Dynamics on Homogeneous Spaces and Interactions with Number Theory

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Explores quadratic forms, symmetric bilinear forms, and their properties.

Introduces differential forms on manifolds, covering tangent bundles and intersection pairings.

Explores pseudo-Euclidean spaces, emphasizing isometries and bases in vector spaces with non-degenerate quadratic forms.

Explores harmonic forms on Riemann surfaces and the uniqueness of solutions to harmonic equations.

Covers the topology of Adeles and their relationship with quadratic forms, polynomial varieties, and finiteness properties.

Covers the joint equidistribution of CM points in algebraic structures and quadratic forms.

Explores Lagrange interpolation, emphasizing uniqueness and simplicity in reconstructing functions from limited values.

Covers the proof of the recursive formula for the optimal gains in LQ control over a finite horizon.

Explores symmetric matrices, diagonalization, and quadratic forms properties.

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.