January 8: Class Policy. Categories
January 10: Categories, Functors, Natural Transformations
January 13: Typing Diagrams in latex. Equivalence of Categories. Groupoids.
January 15: The Method of Algebraic Topology. The Fundamental Groupoid.
January 17: The Fundamental Groupoid. The Effect of Homotopy.
January 22: Based Spaces. Pushout, Limits and Colimits.
January 24: Seifert-Van Kampen Theorem for Fundamental Groupoids. 2-Pushout.
January 27: Pushout vs. 2-pushout of Groupoids. Fundamental group of $S^1$
January 29: Seifert-Van Kampen Theorem for Fundamental Groups. Free Groups.
January 31: Free Groups. Generators and Defining Relations. Algorithmic Aspect. Knots.
February 3: Invariants of Knots. Non-triviality of the Trefoil Knot. Compact Surfaces.
February 5: The Colimit Argument. Fundamental Group of Graphs. Fundamental group of Surfaces.
February 7: Adjoint Functors. Forgetful Functors of Universal Algebras. Abelianization.
February 10: More on Adjoint Functors and Abelianization. Classification of Finitely Generated Abelian Groups.
February 14: Covering Spaces. Homotopy Lifting Property for Fundamental Groupoids.
February 17: Lifting Maps into Covering Spaces with Local Conditions.
February 19: The Category of Covering Spaces and Deck Transformations. The Orbit Category. Fiber Functor.
February 21: Equivalence of Categories of Covering Spaces and $\pi_1$-sets via Fiber Functor.(Classification of Covering Spaces.)
February 24:Classification of Covering Spaces - Finishing the Proof. Geometrization of Surfaces (Statement). Subgroup of a Free Group is Free.
February 26:Free Generators of Subgroups of Free Groups.
February 28: CW Complexes and their Fundamental Groups.
March 10: Singular Homology. Definition, $dd=0$.
March 12: Chain Complex. Chain Map. Exact Sequence.
March 14: Relative Singular Homology. Long Exact Sequence (Exactness Axiom.)
March 17: Eilenberg-Steenrod Axioms. Compactness of Homology.
March 19: Compactness Continued. Reduced Homology. The Homology of a Sphere.
March 21: Chain Homotopy. The Homotopy Axiom.
March 24: The Homotopy Axiom Continued. The Excision Axiom.
March 26: The Excision Axiom Continued.
March 28: The Excision Axiom - Finishing the Proof. The Mayer-Vietoris Sequence.
March 31: The Homology of a CW Complex.
April 2: CW Homology. An Example.
April 4: The Differential in CW Homology. Degree of a Map. Hopf's Theorem.
April 7: The Degree of a Map. Proof of Hopf's Theorem.
April 9: Concluding the Proof of Hopf's Theorem. The Homology of Projective Spaces.
April 11: Euler Characteristic and Trace.
April 14: Lefschetz Fixed Point and Trace Theorems - Statement. Other Contexts.
April 16: Jordan's Separation Theorem in $n$ Dimensions.
April 18: Invariance of Domain.
April 21: The Derived Category of Spaces.