Spezielle Kapitel der Topologie 2023

Here is a list of topics for my set of lectures for the special topics course in Wuppertal this (Summer 2023) semester. The list will be updated after each lecture, so that irregular participants can decide whether they would like to join on short notice. I will also upload handwritten notes after each lecture.

Each topic is intended to cover roughly 40 minutes of the 90 minute time-slot, which will include a 10 minute break.

The usual lecture times are Monday (G.13.18) and Wednesday (G.15.25) between 12:15 and 13:45. In addition, there will also be a lecture on the Fridays 23.06 and 30.06, in G.15.25 between 14:15 and 15:45.

Date           Notes           Topic
14.06(1) Motivation and overview.
14.06(2) Review of module of differentials. Review of derived functors, motivation for cotangent complex.
19.06(1) Review of model structures on chain complexes and simplicial abelian groups. Simplicial commutative rings. Cotangent complex as derived functor and via animation.
19.06(2) Properties of the cotangent complex: base-change and transitivity. The cotangent complex of a regular quotient.
21.06(1) Deformation theory for simplicial commutative rings.
21.06(2) Quillen's cotangent complex formalism, "stabilization is abelianization", Lurie's cotangent complex formalism.
23.06(1) Hochschild homology. First properties and examples.
23.06(2) Structures on Hochschild homology and the HKR-theorem.
26.06(1) Spectral sequences, spectral sequence of a double complex and Quillen spectral sequence, proof of HKR.
26.06(2) Derived Hochschild homology of F_p, divided power structures, further properties of derived Hochschild homology.
28.06(1) Morita theory.
28.06(2) Morita invariance of Hochschild homology, the trace map.
30.06(1) Cyclic module structure on Hochschild homology, Connes' B-operator, compatibility with de Rham.
30.06(2) Negative cyclic and periodic homology via circle action. Examples.
10.07(1) t-structures, Beilinson t-structure.
10.07(2) Perfectoid methods, filtrations on negative cyclic and periodic homology.
12.07(1) Topological Hochschild homology.
12.07(2) Prismatic cohomology via THH, de Rham comparison.

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