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."
"Polar spaces, dual polar spaces and related geometries" "Bart De Bruyn" "Department of Mathematics: Algebra and Geometry" "The intention of the project is to obtain results about (1) classification of dense near polygons with three points per line (2) hyperplanes and projective embeddings of dual polar spaces and related geomtetries like polar Grassmannians (3) substructures of (dual) polar spaces (4) generating rank of dual polar spaces and related geometries."
"Reduction of stochastic noise in physically-based rendering algorithms." "Philip Dutré" "Informatics Section, Human-Computer Interaction (HCI)" "The use of computer graphics has become commonplace in industry, games, films, architectural visualizations, etc. Over the years, there has been an increasing demand for more realism and more graphical fidelity. To achieve this realism, current state-of-the-art illumination algorithms are based on ray tracing, which simulates the physical transport of light through three-dimensional scenes. While ray tracing is conceptually simple, the underlying problem is a recursive integral equation, which has to be evaluated for each pixel. Solving this integral equation is usually done using stochastic methods, like Monte Carlo integration. However, using stochastic algorithms inevitably leads to variance on the numerical outcomes which appears as noise in rendered images.In this dissertation, we aim to reduce this noise by improving the efficiency of the illumination algorithms. First, we present a novel view on the visibility evaluation between two points in a three-dimensional scene. Traditionally, the visibility is evaluated by testing all the geometrical primitives for intersection with a line segment spanned by two points. However, we propose a method to stochastically evaluate the visibility by only testing a random subset of the geometrical primitives, while still converging to the correct solution.Second, we present a new algorithm which accelerates the evaluation of the visibility between two points in a three-dimensional scene by leveraging cheap geometrical proxies. While simplified versions complex geometry are already commonly used to accelerate visibility operations, naively substituting the simplified geometry for the original geometry will introduce artefacts and bias. In contrast, our method remains unbiased by stochastically using either the simplified geometry or a combination of the simplified geometry and the original geometry for intersection testing.Finally, we propose a semi-analytical technique to evaluate the direct illumination. Instead of sampling the light sources using point samples, we propose to use line samples where the contribution of a line sample is evaluated analytically. Using line samples to evaluate the direct illumination instead of point samples reduces the dimension of the integral equation, which can lead to higher orders of convergence when specialized distributions are used to generate the line samples."
"Intriguing subsets of finite projective and polar spaces" "Leo Storme" "Department of Mathematics: Algebra and Geometry" "This research proposal focuses on four types of problems related to subsets in finite projective andpolar spaces, for which the research is currently making a lot of progress. Projective and polarspaces are geometries that arise from vector spaces.The four types of subsets I will study are Erdos-Ko-Rado sets, Cameron-Liebler classes, tight setsand Kakeya sets. There are important links between these subsets.An Erdos-Ko-Rado set is a set of subspaces pairwise meeting in at least a point, e.g. the set of allsubspaces through a fixed point. The Erdos-Ko-Rado problem asks to classify the (large) Erdos-Ko-Rado sets.A Cameron-Liebler line class in a projective space is a set of lines such that each line in the setmeets a fixed number of lines in the set and such that each line not in the set meets an other fixednumber of lines in the set, e.g the set of all lines in a plane. This definition has recently beengeneralised to subspaces. It is asked to classify all small Cameron-Liebler classes.A tight set in a polar space is a set of points combinatorially behaving as a set of pairwise disjointsubspaces of maximal dimension. The main question is to classify the smallest tight sets. Theseresults depend heavily on the type of the polar space.A Kakeya set in a projective plane is a set of lines in a projective plane, one through each point ona fixed 'line at infinity'. Its size is the number of points it covers. It is asked to classify the smallKakeya sets."
"Diophantine problems and algebraic geometry: new connections." "Arne Smeets" Algebra "The core idea of this research proposal is that there is a lot to be gained from an intensified interaction between different subfields of arithmetic/algebraic geometry. The community of people studying rational points on algebraic varieties (to which I belong) is now very big and mature, but it would benefit enormously from more interaction with other, rapidly developing subfields of algebraic geometry, such as birational and logarithmic geometry.The proposal consists of three broad and complementary research lines:1) Reinventing rational points: Campana's orbifolds2) Families of varieties over one-dimensional bases: arithmetic and geometric aspects3) Families of varieties over higher-dimensional bases: geometric aspects"
"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."
