Module 24-M-GT-AT10 Advanced Topics in Geometry and Topology

Faculty

Person responsible for module

Regular cycle (beginning)

Dieses Modul ist Teil einer langfristigen Gesamtlehrplanung für das Masterprogramm, die sicherstellt, dass in allen fünf Gebieten jedes Jahr jeweils mindestens Module im Umfang von 20 LP angeboten werden. Im Rahmen dieser Gesamtlehrplanung wird das Modul in unregelmäßigen Abständen angeboten.

Credit points and duration

10 Credit points

For information on the duration of the modul, refer to the courses of study in which the module is used.

Competencies

Non-official translation of the module descriptions. Only the German version is legally binding.

Students expand and deepen their mathematical knowledge and skills in the field of Geometry and Topology.
The students master the essential contents and methods of a special, research-relevant subject area of Geometry and Topology, in particular they can independently carry out complex proofs in this area requiring a high level of mathematical expertise.
Furthermore, students recognise further-reaching connections to previously acquired mathematical facts. They can transfer and apply the knowledge and methods they have learnt so far to other, deeper mathematical problem areas. Students also expand their mathematical intuition through further and more intensive study.
In the tutorials, students develop their ability to discuss mathematical topics and thus further prepare themselves for the requirements of the Master's module, in particular for the scientific discussion within the Master's seminar presentation and the defence of their Master's thesis.

Content of teaching

The courses in this module lead to current research questions in the field of Geometry and Topology in terms of method and content. Possible contents include:

  • Discrete geometry and aperiodic order
  • Introduction to Kähler geometry
  • Introduction to toric geometry
  • Four-dimensional manifolds
  • Seiberg-Witten invariants
  • Spectral sequences and convergence
  • Serre’s spectral sequence
  • Steenrod’s cohomology operations
  • The cohomology of Eilenberg-Mac Lane spaces
  • The Cartan-Serre algorithm
  • The theory of Serre classes
  • Bockstein spectral sequence
  • Atiyah-Hirzebruch spectral sequence
  • Adams spectral sequence
  • Characteristic classes (Bordism, Spectra, Topological K-theory, Bott-periodicity)

Recommended previous knowledge

Depending on the chosen subject, the requirements will be specified in the course announcement.

Necessary requirements

Explanation regarding the elements of the module

Module structure: 1 SL, 1 bPr 1

Courses

Lecture Advanced Topics in Geometry and Topology
Type lecture
Regular cycle Dieses Modul ist Teil einer langfristigen Gesamtlehrplanung für das Masterprogramm, die sicherstellt, dass in allen fünf Gebieten jedes Jahr jeweils mindestens Module im Umfang von 20 LP angeboten werden. Im Rahmen dieser Gesamtlehrplanung wird das Modul in unregelmäßigen Abständen angeboten.
Workload5 60 h (60 + 0)
LP 2 [Pr]
Tutorials Advanced Topics in Geometry and Topology
Type exercise
Regular cycle Dieses Modul ist Teil einer langfristigen Gesamtlehrplanung für das Masterprogramm, die sicherstellt, dass in allen fünf Gebieten jedes Jahr jeweils mindestens Module im Umfang von 20 LP angeboten werden. Im Rahmen dieser Gesamtlehrplanung wird das Modul in unregelmäßigen Abständen angeboten.
Workload5 90 h (30 + 60)
LP 3 [SL]

Study requirements

Allocated examiner Workload LP2
Teaching staff of the course Tutorials Advanced Topics in Geometry and Topology (exercise)

Regular completion of the exercises, each with a recognisable solution approach, as well as participation in the exercise groups for the module's lecture. As a rule, participation in the exercise group includes presenting solutions to exercises twice after being asked to do so as well as regular contributions to the scientific discussion in the exercise group, for example in the form of comments and questions on the proposed solutions presented. The organiser may replace some of the exercises with face-to-face exercises.

see above see above

Examinations

e-written examination o. written examination o. e-oral examination o. oral examination
Allocated examiner Teaching staff of the course Lecture Advanced Topics in Geometry and Topology (lecture)
Weighting 1
Workload 150h
LP2 5

(electronic) written examination in presence of usually 120 minutes, oral examination in presence or remote of usually 40 minutes, A remote electronic written examination is not permitted.

The module is used in these degree programmes:

Degree programme Profile Recom­mended start 3 Duration Manda­tory option 4
Mathematical Economics / Master of Science [FsB vom 28.02.2025] Mathematics 1. o. 2. o. 3. one semester Compul­sory optional subject
Mathematical Economics / Master of Science [FsB vom 28.02.2025] Economics 1. o. 2. o. 3. one semester Compul­sory optional subject
Mathematics / Master of Science [FsB vom 28.02.2025] 1. o. 2. o. 3. one semester Compul­sory optional subject

Automatic check for completeness

The system can perform an automatic check for completeness for this module.


Legend

1
The module structure displays the required number of study requirements and examinations.
2
LP is the short form for credit points.
3
The figures in this column are the specialist semesters in which it is recommended to start the module. Depending on the individual study schedule, entirely different courses of study are possible and advisable.
4
Explanations on mandatory option: "Obligation" means: This module is mandatory for the course of the studies; "Optional obligation" means: This module belongs to a number of modules available for selection under certain circumstances. This is more precisely regulated by the "Subject-related regulations" (see navigation).
5
Workload (contact time + self-study)
SoSe
Summer semester
WiSe
Winter semester
SL
Study requirement
Pr
Examination
bPr
Number of examinations with grades
uPr
Number of examinations without grades
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