# Applied Topology and Differential Geometry

**Teachers**: dr. Matija Bašić, assistant professor, dr. Željka Milin Šipuš, professor

**Semester**: first or third

**ECTS**:4

Elective course

To equip students to use basic concepts and techniques from (algebraic) topology and differential geometry in applications to biology and biomedicine.

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.

**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.