# PhD course: Indefiniteness (First meeting)

**Time: **
Mon 2021-01-18 12.15

**Location: **
Zoom, meeting ID: 625 9741 7933

**Lecturer: **
Annemarie Luger

### Overview

When inner product spaces are considered in Linear Algebra or Functional Analysis, then these - by definition - are equipped with positive definite sesquiliear forms. In this context we are used to basic properties such as eg that a matrix, which is symmetric with respect to a (positive definite) inner product, has real eigenvalues only.

Interesting new phenomena appear when allowing inner products, not positive definit, but indefinit. This assumption has severe consequences on both the geometry of the space as well as on the spectral properties of linear transformations.

In the PhD-course we are exploring Pontryagin- and, more generally, Krein spaces and self-adjoint operators therein. We will study in particular spectral properties, ie

results corresponding to a spectral theorem and its limitations. Moreoever, we will touch on applications such as PT-symmetry and Nevanlinna-Pick problems.

The concepts are quite fundamental and hence the necessary prerequisites for the course are only knowledge equivalent to "Advanced Real analysis I".

**Course page:** https://kurser.math.su.se/course/view.php?id=982

Zoom meeting ID 625 9741 7933 (** for password see the coursepage or ask Annemarie**)

At this first meeting we fill decide about time for the lectures. The first lecure will then be given at the ordinary time.

*If you are interested in the course but cannot take part in the first meeting, please write me an email before, so that we can take your time restrictions into account.*