Math 583GA: what we've covered so far

• Monday, 1 April. Passed out the syllabus, then passed out the real syllabus. Started an introduction to spectral sequences, using the Serre spectral sequence as an example.
• Wednesday, 3 April. Finished introduction: computed the cohomology of SU(3) and hinted at the computation for SU(n), and analyzed fiber bundles where the base space, total space and fiber are all spheres. Discussed filtrations.
• Friday, 5 April. Constructed the spectral sequence associated to a filtered chain complex. Sketched the proof of everything except convergence.
• Monday, 8 April. A few words about convergence in general, and also about terminology ("homology" spectral sequence vs. "cohomology", and "1st quadrant" (etc.) spectral sequences). Discussed exact couples.
• Wednesday, 10 April. More on exact couples: bigrading and comparison to filtered chain complexes. Example: the Bockstein spectral sequence, constructed from an exact couple.
• Friday, 12 April. Another construction of the Bockstein spectral sequence.
• Monday, 15, April. Example of computing differentials in the Bockstein spectral sequence.
• Wednesday, 17 April. Double complexes and the two spectral sequences that arise from them (Section 2.4). Example: balancing Tor.
• Friday, 19 April. More on Tor: examples, proving that Tor is "balanced". Also defined and discussed Ext, briefly.
• Monday, 22 April. Spectral sequences of algebras, the cohomology Serre spectral sequence. Began the computation of the cohomology of loops on a sphere.
• Wednesday, 24 April. More on the Serre spectral sequence: Euler characteristic, the Gysin sequence, the cohomology of complex projective space.
• Friday, 26 April. Morse theory: finding infinitely many geodesics in a complete connected Riemannian manifold by computing the cohomology of its loop space.
• Monday, 29 April. Corrected part of the proof of the Morse theory thing. Outlined the computation of the cohomology of SU(n). Started discussing algebra cohomology.
• Wednesday, 1 May. Basics on group cohomology: computing Ext over k[x]/(xⁿ), computing the cohomology of the cyclic group C₂.
• Friday, 3 May. More on group cohomology: functoriality, the Lyndon-Hochschild-Serre spectral sequence.
• Monday, 6 May. Applications of the LHS spectral sequence: embedding the mod p cohomology of a finite group into the cohomology of a (normal) Sylow p-subgroup; the change-of-rings isomorphism; splitting the E₂-term if the extension is central.
• Wednesday, 8 May. Example: computing a differential in the spectral sequence for the group of quaternionic units, using naturality of the spectral sequence. Remarks about finite-generation of cohomology and Quillen's theorem.
• Friday, 10 May. Outline of Evens' theorem: the cohomology of a finite group is finitely generated as an algebra.
• Monday, 13 May. Extensions of augmented algebras and the associated spectral sequence. Hopf algebras.
• Wednesday, 15 May. More on Hopf algebras: structure on modules, duality, a few examples.
• Friday, 17 May. The cobar complex; examples.
• Monday, 20 May. Using the cobar complex to compute differentials in the extension spectral sequence.
• Wednesday, 22 May. Computing the cohomology of A(1), the semidihedral algebra of dimension 8 (and also the sub-Hopf algebra of the mod 2 Steenrod algebra generated by Sq¹ and Sq²).
• Friday, 24 May. Massey products. The Grothendieck spectral sequence.
• Wednesday, 29 May. Triangulated categories, closed symmetric monoidal categories.
• Friday, 31 May. The stable category of modules over an algebra.
• Monday, 3 June. Hopf algebroids. An example of the Adams spectral sequence.
• Wednesday, 5 June. Constructing the Adams spectral sequence.

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Last modified: Fri Jun 7 12:54:18 PDT 2002