Sheaf theoretic methods in topology
Joana Cirici (Universität Münster): "Mixed Hodge Theory and the decomposition theorem"
Lecture 1: Derived categories, sheaf cohomology and the 6 operations.
Lecture 2: Generalities on the cohomology of complex algebraic varieties.
Lecture 3: Mixed Hodge theory and the weight filtration as a new invariant.
Lecture 4: Mixed Hodge modules and the decomposition theorem.
Jon Woolf (University of Liverpool): "Topological aspects of perverse sheaves"
Lecture 1: stratified spaces, exit paths, constructible sheaves and constructible functions.
Lecture 2: category of perverse sheaves (constructed via glueing categories of local systems), behaviour of perverse sheaves under duality and six functors, basic algebraic properties of perverse sheaves (finite length, global dimension), simple objects, intermediate extension and intersection cohomology, perverse sheaves on a curve.
Lecture 3: Stratified Morse theory, normal Morse data, purity of perverse sheaves, Lefschetz hyperplane theorem.
Lecture 4: characteristic cycles, behaviour of characteristic cycles under standard functors, Dubson-Kashiwara index formula for Euler characteristic of a perverse sheaf.