# Analysis and PDE seminar: Fall 2024

This is the page for the current semester of the seminars in Analysis and PDE at the University of Bergen. This semester seminars are held on Thursdays in the room Sigma at 12.15 until 13.45.

## Main content

Date | Speaker | Institution | Title |

05.09.2024 | Vegard Hansen | UiB | An Introduction to Geometric Measure Theory and Rectifiable Sets |

26.09.2024 | Erik Ivar Broman | Chalmers/University of Gothenburg, Sweden | Phase transitions of semi-scale invariant random fractals |

10.10.2024 | René Langøen | UiB | Curvature in the group of measure-preserving diffeomorphisms of the Klein bottle |

17.10.2024 | Didier Pilod | UiB | TBA |

31.10.2024 | Irina Markina | UiB | TBA |

14.10.2024 | UiB | ||

21.10.2024 | Torunn Stavland Jensen | UiB | TBA |

### Detailed entries with abstracts

## September 5, Vegard Hansen

**Date and time**: Thursday, September 5, at 14.15

**Place**: Aud. Sigma

**Speaker**: Vegard Hansen, Master student, Department of Mathematics, UiB

**Title**: An Introduction to Geometric Measure Theory and Rectifiable Sets

**Abstract**: Geometric measure theory is rougly speaking the study of geometric objects using the techniques of measure theory. Among the core concepts of this theory is the notion of rectifiable subsets. They provide a generalisation of manifolds to a class with a much less rigid structure. I will present the main ingredients in GMT, namely the hausdorff measures and Lipschitz maps. Using these we will define the recifiable sets, and discuss some of their properties.

-. . ...- . .-. --. --- -. -. .- --. .. ...- . -.-- --- ..- ..- .--. -. . ...- . .-. --. --- -. -. .- .-.. . - -.-- --- ..- -.. --- .-- -.

## September 26, Erik Ivar Borman

**Date and time**: Thursday, September 26, at 12.15

**Place**: Aud. Sigma

**Speaker**: Erik Broman, Senior Lecturer, Chalmers/University of Gothenburg, Sweden

**Title**: Phase transitions of semi-scale invariant random fractals

**Abstract**:

In all semi-scale invariant random fractal models, there is an

intensity parameter $\lambda>0$ of the underlying Poisson process which essentially determines

the nature of the resulting random fractal. As $\lambda$ varies, the models

undergo several phase transitions. One is when the fractal set transitions from containing

connected components, to the phase where it is almost surely totally disconnected.

Another is when the fractal transitions from being totally disconnected to disappearing

completely (i.e. it is empty). As we will explain, this is intimately connected to the classical

problem of covering a fixed set by other random sets (see for example the classical papers

by Dvoretsky or Shepp).

In the talk we will present results concerning both of these phase transitions. In particular,

the results include determination of the exact value of the parameter $\lambda$ at which

the second transition mentioned occurs. Furthermore, we are able to determine the behavior of the

fractal sets at the critical points of both of these phase transitions.

The talk will be non-technical and is aimed at a broad audience.

-. . ...- . .-. --. --- -. -. .- --. .. ...- . -.-- --- ..- ..- .--. -. . ...- . .-. --. --- -. -. .- .-.. . - -.-- --- ..- -.. --- .-- -.

## October 10, René Langøen

**Date and time**: Thursday, October 10, at 12.15

**Place**: Aud. Sigma

**Speaker**: René Langøen, Phd. student @ Department of Mathematics, UiB

**Title**: Curvature in the group of measure-preserving diffeomorphisms of the Klein bottle

**Abstract**: In this talk I introduce the diffeomorphism group of a manifold, with special focus on the diffeomorphism groups of the torus and the Klein bottle. The Lie algebra of a diffeomorphism group of a manifold is given by the vector fields on the manifold. The torus is a double orientation cover of the Klein bottle implying a direct relation between vector fields on the torus and the Klein bottle. We use this relation to calculate curvature in the diffeomorphism group of the Klein bottle, in particular, we calculate sectional curvature and an infinite dimensional version of the Ricci curvature. The talk is based on recent work with Boris Khesin (University of Toronto) and Irina Markina (UiB).

-. . ...- . .-. --. --- -. -. .- --. .. ...- . -.-- --- ..- ..- .--. -. . ...- . .-. --. --- -. -. .- .-.. . - -.-- --- ..- -.. --- .-- -.