# Specialization "Geometry and topology"

This page collects the most important information on the area of specialization "Geometry and topology", especially on possible topics for bachelor and master's theses for all students of mathematics. It should be noted that geometric topics play a role in the specialization algebra as well, in particular in the field of algebraic geometry and geometric group theory. The information is sorted according to (current) study programmes. In addition, you can find a list of possible supervisors and lists of examples of topics for bachelor, master's and doctoral theses from the area of geometry and topology.

## Teacher training programme

In the old teacher training programme as well as in the new teacher training programme (bachelor/master programme) a certain amount of (elementary) geometry is contained but the courses are independent of the area of specialization "Geometry and topology". Nevertheless geometric topics for bachelor and master's theses are possible. In the bachelor programme, apart from elementary geometry, classical differential geometry of curves is a possible topic. In the master programme classical differential geometry of surfaces is another possible topic.

## Bachelor programme

There are no compulsory courses on geometry in the bachelor programme of mathematics but references to geometric topics are contained in the cycle "Linear algebra and geometry" (elementary geometry) and in the course "Advanced analysis and elementary differential geometry". The module where the last-mentioned course belongs to also contains the course "Introduction to topology", which is devoted to point-set topology. The emphasis, however, is less on topology as an area of its own but on notions and methods that are applied in other areas of mathematics. In the area of geometry there are (in the bachelor curriculum valid from WS2015) two elective modules:

• In the elective module Classical differential geometry methods of multidimensional differential calculus are applied to problems of the geometry of curves and surfaces. Its centre are a basic understanding of geometric issues and different notions of curvature.
• The elective module Career oriented mathematics: Algorithmic geometry is devoted to the socalled "computational geometry". It deals with specific algorithmic solutions of problems with a geometric character, culminating in an implementation of these solutions on the computer.

There is an abundance of possible topics for bachelor theses from the field of geometry as well as the field of topology. In topology there is a wide range of topics from point-set topology that follow immediately from the usual topics of the course "Introduction to topology". In the field of geometry topics from elementary geometry (often with references to linear algebra), from classical differential geometry and algorithmic geometry are possible. Both topics based on one of the elective modules and topics that are independent from them are possible.

## Master programme

In the master programme "Geometry and topology" is one of 7 main areas of specialization. You have to choose one of these 7 areas and the chosen main area of specialization results from the completion of the compulsory module group "basic courses in the area of specialization ...". The further modules of the master programme can be divided into courses from the chosen area of specialization and courses from other areas of specialization.

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

• First, in the module differential geometry the methods of multidimensional differential and integral calculus known of open subsets of Rn, known from the basic courses, are expanded to more general objects, socalled manifolds, in the course "Analysis on manifolds". The focus is on operations that can be defined independently of the choice of coordinates, whereby the analysis gets a geometric viewpoint. This course can be taken by all students in the master programme. The immediately following course "Riemannian geometry", where the 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 it. Here diifferential geometry and algebra are linked and the 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 programme. It deals with assigning objects (numbers, groups, vector spaces etc.) to topological spaces in order to make them distinguishable.
• On the one hand, you have to complete the introductory seminar on one of the courses "Analysis on manifolds", "Lie groups", and "Algebraic topology" in the module "Seminars: Geometry and topology" (further introductory seminars can be chosen as advanced courses, their attendence is in any case highly advisable). On the other hand you have to complete two seminars. The offer of seminars in the area of geometry and topology is limited, a coordination of the seminars with the area of the master's thesis may be advisable but is not required and will often not be possible.

The offer of advanced courses for the master programme is closely linked to the research interests of the faculty members in this research area and restricted 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 start thinking about possible topics and a supervisor at an early stage of the master programme. (The standard study period of 4 semesters is short.) When looking for a supervisor and a topic, you should also take into account whether you intend to do the doctoral programme based on the master programme. In this case the choice of a topic is more delicate and the connection to research should be stronger. Otherwise a broader range of topics is possible.

## Doctoral programme

As usual at the faculty of mathematics, there is no real difference between advanced courses for the master programme and courses for the doctoral programme in the specialization "Geometry and topology". The recognition of courses for the doctoral programme will be specified individually in an agreement ("Dissertationsvereinbarung"). In particular it is irrelevant for the recognition whether a course  is announced with a course number for mathematics (25XXXX) or for the doctoral programme (44XXXX). You can find general information on the doctoral programme 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 thesis than for a master's thesis. The topics are usually related to the (more or less) immediate research area of the supervisor. Therefore it does not make sense to give global information on 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 we primarily refer to the web pages of the single faculty members, which contain information on their research interests.