University of Leipzig | Faculty | Institute for Computer Science | Mathematical Institute | MPI für Mathematik in den Naturwissenschaften | Math-Net-Guide

University of Leipzig
Faculty of Mathematics and Computer Sciences
Mathematical Institute

Graduate College- Analysis, Geometry and their Interaction with the Natural Sciences

Lectures, Seminars, Studies

The aim of the study program at the Graduate College is to provide the Doctor students with a good understanding of the various methods and applications of analysis and geometry within the scope of the scientific program. The study program is supposed to counteract an early specialization and enable the students to see their work in a broader context and to get familiar with the research topics of other groups involved. We assume that the doctoral degree can be completed within three years, which can be achieved by an intensive interaction between the student and the professors. Certainly the Doctor students will discuss their mathematical questions not only with their advisor but also with other professors. In intervals the Doctor students report on their work in the colloquium of the Graduate College or in working groups and thus get in contact with professors and other fellows. The study program shall also take into account that the majority of Doctor students have not done graduate studies in Leipzig. Therefore in particular the initial phase is intended to provide the fellows with a common basis of mathematical understanding. To become a fellow of the Graduate College the candicates should have mathematical knowledge including the contents of the following courses:

 Partial differential equations, differential geometry, numerics, algebra, functional analysis.

However the Doctoral students have the opportunity to obtain this knowledge by visiting courses offered by the mathematical institute. The following events associated with the Graduate College shall be performed:

I. Colloquium of the Graduate College

The Colloquium of the Graduate College is the central joint event of the Graduate College. It takes place weekly during the lecture period. Guests will be invited as speakers to report about current developments in the fields of the research of the Graduate College. On the other hand Doctor students and postdocs report on their work. After the colloquium there is time for discussion about the lecture and other questions of scientific interest or organization.

II. Lectures and working groups

The first year in the Graduate College is intended as an initial phase. Two basic lectures of four hours a week are offered, which introduce analytical and geometric methods used in the research projects. A special emphasis is laid on the interaction with the sciences. The participation in one of these lectures is obligatory for the Doctor students. It is strongly recommended to take part also in the other lecture. In addition the Doctor students choose one or two of the offered lectures of two hours a week.

The second year is the specialization phase, in which the Doctor students participate in one of the two hour lectures and at least one working group. The Doctor students should be advised by their supervisors while choosing the appropriate courses.
In the fifth and sixth semester the dissertation shall be completed. The participation in a working group during this period is strongly recommended . These working groups are guided by a professor associated with the Graduate College. The Doctor students can make suggestions for the choice of the subjects, which should be closely related to current research topics pursued in the Graduate College. Also the participation of Diploma students, guests and collaborators of the university and the MPI is greatly desired.

Lectures in the initial phase (4h per week)

• Modern aspects of analysis with applications in the sciences

• Modern aspects of geometry with applications in physics

Suggested topics for lectures in the initial phase (2h per week)

• Geometric analysis

• Neural networks

• Dynamical systems

• Differential topology

• "Steilkurs" algebraic geometry

Suggested topics for lectures in the initial phase (2h per week)

• Variational problems in quantum field theory

• Mathematical models of cognition

• Information theory

• Dynamical systems

• Moduli spaces in algebraic, complex and symplectic geometry

• Asymptotic analysis

• Mathematical models of general relativity

• Conformal geometry and field equations

• Global Riemannian geometry

• Hodge theory

• Hamiltonian dynamical systemes

• Intersection theory

• Capillary surfaces

III. Guest scientists

International leading guest researchers shall be invited for talks and lectures. The suggestions of the Doctor students for inviting particular guests are welcome.

 IV. Summer schools

Lecture series and summer schools for the Doctor students are planned. The fellows of the Graduate College will be involved in the planning process. One postdoctoral member of the Graduate College will organize these events. Graduate and Doctoral students from other universities are welcome.

Suggested topics

• Capillarity - analytic and geometric methods

• Mathematical modeling of complex materials

• Supersymmetry

• Hodge theory

V. Planning meetings

At the beginning or the end of a semester a planning meeting will take place in order to discuss the program of the upcoming semester and to collect suggestions for working groups and guests to be invited.

VI. Phase of Qualification

Candidates holding a bachelor degree or a diploma, yet not having written a diploma thesis, may be offered a qualification grant. Within one year, attending lectures of the graduate program and writing a short thesis, the candidate shall be enabled to take up postgraduate studies within the scope of the Graduate College.

H.B.Rademacher, 2007-03-10