# Analysis - Master programme and Doctoral programme

## Master's program

In the Master's program, "Analysis" 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 "Analysis" consists of the following 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 course 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 attendance 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's program 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. 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 "Analysis". An abundance of advanced lecture courses and seminars from the area of analysis is offered. 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 many research groups are devoted to analysis at our faculty and various research grants for doctoral students are offered.

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.

## Supervisors:

### Olivia Constantin

#### Short CV

#### Short CV

- 2016-present: Associate Professor, University of Vienna
- 2012-2014: Lecturer (permanent), University of Kent at Canterbury, UK
- 2008-2012: Universitätsassistentin, University of Vienna
- 2006-2008: Lecturer (non-permanent), Trinity College Dublin, Ireland
- 2005: Ph.D. University of Lund, Sweden

### Roland Donninger: Dispersive Partial Differential Equations

#### Research Activities

There is a fruitful interplay between Operator Theory and Complex Function Theory, the latter one being used to model Hilbert space operators. Operators play a central role in several branches of physics and engineering. For instance, modern operator theory initially developed as the natural language of quantum mechanics, and, in particular, many operators from quantum mechanics have useful realizations on spaces of analytic functions.

My research interests concern the study of various classes of operators acting on scalar or vector-valued spaces of analytic functions. While the scalar case has been extensively studied and developed, the vector-valued framework gained prominence more recently, since it offers new challenges as well as new insights, as many classical problems become very complicated in this setting and require a completely novel approach.

#### Short CV

#### Short CV

- as of 2018: Associate Professor, University of Vienna
- 2017-2018: Assistant Professor, University of Vienna
- 2014-2017: Research Group Leader, University of Bonn
- 2014: Sofia Kovalevskaja-Award
- 2011-2014: Postdoc, EPF Lausanne
- 2009-2010: Postdoc, University of Chicago
- 2008: PhD, University of Vienna

#### Research Activities

My field of research is the theory of dispersive partial differential equations (PDEs). This is a subfield of modern analysis with many crosslinks to mathematical physics and general relativity. During the last 40 years the field has become a cornerstone of modern PDE theory. In particular, two Fields Medals (Jean Bourgain and Terence Tao) have been awarded for contributions to dispersive PDEs. My research group is mainly focusing on the rigorous analysis of nonlinear wave and Schrödinger equations, wave maps, Yang-Mills models, and other related evolution problems arising in physics and/or geometry. We study these equations by developing new tools based on spectral theory, harmonic analysis, nonlinear functional analysis, and rigorous computer-assisted methods. For more information please visit my homepage.

### Günther Hörmann: Hyperbolic (pseudo)differential equations with non-smooth data and/or coefficients

#### Short CV

#### Short CV

- since 2004: a.o.Univ.Prof., University of Vienna
- 2001-2004: PostDoc, University of Innsbruck
- 1999-2001: PotDoc, Colorado School of Mines
- 1994-1999: computer system administrator, University of Vienna
- 1993: Dr.rer.nat., University of Vienna Research

#### Research Activities

My main interest is in the analysis of partial (or pseudo-) differential equations with a strong motivation from models of wave propagation in physics, geophysics or mechanics. I am studying non-classical solution concepts required due to singularities (in the sense of non-smoothness) typically with methods based on functional and Fourier analysis.

### Michael Kunzinger: Analysis and Geometry in Low Regularity

#### Short CV

#### Short CV

- Since 2001: Associate Professor, University of Vienna
- 2004: START-Prize
- 2003: ÖMG-Prize
- 2001: Visiting Scientist, University of Southampton
- 1996: PhD, University of Vienna
- Study of Mathematics and Physics, University of Vienna

#### Research Activities

The overarching theme in my mathematical research interests is the connection to the fundamental theories of physics. Ultimately, my goal is to gain a deeper understanding of the mathematical foundations of physics and to contribute in my research to furthering this understanding. This has driven the choice of my fields of interest towards Functional Analysis, Differential Geometry, and Partial Differential Equations. An additional unifying theme is my interest in low regularity situations and in conceptual developments that help to extend the validity of analytic or geometric approaches beyond the smooth setting. Concretely, I work on linear and nonlinear theories of generalized functions, geometric analysis, Lorentzian geometry, and applications to the study of singularities in General Relativity. For more information, see my homepage.

### Bernhard Lamel: Complex Analysis

#### Short CV

#### Short CV

- as of 2018: Professor of Complex Analysis at University of Vienna
- 2007: START-Prize
- 2001-2002: J.L. Doob Research Assistant Professor UIUC
- 2000-2001: Postdoc KTH
- 2000: Ph.D. University of California, San Diego

#### Research Activities

In the analysis of several complex variables, the geometrical properties of boundaries of domains and the analytical properties of the domains themselves play a crucial role. My main interest is in the interplay between the geometry of real objects in complex space and properties of functions and maps which satisfy the "tangential Cauchy-Riemann equations", usually dubbed as "CR-geometry". The theory has many connections to the theory of PDEs, algebraic geometry, dynamical systems. For a list of current research projects, papers, and so on, please consult my homepage.

### Roland Steinbauer: Nonregular spacetime geometry

#### Short CV

#### Short CV

- as of 2003: Associated Professor, University of Vienna
- 2016: Ars Docendi (Austrian National Award in University Teaching)
- 2000: PhD (Mathematics, sub auspiciis praesidentis), University of Vienna
- 1996: MSc (Physics), University of Vienna

#### Research Activities

My current research interests are focused on issues of regularity in geometry and questions such as: How does low differentiability of basic quantities such as e.g. a (semi-)Riemannian metric change the geometric properties that usually hold in the "smooth world". Main motivations come from Mathematical General Relativity especially the singularity theorems and exact solutions such as impulsive gravitational waves. There are strong connections with analysis (generalised functions), metric geometry (Alexandrov spaces) and, of course, mathematical physics. For further information please visit my homepage.

### Gerald Teschl: Spectral Theory and Mathematical Physics

#### Short CV

#### Short CV

- currently: Professor, University of Vienna
- 2006: START Prize of the Austrian Science Fund (FWF)
- 1999: Prize of the Austrian Mathematical Society
- 1997: Ludwig Boltzmann Prize of the Austrian Physical Society
- 1996-1997: Postdoc, Department of Mathematics, RWTH Aachen, Germany
- 1995: PhD University of Missouri, Columbia, US

#### Research Activities

My current research area concerns mathematical physics and dynamical systems. I am interested in direct and inverse spectral theory and connections with completely integrable nonlinear wave equations (soliton equations). For more information please visit my homepage.