**Elena Dematte: On some properties of the radiative transfer equation**

Abstract: The radiative transfer equation is the kinetic equation which describes the distribution of energy of photons, which can be absorbed, emitted and scattered by a material. In particular, the radiative transfer equation is used in order to describe the transfer of heat in a medium due to radiative processes. It plays an important role in astrophysics. In this talk we will first introduce the radiative transfer equation in its general form. Afterwards we will focus on the stationary equation and we will show that this can be reduced to a non-local integral equation for the temperature.

**Luise Puhlmann: Planning routes without (exactly) knowing where to go -- improved guarantees for the a priori TSP**

Abstract: Imagine a small delivery service with a fixed set of customers that drives daily delivery tours. However, not every customer needs a delivery every day. So each day we need a tour visiting a subset of the customers. In order to save cost and time, we want to optimize the roundtrip starting at the depot, visiting all customers that need a delivery that day, and returning to the depot. However, the information which customers have to be visited might only be available on very short notice; thus it might be too late to compute a shortest tour for that day. Furthermore, the driver might actually prefer driving a similar tour each day. Therefore, instead of computing daily tours, we want to fix an ordering of all the customers in which they have to be visited; and if one customer doesn't need a visit one day, they are skipped and the tour takes a shortcut to the next customer in the order. Then of course we want the resulting tours each day to be preferably short.

The a priori TSP gives a mathematical model of this problem: Given a set of customers, pairwise distances between these customers, and activation probabilities for each customer, compute a round-trip tour through all the customers that is in expectation as short as possible when cut short to the active customers. We consider approximation algorithms for this problem and prove better guarantees than previously known. This is joint work with Jannis Blauth, Meike Neuwohner, and Jens Vygen. (\url{https://arxiv.org/pdf/2309.10663.pdf})

**Leonie Scherer: Generators and Relations of the Étale Fundamental Group**

Abstract: The fundamental group of a topological space is the group of equivalence classes under homotopy of loops.

It classifies covers of "nice'' topological spaces.

In this talk, we will take a look at the étale fundamental group of a scheme which classifies finite étale covers. For example, the étale fundamental group of a field is its absolute Galois group, it classifies finite separable field extensions.

We will investigate the number of topological generators and relations of the étale fundamental group of a connected, smooth, projective curve over a field.

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