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Explore a comprehensive lecture examining the unification of conformal and fuzzy gravities with internal interactions through a gauge theoretic framework. Learn how the tangent group of a curved manifold and the manifold itself can have different dimensions, and discover how both gravitational theories can be formulated using gauge theory approaches. Begin with a review of the gauge theoretic approach to Einstein gravity, then progress through the construction of Conformal Gravity and Noncommutative (Fuzzy) Gravity within this theoretical framework. Conclude by understanding how an extension of the four-dimensional tangent group enables the unification of both gravitational theories with internal interactions, providing insights into advanced theoretical physics concepts that bridge general relativity with modern gauge theory approaches.
