Credits: 5

Schedule: 07.01.2020 - 14.02.2020

Teaching Period (valid 01.08.2018-31.07.2020): 

Not lectured (2018-2019)

III Spring (2019-2020)

Lectured every other year

 

Learning Outcomes (valid 01.08.2018-31.07.2020): 

This course is an introduction to the basic machinery behind the modern differential geometry: tensors, differential forms, smooth manifolds and vector bundles. The geometries lying above these structures are involved in several applications through mathematical analysis, physics, stochastics and statistical modells. The central goal is to become familiar with this particular language of abstract mathematics that opens the venue to apply geometric methods in different applications. A modern viewpoint to some of the classical Riemann, Finsler or Kähler model geometries is served in addition to the possibility to open the door to the beautiful worlds of contact and symplectic geometry that are present in the most recent progress of geometrization of applications. The course provides basic skills to recognize geometric phenomena in mathematical analysis and applications.

 

Content (valid 01.08.2018-31.07.2020): 

Topics related to differential geometry varying from classical Riemannian geometry to modern geometries. More specified topics will be announced later.

 

Assessment Methods and Criteria (valid 01.08.2018-31.07.2020): 

Active participation in lectures and weekly exercises. Individual research projects that are related to the topics of the course. Always discuss beforehand with the lecturer before starting such a project. A traditional exam is also possible.

 

Workload (valid 01.08.2018-31.07.2020): 

36 + 18 (4 + 2)

 

Study Material (valid 01.08.2018-31.07.2020): 

All material related to the course can be found from MyCourses pages of the course. There is no special book the course is following but excellent treatments in the spirit of the lectures are provided by:
- John M. Lee: Introduction to Smooth Manifolds, Springer
- John M. Lee: Riemannian Manifolds: An Introduction to Curvature, Springer.

Substitutes for Courses (valid 01.08.2018-31.07.2020): 

Mat-1.3531

 

Course Homepage (valid 01.08.2018-31.07.2020): 

https://mycourses.aalto.fi/course/search.php?search=MS-E1531

Prerequisites (valid 01.08.2018-31.07.2020): 

MS-A02XX, MS-A03XX, MS-C1530, MS-C1540

 

Grading Scale (valid 01.08.2018-31.07.2020): 

0-5

 

Further Information (valid 01.08.2018-31.07.2020): 

The content of the course is part of a good mathematical education, which should self-evidently belong to the curriculum of every math major student. A highly open mind is necessary to gain the capability to apply methods provided by differential geometry to other sciences. Suitable to everybody interested in geometrization, especially those with a focus on fields in natural sciences where the connection is most visible like in general relativity and electromagnetism. Other potential fields are all sciences that make use of statistical or stochastic methods.

 

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