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Topics in Algebraic Geometry (Intersection Cohomology)
Schedule: Wednesday 10:15-12 Friday 14:15-16 (depending on the participants I will add an exercise session - the precise format will depend on the pandemic situation)
For details and more information on the content of the course please register to the moodle-page of the course using TopicsAG - registration is important to organize the format of the course, in case of an in person lecture we will need to know the list of participants in advance.
Content: I would like to give an introduction to intersecton cohomology and perverse sheaves. Mark Goresky and Robert MacPherson originally defined (or found) these in the form of a beautiful geometric construction. The starting point is that classcial (singular) cohomology of compact manifolds (or smooth projective varieties) has a number of beautiful symmetries (Poincaré-duality, Lefschetz decomposition) that fail for singular spaces - in case you have never heard of Poincaré duality the course will start by explaining what this is. Surprisingly, they found that for many singular spaces there is a way to construct a version of cohomology (concretely described by geometric conditions as a subcomplex of the singular chain complex) that enjoys these symmetries. This was quite unexpected.
It turned out, that they even found something much more powerful. There is a natural categorical reformulation of the construction that gives a new toolkit for the study of the cohomology of algebraic varieties and the structure induced on it by morphisms. The first applications of this were in representation theory, where the theory has become indespensable but it also allowed to prove conjectures in combinatorics that looked like simple linear algebra questions on the number of different subspaces spanned by a finite collection of vectors.
Part of the beauty of the construction is that it was conceived from geometric observations but then turned into a conceptual package that is meaningful in many different settings from topology to artihmetic, combinatorics and representation thoery.
If all goes well, I would like to start by motivating the classical construction, then give an intorduction to the categorical approach, if possible prove the famous decomposition theorem and give some of the applications mentioned above.
Prerequisites: You should have encountered some version of cohomology groups before. I will try to adjust the course to accomodate for different types of background. You can help me with this by filling in a short poll on the moodle page.
Contact: If you have questions regarding the course, you can simply send me an email.