Title Promoter Affiliations Abstract "Structurable algebras, representation theory and related point-line geometries" "Tom De Medts" "Department of Mathematics: Algebra and Geometry" "The goal of the project is to investigate connections between algebraic structures (linear algebraic groups, Jordan pairs, structurable algebras, Lie algebras) and geometric structures (especially point-line geometries, but also so-called root filtration spaces). We will often rely on the representation theory of the underlying groups." "Buildings of exceptional type: their geometries and their representations." "Hendrik Van Maldeghem" "Department of Mathematics: Algebra and Geometry" "My project is about buildings, abstract geometric objects introduced to get a grip on certain classes of groups. Classical examples are projective and polar spaces. Exceptional examples are buildings of type G2, F4, E6, E7 and E8, widely studied in mathematics and physics.  The above buildings are the spherical buildings; a second important class is given by the affine buildings. Overall, my project contributes to the study of (exceptional) spherical buildings and the related affine ones. As a geometer, I work with point-line geometries naturally associated to buildings. For spherical buildings, these geometries are parapolar spaces. This general geometric framework allows to extract more properties. My first goal is to extend the definition of parapolar spaces to include affine buildings in this theory. I also want to classify some interesting families of parapolar spaces. A second goal consists of classifying inclusions of buildings in exceptional buildings via parapolar spaces, yielding valuable information on the buildings. The ultimate goal in the theory of spherical buildings is to understand E8. A means for this is given by the Magic Square. From left to right and top to bottom, this 4x4 array contains increasingly intricate, but consecutively related, classical and exceptional buildings, culminating in E8. Unraveling this square is a long-term project in which my third goal fits: studying dualized projective representations of its third row." "Stability and singular foliations" "Marco Zambon" Geometry "This proposal addresses questions in geometry using some tools from algebra.The first part of the project involves geometric objects with singularities, called singular foliations and singular subalgebroids; think of a field of arrows of the plane, which at certain points degenerate to have length zero. It is known that when no singularities are present, such an object has a global counterpart, which is a smooth object that describes how one can flow along the field of arrows. We address properties of the global counterpart when singularities are present.The second and most important part of the proposal addresses stability questions, in a similar setting, involving objects called Lie algebroids. One such question is: given a field of lines on the plane and a closed curve (a circle) tangent to the lines, is it true that for any small perturbation of the field of lines one still has a closed curve tangent to the field of lines? When this happens, one says that the circle is stable." "Reductions, deformations & resolutions in representation theory and noncommutative geometry" "Theo Raedschelders" Mathematics "Quantum mechanics is a staple of 20th century science, and has led to the realisation that  physical quantities are governed by noncommutative algebra. More precisely, Werner Heisenberg replaced classical mechanics, where observable quantities pairwise commute, with matrix mechanics, where crucial observables like position and momentum no longer commute with each other. To study quantum mechanics, it is therefore natural to also try and extend the classical geometry of points, lines, planes etc. to the noncommutative world. This gives rise to the mathematical field of noncommutative geometry. Later on, the mathematician Hermann Weyl realised that the operators corresponding to position and momentum satisfied relations that occurred in another area of mathematics called representation theory, which studies the ""symmetries"" of abstract mathematical objects.  In this project we analyse several spaces appearing in (noncommutative) geometry by looking at their symmetries and deformations, and use representation theory to say something new about them. The fundamental idea, which goes back to Alexander Grothendieck, is to associate to a possibly noncommutative space an algebraic invariant (its derived category), which is rich enough to capture a lot of the geometry of the space while at the same time being sufficiently flexible, moving the focus from geometry to a more algebraic point of view." "Hensel minimality" "Wim Veys" Algebra "Studying rational points on algebraic varieties is one of the most fundamental topics in algebraic geometry and number theory. The proposed project fits into this research topic. For a rational number q=a/b with a and b coprime, we define the height of q to be the maximum of |a| and |b|. In some sense, this measures the complexity of the number q. For a tuple (q_1, ..., q_n) of rational numbers, the height is defined to be the maximum of the heights of the q_i. For a polynomial equation f(x_1, ..., x_n) = 0 in several variables, we are interested in upper bounds for the number of solutions of height at most B, say. This line of research was initiated by Bombieri and Pila and further developed by Heath-Brown, Salberger and Walsh. It has since found many applications within algebraic geometry and number theory.Instead of working with rational numbers, one could also work with rational function fields in one variable over a finite field, F_q(t), and ask the same type of question. There are many analogies between these two fields, and theorems in one setting often have a counterpart in the other. Therefore, it seems believable that these results on upper bounds on points of bounded height in characteristic 0 can be extended to positive characteristic as well. The project will study these questions and applications to cryptography in more detail." "Rational points on varieties over discretely valued fields." "Johannes Nicaise" Algebra "The general theme of this thesis is the interaction between the behaviour of rational points on certain classes of algebraic varieties over global and local fields, and the étale cohomology of these varieties.Part I studies cohomological obstructions to the validity of local-global principles (such as the Hasse principle and weak approximation)coming from the Brauer group of a variety. The BrauerManin obstructionhas become an essential tool to understand when such local-global principles can (or cannot) hold. Our contribution to this line of research istwofold. In a first piece of work, we investigate families of torsors under a constant torus defined over a number field using different methods (such as the descent method and the fibration method), thereby provingthat the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one for certain families. In a second piece of work, we construct new examples of varieties for which the étale Brauer-Manin obstruction (a refinement of the classical Brauer-Manin obstruction) does not suffice to explain the failure of the Hasse principle. We construct the first such examples with trivial Albanese variety; assumingthe abc conjecture, we even construct examples which are geometrically simply connected.In Part II, we investigate the relationship between the rational volume of a smooth, projective variety defined over a strictly local field, and the trace of the tame monodromy operator on the étale cohomology of this variety. The motivation is work of NicaiseSebag on a trace formula for the motivic Serre invariant, inspired by the GrothendieckLefschetz trace formula for varieties over finite fields. We study this relationship using the framework of logarithmic geometry. We give explicit formulae for both the rational volume and the tamemonodromy zeta function in terms of the logarithmic stratification of alog smooth integral model (assuming that such a model exists). The maintools which we use in the proofs are Nakayamas description of the complex of nearby cycles in the log smooth case, and Katos formalism of logblow-ups." "“Submanifolds of Nearly Kaehler spaces”" "Joeri Van der Veken" Geometry "The proposed project is about the study of differential geometry. The objects that we study are called manifolds and are more complicated than the points, lines and curves from basic geometry. They could be seen as surfaces of higher dimensions in different ambient spaces. For example, one could think of the blackboard in the classroom as a flat surface (the blackboard) in the 3-dimensional Euclidean space (the classroom), which is the ambient space. If one slightly bent the blackboard without breaking it, one could obtain a more general surface. In geometry, the surface doesn't always have 2 dimensions, nor the ambient space 3, but higher arbitrary dimensions. Endowed with particular mathematical properties, these manifolds may correspond to objects of high interest in classical mechanics or advanced physics, for example. Mathematics offers a whole set of tools in order to be able to study these objects. By means of analysis, algebra, topology, etc., we can do computations on the manifolds, in order to understand their properties and describe them by formulas. We are also interested about relations between such objects. Manifolds that stay inside other manifolds are called submanifolds (one could think about the trace of a river through a field as a curve (a submanifold) in a plane (a manifold)). A particular class of submanifolds (called 'Lagrangian') that live inside other ambient manifolds called 'nearly Kaehler manifolds' are of central interest in the project."