Dade's Conjecture for finite reductive groups

• Applicant: Dr. Damiano Rossi
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 525464727

The effect of symmetries on our perception of the world is ubiquitous in our daily life: from the recognition of human emotions through facial expressions to the appreciation of art and architecture. Simply put, our brains process nature by means of symmetries. Mathematicians have developed a rigorous and elegant framework, known as group theory, to better grasp the natural idea of symmetry. Besides its importance within pure mathematics, group theory has applications to physics, genetics, chemistry, materials science, crystallography and cryptography. Unfortunately, groups are complicated in nature and it is therefore extremely challenging to describe their features. To circumvent this difficulty, it is customary to study groups through representations provided by their actions on certain geometric spaces. Hence, it is fundamental to understand group representation theory in depth. At present, the main challenge in this research area is to provide an explanation for the local-global principle. To illustrate this idea, recall that natural numbers can be decomposed as a product of prime powers. Similarly, finite groups can be decomposed into local structures associated with prime numbers. According to the local-global principle, the representation theory of a given finite group is strongly determined by the representation theory of such local structures. Dade’s conjecture lies at the heart of this problem and provides fundamental evidence for the local-global principle by suggesting a precise formula for determining the number of certain types of representations in terms of invariants associated with these local structures. The aim of this project is to introduce a novel approach to the study of Dade's conjecture by exploiting powerful algebro-geometric techniques and by further developing the framework of Deligne--Lusztig theory. These ideas will be applied to settle a geometric interpretation of Dade's conjecture for the important family of finite reductive groups. This research project is part of a long-term plan to settle Dade's conjecture and will bring us closer to understanding the local-global principle.

• DFG Programme: WBP Position