MS-E1531 - Differential Geometry D, Lecture, 11.1.2022-18.2.2022

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Please note! Course description is confirmed for two academic years, which means that in general, e.g. Learning outcomes, assessment methods and key content stays unchanged. However, via course syllabus, it is possible to specify or change the course execution in each realization of the course, such as how the contact sessions are organized, assessment methods weighted or materials used.

LEARNING OUTCOMES

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.

 

Credits: 5

Schedule: 11.01.2022 - 18.02.2022

Teacher in charge (valid for whole curriculum period):

Teacher in charge (applies in this implementation): Kirsi Peltonen

Contact information for the course (applies in this implementation):

CEFR level (valid for whole curriculum period):

Language of instruction and studies (applies in this implementation):

Teaching language: English. Languages of study attainment: English

CONTENT, ASSESSMENT AND WORKLOAD

Content

  • valid for whole curriculum period:

    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 for whole curriculum period:

    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 for whole curriculum period:

    36 + 18 (4 + 2)

     

DETAILS

Study Material

  • valid for whole curriculum period:

    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
Prerequisites

FURTHER INFORMATION

Further Information

  • valid for whole curriculum period:

    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.

     

    Teaching Period:

    (2020, 2021) - No teaching

    2021-2022 Spring III

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

    Registration for Courses: In the academic year 2021-2022, registration for courses will take place on Sisu (sisu.aalto.fi) instead of WebOodi.

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