This course explores connections between sheaf theory and persistent homology. We will discuss cellular sheaf cohomology, level-set persistence modules, and derived sheaf theory over the real line. Using results of Kashiwara-Schapira, we will study a derived convolution of sheaves and the resulting derived interleaving distance. The goal is to obtain working knowledge of a cellular description of the derived category of sheaves on the real line and to study barcodes of such objects. As an application, we will aim to compute the graded barcode corresponding to the derived push-forward of the constant sheaf along a discrete Morse function. These structures are readily generalizable, and present many exciting directions to explore in the theory of persistence modules. While this course will expand on discussions from the 2020 course ‘Sheaf theory and applications’, it will be accessible for anyone with prior experience with topology and linear algebra.
Target group: PhD students and postdocs in mathematics and computer science.
Prerequisites: Algebraic topology and linear algebra.
Evaluation: Participation, written assignments, problem sets.
Teaching format: Lectures and recitations
ECTS: 3 Year: 2020
MAT-GEO Mathematics - Geometry and Topology
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- Teacher: Adam BROWN