Title Promoter Affiliations Abstract
"Subgraphs and codes in distance-regular graphs in incidence geometry" "Frank De Clerck" "Department of Mathematics: Algebra and Geometry" "My project includes the application of techniques in algebraic graph theory, to obtain new bounds and non-existence results in geometry. Conversely, finite geometries can yield new (distance-regular) graphs with certain properties, and algebraic techniques can yield a characterization of them. In particular, I intend to further examine a proposed construction relying on substructures in a specific dual polar graph."
"Study of m-systems in finite projective spaces en associated incidence geometries" "Frank De Clerck" "Department of Mathematics: Algebra and Geometry" "From m-systems one can construct incidence geometries in particular, by using generalised linear representation. It is already known that in certain cases this method leads to semipartial geometries, and it is likely that in other similar cases, weaker properties can be found. We also endeavour to obtain more results on existence and non-existence of m-systems in some specific polar spaces."
"The Geometry of Absolute Arithmetic" "Koen Thas" "Department of Mathematics: Algebra and Geometry" "We propose a study of Algebraic Geometry over the field of characteristic 1, from the viewpoint of a class of incidence geometries which explicitly relates (automorphisms of) certain combinatorial objects to (automorphism theory of) Absolute Arithmetic. We aim at new insights in DeningerU+2019s program on the Riemann hypothesis."
"Intertwining Extremal Combinatorics and Finite Geometry" "Leo Storme" "Department of Mathematics: Analysis, Logic and Discrete Mathematics" "Extremal combinatorics investigates finite objects such as graphs or set systems with extremal properties. Finite geometry investigates finite incidence structures. For decades there have been interesting interactions between both areas: (1) Finite geometry provides examples of graphs and hypergraphs with extremal properties for extremal combinatorics. (2) Problems in extremal combinatorics on families of finite sets generalize naturally to questions on families of subspaces in finite vector spaces.This proposal will investigate some of these connections: (1) Low degree Boolean functions on vector spaces. (2) The investigation of q-analogs of hypergraph Turán problems. (3) Pseudorandom clique-free graphs and Ramsey numbers.These particular topics have broad relevance with applications in combinatorics and computer science."
"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."
"Exceptional Groups and Exceptional Geometries in the Theory of Tits Buildings" "Hendrik Van Maldeghem" "Department of Mathematics: Algebra and Geometry" "The aim is (1) to write some books on buildings and geometries, focusing on the exceptional types, making use of the Freudenthal-Tits Magis Square, and (2) continue and intensify my research around the same Magic Square. "
"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 mathem"
"A functorial approach to projective representations of Tits-buildings and the associated groups of Lie type." "Hendrik Van Maldeghem" "Department of Mathematics: Algebra and Geometry" "The aim of the project is to take the axioms of Mazzocca-Melone describing Veronese varieties as a solid base for a functor mapping classical Tits-buildings to other classical or exceptional Tits-buildings of higher rank. This we, we lay the foundations for a systematic treatment of representations of Tits-buildings."
"Asymptotics for Orthogonal Polynomials and High-frequency Scattering Problems" "Daan Huybrechs" "NUMA, Numerical Analysis and Applied Mathematics Section" "The goal of this thesis is to exploit asymptotic behaviour in high-degree orthogonal polynomials and high-frequency acoustic scattering problems to obtain %[Dave] problems. This leads toa lower computational cost. The code is made publicly available and validates this goal as well as the accuracy of the results of this thesis. We are interested in the higher-order asymptotic behavior of orthogonal polynomials of Jacobi-, Laguerre- and Hermite-type as their degree n goes to ∞. These are orthogonal with respect to the weight functions w(x) = (1-x)^α (1+x)^β h(x) for x ∈ [-1,1], w(x) = x^α exp[-Q(x)] for x ∈ [0,\infty), and w(x) = exp[-Q(x^2)] for x ∈ (-\infty, \infty), respectively, with α and β > -1. The functions h(x) and log Q(x) are real analytic and strictly positive on the interval of orthogonality. This information is implicitly available in two papers, where the authors use the Riemann-Hilbert formulation and the Deift-Zhou non-linear steepest descent method. We show that the computation of higher-order terms can be simplified analytically in a jump relation, leading to their efficient, fully explicit and automatic construction. For Laguerre- and Hermite-type polynomials, we need the derivation of an asymptotic expansion of a zero of a general polynomial with respect to its constant coefficient to obtain more explicit expressions when Q(x) is a polynomial. The resulting asymptotic expansions in four different regions of the complex plane are implemented both symbolically and numerically, along with heuristics to determine these regions and the number of terms to use for a certain accuracy. The main advantage of these expansions