"Interactive Robotic Architecture" "Wim Dewulf" "Mechanical Engineering Technology, Group T Leuven Campus" "This thesis will focus on high accuracy control of collaborating robot systems, aiming to provide accurate positioning and orienting ability for a flexible robotic X-ray CT system. Industrial X-ray ct system has gained more and more attention because of the outstanding ability to perform quality control on both internal and external structure. However, current CT systems can only perform on fixed circular trajectories, which restrict its application in inline environment or on large targets. This thesis will develop a high accurate robot system which can be used to perform flexible and accurate movement of CT system. To improve the accuracy of robot manipulators, five aspects should be investigated: * optimal motion planning method * absolute positioning method based on laser tracker * relative positioning method based on radiograph. optimal motion planning method This task aims to develop optimal path planning method. For a given scanning geometry sequence，optimal path planner will provide feasible and optimal solution for the collaborating robots to move from the initial configuration to each of the goal state defined by the scanning geometry sequence. Objective: * execution uncertainty * execution time (path length) * path safety Different path characteristics (such as geometry, velocity, acceleration, joint type and the number of joint involved) can signification affect the execution effect, such as time and uncertainty. In order to achieve an optimal trajectory, we need to cope with following sub-tasks: * For a given scanning geometry sequence, which can be seen as Tool Center Point (TCP) sequence, inverse kinematic equations need to be solved to get the configurations of the whole robot system. * For a given start TCP/configuration and a goal TCP/configuration, how many joints should be occupied at least? * Which one pose more impact on the path uncertainty, the number or the type of joints. * How to adopt the answer of questions above to guide the selection of multi solution of inverse kinematic equations. * Combining concerns about velocity, acceleration, geometry and information above, to generate an optimal feasible path. absolute positioning method based on laser tracker The accuracy can also be improved by incorporating extra sensors. Generally, the actual robot cannot fit in its system model ideally, because of the deviation in assembling and manufacturing. This sub-task will first develop an automatic method to calibrate the robot system model (for example D-H model) by generating and tracking a series of specific movements. Then, the system will perform absolute static calibration of both TCP start positions and orientations. Dynamic calibration will also be investigated to track TCP positions and orientations, as well as improve the control accuracy. Since line-of-sight limitations are associated with external sensors, investigation on off-line calculation of trajectory pre-compensation of the effects of the nonlinear dynamics (e.g. robot joints compliances) is also important. The final accuracy of both TCP configurations should be ca. 1mm or better. relative positioning method based on radiograph. The challenge with collaborating robots and robotized X-ray CT lies in the importance of both source and detector position with respect to each other. In fact, X-ray CT systems can also be seen as an extra sensor with superior accuracy ( up to a few µm), which can be used to further calibrate the relative position accuracy between X-ray source and detector. To this aim, calibrated reference objects in between source and detector will be introduced. The principle will first be evaluated using both simulations and two complementary validation platforms: two collaborative robots with visual light and a XCT machine with additional rotation axis."
"ManiFactor: Factor analysis for maps into manifolds" "Nick Vannieuwenhoven" "Numerical Analysis and Applied Mathematics (NUMA), Geometry" "Factor analysis, principal component analysis or singular value decomposition (SVD) of a collection of points in a Euclidean space, which can be represented by a matrix, is a fundamental and omnipresent technique to reveal latent factors in the data. Factor analysis was generalized in psychometrics in the 1960s to data that consists of multiple modes, leading to the nowadays popular canonical polyadic decomposition (CPD) of tensors. Taking the equivalent viewpoint that matrices and tensors represent, respectively, linear and multilinear maps in coordinates, the aforementioned decompositions (SVD and CPD) can be viewed as approximating them by simpler, low-rank versions of these maps.In this project, motivated by concrete applications in computer vision and multiscale simulations, we will vastly generalize factor analysis to maps into manifolds. An inherent obstacle in this setting is that linear combinations of factors, yielding straight-line approximations, as used in the linear and multilinear case, are fundamentally incompatible with the curved geometry of smooth manifolds. This project will investigate several generalizations suited for combining (or joining) factors in the manifold settings, including replacing straight lines by geodesics. As in the matrix and tensor case, we say that maps that can be expressed with a small number of factors admit a low-rank ManiFactor decomposition. We will develop practical Riemannian optimization algorithms for approximating an implicitly given smooth map by such a decomposition. The functional approximation properties of the proposed model will be investigated."
"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."