## Programme

The event takes place **online**.

Time | Title | Speaker(s) |
---|---|---|

14h00 - 14h05 | ||

14h05 - 14h30 | Sparse exponential analysis | |

14h30 - 14h55 | Algebras with involution | |

14h55 - 15h25 | Women in science | |

15h25 - 15h40 | Reinsurance pricing | dr. Tim Janssens |

15h40 - 16h10 | Virtual coffee break | |

16h10 - 16h25 | Data Science: what it is and what are today's challenges. | dr. Matthias Roels |

16h25 - 17h10 (Pitch presentations) | The geometry of classical mechanics | |

Linearized scaffolds of spaces: an invitation to homological algebra | ||

How much information can we recover from a space if we ignore its points? | ||

Regularization of ill-conditioned linear systems | ||

No spins attached? | ||

17h10 - 17h35 | The N-body problem of celestial mechanics | |

17h35 - 17h40 | Closing remarks | |

17h40 - 18h10 | Discussion with speakers |

## Abstracts

## Sparse exponential analysis (Ferre Knaepkens)

Exponential analysis is termed an inverse problem because it consists in extracting an exponential model’s linear and nonlinear parameters from a limited number of observations of the model’s behaviour.

Exponential analysis may sound remote but touches our lives in many ways: exponentials with real exponents appear in chemical reactions and physical phenomena, while signal processing is all about exponentials with complex exponents. For this presentation, we focus on direction of arrival (DOA) estimation. Here, a uniform linear antenna array receives some signals from several sources and the objective is to obtain the angles between the sources and the antenna array.

The problem of DOA naturally leads to two additional questions, namely, how to estimate the number of signals and how to avoid mutual coupling. To formulate the answers to both questions, we take the journey from exponential analysis to Padé approximation theory and sub-Nyquist sampling.

## Algebras with involution (Fatma Kader Bingöl)

Quaternion algebras are the first non-trivial examples appearing in the literature of central simple algebras defined over a field that carry an involution. They have a very simple description (as a 4-dimensional vector space with a basis subject to some rules) and allow us to construct new examples of such algebras.

We will discuss these examples and the question whether the use of quaternion algebras is the only way to construct central simple algebras carrying an involution.

## Women in science (Prof. dr. Katrien Kolenbergs & Sarah Ahannach)

**Sarah Ahannach**: While an approximately equivalent numbers of women and men start their careers in science even in liberal societies far less women make up the top level scientists. Throughout their career various obstacles referred to as the "leaky pipeline" require attention. Next to many societal restructuring, positive depictions and representations of women in STEM as role models can help to inspire girls around the world to try out STEM-activities. This will hopefully encourage them to stick with it into their post-secondary education and beyond.

## Reinsurance pricing (dr. Tim Janssens)

An insurance company can protect itself from large loss accumulations, e.g. natural catastrophes, by ceding risk to reinsurers. The precise nature of the cession is determined by both the risk appetite of the insurer and the premium that has to be paid to the reinsurer for the coverage. One therefore has to predict future losses, which is particularly challenging for long tail business (e.g. liability business). The aim of this talk is to give a brief introduction to the principles of reinsurance and to explain a possible approach to parametrising liability risks.

## Data Science: what it is and what are today's challenges. (dr. Matthias Roels)

In this talk, we dive into the field of data science. First, we will start by explaining what it is and how it is related to machine learning. Next, we will go through the typical use-cases a data scientist encounters in a company and try to situate data science within a company strategy. Finally, we will discuss some of today's challenges a data science team faces when moving beyond a proof-of-concept in a project.

## The geometry of classical mechanics (dr. Joseph Palmer)

Tie a ball to a string, hold the string up, and give the ball a push. Can you predict how the ball will swing? The field of classical mechanics is concerned with studying the behavior of physical systems, such as this pendulum. It turns out that to understand the way this pendulum moves, we instead have to look at a certain 4-dimensional geometric space called the phase space of the system. We will discuss this specific example, and more generally how questions about classical physical systems can be rephrased into questions about the geometric properties of their phase spaces. This relationship builds a bridge between concrete physically motivated questions and problems in (often high dimensional) geometry.

## Linearized scaffolds of spaces: an invitation to homological algebra (dr. Francesco Genovese)

It is common to treat complicated mathematical objects by reducing them to something which is "linear", so that one can use the powerful tools of linear algebra - think of differentials of functions, for example. In this short talk I will show you that even topological spaces can be in a certain way "linearized" and reduced to nice algebraic gadgets called *chain complexes*, which we can conveniently manipulate to extract a useful invariant - namely, homology.

## How much information can we recover from a space if we ignore its points? (Wouter Van Den Haute)

Since a point is an entity with a position but no extent, it makes sense to only study (small) spots and their interaction. To this end, we introduce the theory of frames, which only looks at the open sets of a space. This is done in an order theoretic and hence algebraic way. We show a way to reconstruct the original space and how some classical properties can be translated to properties of frames.

## Regularization of ill-conditioned linear systems (Jeffrey Cornelis)

Solving a linear system of equations is one of the most fundamental building blocks in applied linear algebra. The data that you obtain in practice is often contaminated with measurement errors or noise. This means that simply using one of the well-established techniques such as Gaussian elimination will result in a very bad and noisy solution. In this talk we explain why this is the case and what techniques we can use to obtain some meaningful solution.

## No spins attached? (Güner Muarem)

Solving equations is maybe one of the most prior tasks of a mathematician. An important class consists of the differential equations and related differential operators which mostly describe some kind of physical object. Think of Newton’s equation, describing the motion of a particle in classical mechanics or the Dirac equation, which describes the movement of an electron. From a mathematical point of view, a lot of these equations are difficult to solve exactly. Therefore, it can be useful to check whether the system has any symmetry to reduce the problem in a way. These symmetries then determine an algebraic structure which allows a representation theoretical approach, opening the doors to ‘special’ functions.

## The N-body problem of celestial mechanics (dr. Marine Fontaine)

The N-body problem is a second-ordered system of differential equations that govern the motion of N celestial bodies of given masses moving in space.

For N=2, this problem was solved completely by Bernoulli in 1710. However, Poincaré showed that for N=3, the equations cannot be solved by the standard methods of reduction using first integrals. This problem has survived more than 300 years and mathematicians are still trying to construct new meaningful solutions for N ≥ 3. I will discuss some solutions of the N-body problem, namely homographic solutions, relative equilibria and braids, as well as related open questions attached to them.