250129 VO Topics from Habilitations (2024W)

Minicourse by Kamran Sadiq: Introduction to the Mathematics of X-Ray Imaging

One of the main topic of this minicourse is the study of the X-ray transform arising in medical imaging, in particular in Computerized Tomography (CT), and many other fields. The X-ray tomography is one of the basic inverse problems and consists of determining the density of tissue by measuring the attenuation of X-rays passing through the body. The measurements are modeled by the X-ray transform and the inverse problem is to invert this transform. This mini-course will present in detail the essential properties of the transform, including some inversion formulas.

Minicourse by Shokhrukh Kholmatov: Minimizing movements for mean curvature flow

The minimizing movements approach was
originally proposed by Almgren, Taylor, and Wang in the
1990s to study weak mean curvature evolution of sets.
Today, this concept is applicable to a wide range of
evolution problems, including ordinary differential equations,
parabolic and hyperbolic partial differential equations,
differential inclusions, and gradient flows in metric spaces.
In this mini-course, we will briefly explore these applications,
with the core focus on its application to mean curvature flow.
We will discuss various properties of minimizing movements
solutions that align with the smooth mean curvature flow,
and explore potential generalizations.

Minicourse by Damian Sobota: On complemented copies of the space c_0 in Banach spaces C(K)

In this minicourse we will study the issue of the existence
of complemented copies of the standard Banach space c_0 in the Banach
spaces C(K) of continuous real-valued functions on compact spaces K.
We will prove among others a theorem of Cembranos and Schachermayer
stating that a space C(K) contains such a copy if and only if there is
a weak* null sequence of Radon measures on K which is not weakly null.
Consequently, we will get that for every metrizable compact space K
the space C(K) contains a complemented copy of c_0. On the other hand,
we will also prove a celebrated theorem due to Grothendieck asserting
that for an extremally disconnected compact space K the Banach space
C(K) does not contain any such copy. The minicourse is dedicated to
all students who would like to deepen their knowledge on the structure
of Banach spaces of continuous functions.