is that the accuracy improves as the degree of the polynomials rises, at a computational cost that is practically independent of the degree or even decreasing due to requiring fewer terms. They allow more accurate approximations for modest degrees n than the leading order term (or some higher order terms) that already existed in the literature. In contrast, the typical use of the recurrence relation for orthogonal polynomials in computations leads to a cost that is at least linear in the degree and a decreasing accuracy.A smooth integral can be approximated with a weighted sum of n integrand evaluations which is stable and optimal for polynomial integrands. The nodes of these Gaussian quadrature rules are zeros of the corresponding orthogonal polynomials corresponding to the weight function in the integral. Those can be computed via a Newton method on the recurrence relation in O(n^2), or on the asymptotic expansions mentioned before. In order to reduce the constant for the latter O(n) algorithm, to reduce code length and to solve possible convergence as well as other numerical issues, we set up explicit asymptotic expansions of the nodes and weights of Gauss-Laguerre and -Hermite quadrature rules with a high order in the relevant regions. In another part of our research, Boundary Element Methods are suited to solve wave propagation and scattering problems in acoustics, as they only discretise the boundary of the acoustic domain to obtain linearised sound waves in the frequency domain. They lead to a discretisation matrix that is typically dense and large: its size and condition number grow with increasing frequency. Yet, high frequency scattering problems are intrinsically local in nature, which is well represented by highly localized rays bouncing around. Asymptotic methods can be used to reduce the size of the linear system, even making it frequency independent, by explicitly extracting the oscillatory properties from the solution using ray tracing or analogous techniques. However, ray tracing becomes expensive or even intractable in the presence of (multiple) scattering obstacles with complicated geometries. Instead, we start from the same discretisation that constructs the fully resolved, large and dense matrix, and achieve asymptotic compression by explicitly localizing the Green function. This results in a large but sparse matrix, with a faster associated matrix-vector product and, as numerical experiments indicate, a much improved condition number. Although an appropriate localisation of the Green function also depends on asymptotic information unavailable for general geometries, we can construct it adaptively in a frequency sweep from small to large frequencies in a way that automatically takes into account a general incident wave. We approximately maintain the discretisation error and do not use a priori knowledge on the solution. We show that the approach is robust with respect to non-convex, multiple and even near-trapping domains, though the compression rate is clearly lower in the latter case. It can be applied to general incident waves, boundary conditions, dimensions and integral formulations. Furthermore, in spite of its asymptotic nature, the method is robust with respect to low-order discretisations such as piecewise constants, linears or cubics, commonly used in applications. On the other hand, we do not decrease the total number of degrees of freedom compared to a conventional, classical discretisation. The combination of the sparsifying modification of the Green function with other accelerating schemes, such as the Fast Multipole Method, appears possible in principle. A visibility approach was shown to be inferior, we have extended the principle of Fermat, and regions that are important for insight into the high-frequency behaviour of the problem are adaptively provided. Finally, we focus on multiple scattering obstacles, where the wave pattern becomes very complicated, but wavenumber-independent simulation schemes have been proposed in the literature based on ray tracing. In such schemes, one can note that the phases of the corresponding densities on each of the obstacles converge to an equilibrium after a few iterations. The equilibrium is given by periodic orbits between a subset of the obstacles, on which the existing analysis of ray tracing schemes focuses. We approximate the phase of such a limiting mode representing a full cycle of reflections with a Taylor series. We exploit symmetry in the case of two circular scatterers, but also provide an explicit algorithm for an arbitrary number of general 2D obstacles. The coefficients, as well as the time to compute them, are independent of the wavenumber and the incident wave. We sketch the principle to obtain a series expansion of the smoothly varying part of this mode using the method of steepest descent. The results may be used to accelerate ray tracing schemes after a small number of initial iterations."
"Affine buildings and CAT(0)-spaces" "Hendrik Van Maldeghem" "Department of Mathematics: Algebra and Geometry" "The theory of buildings (developed by Jacques Tits) studies certain (incidence) geometric objects with close ties to Algebra and Geometric Group Theory. In our project we want to study the class of affine buildings and various closely related objects using these different points of view."