Geometry and Topology - Master programme and Doctoral Programm

Master's program

In the Master's program, "Geometry and Topology" is one of 7 main areas of specialization. If this is the chosen main area of specialization, there is a compulsory module group of foundational courses. (The further modules of the Master's program can be divided between courses from the chosen area of specialization and courses from other areas of specialization.)

The basic courses in the area of specialization "Geometry and Topology" consists of 4 compulsory modules:

• First, in the module Differential Geometry, the course "Analysics on Manifolds" will expand upon basic courses' methods of multidimensional differential and integral calculus on open subsets of Rn to obtain methods on more general objects, so-called manifolds. The focus is on operations that can be defined independently of the choice of coordinates, wherefrom the analysis gets a geometric viewpoint. This course can be taken by all students in the Master's program. The subsequent course "Riemannian Geometry", where analytic methods are applied to geometric problems, forms the second part of the module.
• The module Lie groups is based on the analysis of manifolds, and therefore, should be completed (if possible immediately) after the homonymous course. Here, differential geometry and algebra are linked, and its most important application is the theory of symmetries.
• The module Algebraic Topology is independent of the two preceding modules, and therefore, can be chosen by all students in the Master's program. It deals with assigning objects (numbers, groups, vector spaces, etc.) to topological spaces in order to distinguish them or find invariants.
• Two seminars need to be completed for the module Seminar: Geometry and Topology. One of those is required to be a seminar based on one of the courses "Analysis on Manifolds", "Lie Groups", or "Algebraic Topology". The offering of seminars in the area of geometry and topology is limited, so although a coordination of the seminars with the area of the Master's thesis may be advisable, it is not required and will often not be possible. Further introductory seminars can be chosen as advanced courses, with their attendance being, in any case, highly-advisable.

The offering of advanced courses for the Master's program is closely linked to the research interests of the faculty members in this research area and limited by budgetary constraints. Apart from differential geometry and topology, links to functional analysis (infinite-dimensional differential geometry, algebras of generalized functions, partial differential equations of geometric origin), algebra (Lie groups, Lie algebras and representation theory, algebraic geometry), and theoretical physics (general relativity) are topics of advanced courses.

The research interests of the faculty members play an important role in the question of topics for Master's theses. In any case, it is advisable to think about a possible topics and appropriate supervisors of your Master's thesis at an early stage of the Master's program. (The standard study period of 4 semesters is short.) When looking for a topic and supervisors, you should also take into account whether you intend to continue onto the doctoral program. In this case, more consideration should given so that the topic has a strong connection to contemporary research. Otherwise a broader range of topics is possible.

Doctoral program

As usual at the Faculty of Mathematics, there is no real difference between advanced courses for the Master's program and courses for the doctoral program in the area of specialization "Geometry and Topology". The recognition of courses for the doctoral program will be specified individually in the "dissertation agreement" (Dissertationsvereinbarung). In particular, it is irrelevant for the recognition of a course whether the course  is announced with a course number for the Master's in mathematics (25XXXX) or for the doctoral (51XXXX) program. You can find general information on the doctoral program on the web pages of the SSC Mathematics and the Center of Doctoral Studies of the University of Vienna.

The research interests of the individual faculty members play a much larger role in the choice of a topic and supervisor for a doctoral dissertation than for a Master's thesis: dissertation topics are usually adjacent to the research area and interests of the supervisor. Therefore, it does not make sense to give general information regarding these questions. It is worth mentioning that hardly any research on topology is carried out at our faculty, but there are definitely topological aspects in many areas of differential geometry. Otherwise, primarily refer to the webpages of the single faculty members, which contain information about their research interests.

It is extremely important that you contact a potential supervisor before starting the doctoral program and talk about a possible supervision. It does not make sense to enroll for the doctoral program first and then look for a supervisor.