1. komponenta

Lecture type	Total
Lectures	30

* Load is given in academic hour (1 academic hour = 45 minutes)

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


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

1. Topology and Data, G. Carlsson, Bulletin of the American Mathematical Society, vol 46, Number 2, 2009.
2. Topology of viral evolution, J. M. Chan, G. Carlsson, R. Rabadan, Proceedings of the National Academy of Sciences, vol 110, 46, 2013.
3. Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival, Monica Nicolau, Arnold J. Levine, and Gunnar Carlsson, Proceedings of the National Academy of Sciences, 2011.
4. Topological Features in Cancer Gene Expression Data, S. Lockwood, B. Krishnamoorthy, Pacific Symposium on Biocomputing, 2015.
5. Topology in Molecular Biology, M. I. Monastyrsky (Ed.), Springer, 2007.
6. Elementary Applied Topology, R. Ghirst, CreateSpace Independent Publishing Platform, 2014.
7. Discrete Differential Geometry, A.I. Bobenko, P. Schröder, J.M. Sullivan, G.M. Ziegler (Eds.), Springer, 2008.
8. Computational Topology for Biomedical Image and Dana Analysis Theory and Applications, R. R. Moraleda, N. A. Valous, W. Xiong, N. Halama, CRC Press, 2019.
9. Cell Groups Reveal Structure of Stimulus Space, C. Curto, V. Itskov, PLoS Computational Biology, vol. 4(10), 2008.
10. Identification of type 2 diabetes subgroups through topological analysis of patient similarity, Li L, Cheng WY, Glicksberg BS, Gottesman O, Tamler R, Chen R, Bottinger EP, Dudley JT, Sci Transl Med, 2015.
11. Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function, M. W. Reimann, M. Nolte, M. Scolamiero, K. Turner, R. Perin, G. Chindemi, P. L. R. Dłotko, K. Hess, H. Markram, Frontiers in Computational Neuroscience, Vol. 11, 2017.
12. Discrete Differential Geometry: Integrable Structure, A. I. Bobenko, Y. B. Suris, AMS, Graduate Studies in Mathematics, 2009.
13. Advances in Discrete Differential Geometry, A. I. Bobenko (Ed.), Springer, 2016.
14. A curvature theory for discrete surfaces based on mesh parallelity, A. I. Bobenko, H. Pottmann, J. Wallner, Mathematische Annalen 348(1), 2009.

Elective courses 1, 2 - Regular study - Biomedical Mathematics

Elective courses 1, 2 - Regular study - Biomedical Mathematics

Elective course 3 - Regular study - Biomedical Mathematics