Within the research group MaDs we tackle both conceptual problems in mathematics as well as concrete problems in the mentioned application domains, by translating their data or system behavior into suitable abstract representations, continuous and/or discrete. We exploit the richness of these concepts and their theories and develop new ones within discrete mathematics, functional analysis, category theory, geometry and probability theory, also combined with machine learning, to further model and solve problems. This upfront modelling approach improves in particular the understanding of the underlying mechanisms and interpretability of predictive data science models, especially when using the current wave of powerful machine learning architectures, such as deep learning, transfer learning, graph neural networks, generative models and attention mechanisms.

We work in particular on:

- Discrete mathematics with focus on graphs, algebraic combinatorics, groups, rings and skew braces;
- Functional analysis, including operator algebras, partial differential equations, harmonic analysis (in particular, Gabor, wavelet, and shearlet frames);
- Category theory in the context of representation theory, abstract approximation theory and topology;
- Geometry, including finite and differential approaches;
- Statistics and stochastic processes.

Specific application domains include, but are not limited to:

- Coding theory & Cryptography, in particular post-quantum;
- Computer vision, also in combination with Natural Language Processing;
- Mathematical foundations of Data Science, in particular explainable Artificial Intelligence.

## Organised by PI

## Prof. Andreas DEBROUWERE

###### Yarne Tranoy, Michiel Huttener (joint with UGent)

Our research specializes in functional analysis. We are particularly interested in applying tools from the theory of locally convex spaces (e.g. duality theory, the derived projective limit functor, splitting theory) to solve various concrete problems in analysis (e.g. surjectivity of partial differential operators and moment type problems). Furthermore, we also study function and distribution spaces from the point of view of time-frequency analysis (e.g. via the short-time Fourier transform, Gabor frames, and wavelet frames).

## Prof. Jan DE BEULE

###### Dr. Sam MATTHEUS , Sam ADRIAENSEN, Jonathan MANNAERT, Jim WITTEBOL, Leen DEMUYS

On a fundamental level, we mainly investigate structures from finite geometry, with or without using (algebraic) graph theory. On the applied side, we study applications of these structures in data science, more specifically, in coding theory and cryptography.

## Prof. Kenny DE COMMER

###### Joeri De Ro, Joel Right Dzokou Talla, Dr. Geoffrey Janssens, Jacek Krajczok

Our research specializes in different branches of representation theory through analytic, algebraic and categorical techniques. A major focal point is the theory of quantum groups, with a strong connection to the theory of operator algebras (C*-algebras, von Neumann algebras). Various generalizations of these concepts are explored by categorical methods (tensor categories, module categories, ...) These structures provide novel insights into, as well as vast generalizations of the classical notions of space and symmetry, and are of fundamental importance both in pure mathematics and in physics (quantum mechanics, quantum field theory, ...).

## Prof. Ann DOOMS & Prof. Tan LU

###### Geert Braeckman, Finian CAREY, Leen DEMUYS, Carlo EMERENCIA, Lukas GUTBRUNNER, Tanya KLOWDEN (joint with Courtauld), Willy Carlos TCHUITCHEU, Jos VAN DOORSSELAER, Dr. David VERRIER

The team is specialised in *Digital Mathematics*, DIMA for short, which deals with the representation, manipulation, and analysis of mathematical objects using digital computers. The word "digital" is derived from the Latin word "digitus," which means finger. The term was initially used to describe numerical systems based on counting with fingers, which is a natural and intuitive way of representing countable numbers. Over time, the meaning of "digital" evolved to refer to anything related to discrete quantities or representations, as opposed to continuous ones. The team focusses on the mathematical foundations for digital data science, including pattern recognition, computer vision, Natural Language Processing, machine learning and information forensics & security with applications in, amongst others, document processing, painting analysis, medical imaging, explainable AI and post-quantum cryptography.

We are currently setting up the spin-off Fleuron on automating document processing by imitating human layout recognition.

The DIMA team is affiliated with the VUB __AI Lab__ and the __Royal Library of Belgium__ through the __KBR Data Science Lab__.

## Prof. Mark SIOEN

###### Prof. Dr. Em. Eva COLEBUNDERS, Wouter VAN DEN HAUTE, Tomas EVERAERT

We investigate questions in general topology using methods form category theory and universal algebra, more specifically pointfree topology (i.e. the theory of frames/locales) and approach theory (a suitable context for quantitative topology) and their interactions with a.o. analysis and algebra.

## Prof. David TEWODROSE

###### Nicolo’ Cont, Manuel Dias, Dr. Susovan Pal

The team specializes in geometric analysis. The focus is made on the interplay between curvature conditions and spectral analysis on smooth Riemannian manifolds, degenerate Gromov-Hausdorff limits or more general singular metric measure spaces. One special interest of the team is the notion of Ricci curvature which governs in various ways the second-order analysis that can be developed on a space. Applications in manifold learning are also deeply investigated.

## Prof. Leandro VENDRAMIN

###### Dr. Arne Van Antwerpen, Dr. Carsten Dietzel, Dr. Kevin Piterman, Silvia Properzi, Thomas Letourmy, Senne Trappeniers, Davide Ferri, Charlotte Roelandts

The team specializes in quantum symmetries. Symmetry, as formalized by group theory, is ubiquitous across mathematics and science. In quantum settings, the notion of a group is no longer enough to capture all symmetries. The language of tensor categories provides a unified framework to discuss these notions of quantum symmetry. Within the framework of studying the various guises of quantum symmetries and their interactions, we focus on the following areas: tensor categories and representation theory, pointed Hopf algebras and their actions on rings, knots, and 3-manifolds, and algebraic structures related to the Yang-Baxter equation.