Demonstrates the equivalence between simplicial and singular homology, proving isomorphisms for finite s-complexes and discussing long exact sequences.

Homotopy Invariance: Homology Groups

Explores homotopy invariance and its application to homology groups of quotients, showcasing isomorphism and chain homotopy.

Differential Forms on Manifolds

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

Connections: motivation and definition

Explores the definition of connections for smooth vector fields on manifolds.

Smith Theory in Floer Persistence and Dynamics

Covers Smith theory in Floer persistence and dynamics, exploring classical mechanics invariants, Poincaré-Birkhoff theorem, and the Hofer-Zehnder conjecture.