# Geometry and Topology - Master programme and Doctoral Programm

This page collects the most important information about the specialization "Geometry and Topology", for Master's und doctoral programs. 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 the (current) study program. In addition, you can find a list of possible supervisors and lists of examples of topics for Bachelor's, Master's, and doctoral theses in the area of geometry and topology.

## 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 D
*ifferential 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 A
*lgebraic 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.

## Supervisors:

### Andreas Cap: Differential geometry

#### Short CV

#### Short CV

- since 2011: Professor of Mathematics at the University of Vienna
- 1991: PhD University of Mathematics

#### Research Activities

I am interested in the study of geometric structures and of differential operators naturally associated to such structures. My main focus is the class of parabolic geometries that contains examples like conformal structures and CR structures and many others. Crucial input for the study of these structures is provided by theory of Lie groups and Lie algebras and their representations, so my research involves a mix of geometric, algebraic, and analytic techniques. I am particularly interested in applications of these geometric structures to other areas of mathematics and of theoretical phyiscs, for example to complex analysis, the geometric theory of differential equations, and general relativity. For more information, please visit my homepage .

### Herwig Hauser: Algebraic Geometry and Singularity Theory

#### Short CV

#### Short CV

- 1956 Born in Innsbruck, Austria. Austrian citizen. Married, three children
- 1974 - 80 Studies at the Universities of Innsbruck and Paris
- 1980 - PhD in Paris with Teissier and Douady on the semi-universal deformation of singularities
- 1980 - 88 Assistent professor at the University of Innsbruck
- 1982 - 84 Visiting Brandeis and Northeastern University
- 1988 - Habilitation
- 1988 - 2007 Associate Professor at the University of Innsbruck
- 1992 - 94 Visiting Professor at the Universidad Autónoma de Madrid
- 2007 - present Professor at the University of Vienna, on extended leave from the University of Innsbruck
- 2015 Chaire Jean Morlet, Aix-Marseille University und CIRM, Spring 2015

#### Research Activities

My field of interest is the interplay between geometry and algebra, more precisely, algebraic geometry and commutative algebra. Within this area, we look at singularities (points where varieties do not look like a manifold) and at infinite dimensional geometry (for instance, spaces of power series solutions to algebraic equations). Apart from this, we study algebraic functions and power series, a field which has strong connections to combinatorics (lattice walks), commutative algebra (Henselization and étale topology) and complex analysis (holomorphic functions). Through our FWF-research project we are able to support interested students in their Master's or PhD studies. For more information, see my homepage.

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

#### 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

- 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 Interests

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

- 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

- 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

- 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.