Semester: first or third

ECTS:4

Elective course

Course objectives

Expected learning outcomes

After completing the course, the students will be able to:

- define simplicial complexes and describe their application in biomedicine,
- compute persistant homology in simple examples,
- determine Betti numbers and interpret them in the context of molecular biology and biomedicine,
- apply algorithms from topological data analysis,
- compare different definitions and properties of curvature of curves and surfaces in discrete setting,
- apply techniques of discrete differential geometry to geometrical modelling.

Course content

**Metric spaces.**Concept of metric and topology. Topological and differential manifolds. Tangent vectors. Bundles.**Simplicial complexes.**Definition of simplicial complexes and examples in biomedicine and data analysis. Triangulations and simplicial approximation. Simplicial maps.**Topological data analysis.**Betti numbers. Introduction to persistent homology.**Applications of topology in molecular biology and biomedicine.**Topology of viral evolution. Topology of cancer. Visualization and clustering algorithms.**Discrete differential geometry.**Discretization of curves and surfaces. Special classes and parametrizations.**Curvatures of discrete planar curves.**Characterizations of curvature in the smooth setting. Principle of preservation of properties.**Curvatures of discrete surfaces.**Conjugate nets. Orthogonal nets. Surfaces parametrized with curvature or asymptotic lines. Discrete surfaces with constant negative Gaussian curvature.**Geometric modelling with discrete differential geometry.**Application of discrete curves and surfaces and analysis of their properties.