies and Sites

63-66: Étale Cohomology and `-adic Cohomology

67-70: Applications of De-compactification and Localization in Algebraic Geometry

Dimensional Reduction in Field Theory (30 modules):

71-74: Kaluza-Klein Theory and Dimensional Reduction

75-78: Dimensional Reduction in Gauge Theories

79-82: Dimensional Reduction in Supergravity Theories

83-86: Scherk-Schwarz Dimensional Reduction and Supersymmetry Breaking

87-90: Consistency Conditions for Dimensional Reduction

91-94: Effective Field Theories from Dimensional Reduction

95-100: Dimensional Reduction in String Theory and M-Theory

Dimensional Expansion and Oxidation (20 modules):

101-104: Oxidation and Lifting in Supergravity

105-108: Exceptional Field Theory and U-Duality

109-112: Dimensional Expansion in Double Field Theory

113-116: Generalized Geometry and Extended Spacetime

117-120: Applications of Dimensional Expansion in String Theory and M-Theory

Compactification and Dimensional Reduction in Cosmology (30 modules):

121-124: Cosmological Implications of Extra Dimensions

125-128: Brane-World Scenarios and Large Extra Dimensions

129-132: Randall-Sundrum Models and Warped Compactifications

133-136: Compactification and the Cosmological Constant Problem

137-140: Observational Constraints on Extra Dimensions and Compactification Scales

141-144: Inflation and Dimensional Reduction

145-150: Dimensional Reduction in Quantum Cosmology

Dualities and Dimensional Reduction (20 modules):

151-154: T-Duality and Mirror Symmetry

155-158: S-Duality and Electromagnetic Duality

159-162: U-Duality and Exceptional Groups

163-166: AdS/CFT Correspondence and Holographic Dimensional Reduction

167-170: Duality Cascades and Confinement

Advanced Topics and Applications (30 modules):

171-174: Non-Compact Calabi-Yau Manifolds and Toric Geometry

175-178: G-Structures and Generalized Compactifications

179-182: Flux Compactifications and Moduli Stabilization

183-186: Compactification and Dimensional Reduction in Matrix Models

187-190: Dimensional Reduction and Entanglement Entropy

191-194: Compactification and Dimensional Reduction in Quantum Gravity

195-200: Future Directions and Open Problems in Compactification and Dimensional Reduction