00478 - Advanced Geometry

Learning outcomes

At the end of the course, with regard to the first learning module the student acquires knowledge of the basic principles in computational topology and of the main theoretical and computational methods used in topological data analysis, with particular reference to topological and homological persistence. As for the second learning module, the student acquires advanced knowledge of the differential calculus, the varieties and its applications and a knowledge of the main problems and methods that arise from the study of geometric structures such as Riemannian, symplectic and complex structures.

Course contents

FIRST LEARNING MODULE:

Geometric and abstract simplicial complexes. Čech complexes, Vietoris-Rips complexes, Delaunay complexes, Alpha-complexes. Vertex maps. Simplicial maps and PL maps induced by vertex maps. Simplicial homology and cohomology with coefficients in Z_2 and their matrix computation. Basic principles in persistent topology and homology.

SECOND LEARNING MODULE:

Differentiable manifolds. Definition and examples. Differential calculus on manifolds. The external differential. Vector bundles. Tangent bundle. Distributions in the tangent bundle. Elements of transversality theory.

Readings/Bibliography

Herbert Edelsbrunner and John L. Harer, Computational Topology: An Introduction.

Allen Hatcher, Algebraic Topology.

L. Tu, Introduction to manifolds, L. Tu.

F. Warner, Foundations of differentiable manifolds and Lie groups.

Teaching methods

Frontal lessons

Assessment methods

For the first module, the examination consists in the passing of a written test and an oral one. In the written test the candidate must solve exercises using the skills acquired in the course.

For the second module the examination consists in the passing of an oral test.

Links to further information

http://www.dm.unibo.it/~frosini/DIDMAT.shtml

Office hours

See the website of Sergio Venturini

See the website of Patrizio Frosini