# Specialization "Analysis"

Here you can find important information on the area of specialization "analysis" and hints for possible topics for bachelor and master's theses for all students of mathematics. Analysis plays an important role in almost all other areas of specialization and is the foundation for many fields. The information is sorted by study programme and you can also find lists of possible supervisors and examples of topics for bachelor, master's and doctoral thesis from the area of analysis.

## Teacher training programme

Analysis is a compulsory module in the curriculum for the teacher training programme in mathematics and it is also crucial in other compulsory modules like stochastics and applied mathematics. It is also an important ingredient of the upper cycle of secondary schools and therefore also important for school mathematics. There are numerous topics for bachelor and master's theses in analysis for students of the teacher training programme.

## Bachelor programme

In the bachelor programme of mathematics analysis is contained in the compulsory modules introduction to higher mathematics, analysis, advanced analysis, complex analysis, differential equations, topology, and functional analysis. In addition, it can be found in most of the compulsory and elective modules in applied mathematics.

• In the compulsory modules introduction to higher mathematics, analysis and advanced analysis the foundations of real analysis are laid, the basics of topology are explained and the differential and integral calculus for real functions of one and several variables are covered.
• The compulsory module complex analysis is devoted to line integrals and differentiable (holomorphic) complex-valued functions, culminating in Cauchy's integral theorem and its far-reaching implications.
• In the compulsory module differential equations basic theoretical results (existence and uniqueness theorems) for ordinary and partial differential equations are taught and elucidated with numerous examples.
• The compulsory module topology and functional analysis illustrates the abstract foundations of the preceding compulsory modules of analysis (foundations of topology and of infinite-dimensional normed vector spaces).

An abundance of possible topics for bachelor and master's theses arises from the lecture courses of these compulsory modules. Likewise the elective modules from the area of applied mathematics offer various topics that are related to analysis.

## Master programme

In the master programme  "Analysis" 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 area of specialization analysis consists of 4 compulsory modules: The module advanced functional analysis deals with the theory of locally convex vector spaces. Moreover a deeper representation of bounded and unbounded operators on Hilbert spaces is conveyed. Another lecture ourse in this module is devoted to Lebesgue integration theory and the foundations of Fourier analysis.
• The compulsory module advanced complex analysis deals with advanced topics of complex analysis of a single variables, culminating in Runge's approximation theorem and  the Riemann mapping theorem.
• In the theory of partial differential equations methods of functional analysis are taught for the approach of different aspects of differential equations.
• On the one hand, in the module "Seminars: Analysis" you have to complete the introductory seminar on one of the lecture courses advanced functional analysis, advanced complex analysis or theory of partial differential equations (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 analysis is various, a coordination of the seminars with the area of the master's thesis is advisable.

The offer of deepening courses for the master programme is closely linked to the research interests of the faculty members in this research area. It comprises lecture courses from the areas differential equations, functional analysis, complex analysis, distribution theory and generalized functions, harmonic analysis, global analysis, stochastic calculus, and variational calculus.

There are research groups pertaining to each compulsory module and they offer a solid basis for supervising 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.

## 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 "Analysis". An abundance of advanced lecture courses and seminars from the area of analysis is offered. 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 many research groups are devoted to analysis at our faculty and various research grants for doctoral students are offered.