Research team

Fundamental Mathematics

Expertise

Categorical topology, approach theory and metriscally generated theories with applications to analysis, approximation theory, hyperspace theory and measure and probability theory.

Approach theory meets likelihood theory. 01/10/2014 - 30/09/2017

Abstract

Let the density of an unknown probability distribution belong to a family of densities which are determined up to an unknown parameter vector. Based on a sample of observations, the maximum likelihood estimator (MLE) then provides an estimate for the unknown parameter vector by picking the vector under which the chance of observing the particular given sample is maximal. Statisticians value the MLE because it behaves well asymptotically. This roughly means that the estimates given by the MLE will converge to the true value of the unknown parameter vector as the sample size grows to infinity. However, doubts about the applicability of the MLE have emerged as `misspecified models', i.e. models in which the density of the unknown probability distribution fails to belong to the family of densities producing the MLE, are common in realistic settings. Many researchers have investigated under which additional regularity conditions the MLE in a misspecified model continues to behave well asymptotically. Here we want to follow a different route. More precisely, instead of adding extra regularity conditions on the model (which may again destroy the applicability), we will try to establish results in which we measure how irregular a model is and then use this information to assess how asymptotically well the MLE behaves. To this end, we will use approach theory, a mathematical theory which is designed to cope rigorously with notions such as `almost behaving well asymptotically'

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Approach structures in probability theory. 01/10/2012 - 30/09/2014

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In this project we aim to develop a comprehensive and universally applicable theory of quantitative analysis of hitherto only topological structures (e.g. weak convergence, finite dimensional convergence, convergence in probability and in law) on spaces of probability measures and random variables (in particular continuous and cadlag stochastic processes) by replacing the topologies by canonical and intrinsically richer isometric counterparts, eventually aiming to prove quantitative versions of the fundamental results of stochastic analysis such as e.g. Prohorov's theorem and various important limit theorems.

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Approach structures in probability theory. 01/10/2010 - 30/09/2012

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In this project we aim to develop a comprehensive and universally applicable theory of quantitative analysis of hitherto only topological structures (e.g. weak convergence, finite dimensional convergence, convergence in probability and in law) on spaces of probability measures and random variables (in particular continuous and cadlag stochastic processes) by replacing the topologies by canonical and intrinsically richer isometric counterparts, eventually aiming to prove quantitative versions of the fundamental results of stochastic analysis such as e.g. Prohorov's theorem and various important limit theorems.

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Lax monads, lax algebras and applications. 01/10/2009 - 30/09/2011

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The theory of lax algebras is a recent theory which helps us to describe certain categories in a different way. We obtain new characterisations for Top and Ap. I look closer to Ap and describe this category by using functional ideals. Now I try to adapt the technique to Lip (Lipschitz spaces), again I consider ideals of functions, but with another saturation condition. Later on the question will be: "Can we describe a category of lax algebras, starting from an arbitrary operation and saturation condition?"

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A study of new quantitative convergence structures in probability theory and their application in stochastic analysis and parametric and non-parametric statistics. 01/10/2009 - 30/09/2010

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The first cornerstone is the study of new quantitative convergence structures in measure theoretic context, in particular on spaces of random variables and probability measures. We will mainly be concerned with structures strongly related to the p-Wasserstein distance, a topic popular for both applications and theoretical aspects. The second cornerstone is the application of the theory to stochastic analysis (convergence of Feller processes, martingales and solutions of stochastic differential equations) and statistics (convergence of estimators in parametric and non-parametric models).

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Lax monads, lax algebras and applications. 01/10/2007 - 30/09/2009

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Local metrically generated theories. 01/10/2006 - 30/09/2008

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  • Promotor: Lowen Bob
  • Fellow: Van Geenhoven Anneleen

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Further study of the interaction between frame theory and approach theory. 01/10/2006 - 30/09/2007

Abstract

Approach frames are a pointfree abstraction of approach spaces. We research the properties of this category, which concepts from approach theory and frame theory form natural constructions in this new framework and how this interaction between frame theory and approach theory gives us new concepts to study approach spaces.

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  • Promotor: Lowen Bob
  • Fellow: Van Olmen Christophe

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The efficiency of OFDM modulation for digital acoustical underwater communication systems. 01/01/2006 - 31/12/2007

Abstract

In this project the usability of OFDM modulation for digital acoustical underwater communication will be investigated. In addition, spectral noise characteristics will be determined, as these will enable us to specify a number of system parameters of the OFDM communications system.

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Metrically generated theories. 01/01/2005 - 31/12/2008

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A study of the interaction between frame theory and approach theory. 01/10/2004 - 30/09/2006

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Approach frames are a pointfree abstraction of approach spaces. We research the properties of this category, which concepts from approach theory and frame theory form natural constructions in this new framework and how this interaction between frame theory and approach theory gives us new concepts to study approach spaces.

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  • Promotor: Lowen Bob
  • Fellow: Van Olmen Christophe

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Research into feasibility and efficiency of underwater ultrasonic communication. 01/02/2004 - 31/12/2005

Abstract

In this project, the limits of ultrasonic underwater datacommunication concerning bandwidth and distance will be investigated. Parameters will be caracterized using theoretical models and empirical measurements. Ultrasonic transducers must be selected. A simulation model of the system will be built. With this model engineering choices can be made to build a prototype communication system.

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A categorical treatment of compactness, separation and completeness, inspired by a version for approach spaces and its implementation to categories of objects modelled over an algebra. 01/01/2004 - 31/12/2004

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01/01/2004 - 31/12/2004

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A study of the interaction between frame theory and approach theory. 01/10/2002 - 30/09/2004

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  • Promotor: Lowen Bob
  • Fellow: Van Olmen Christophe

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01/10/2001 - 30/09/2004

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01/10/2000 - 30/09/2003

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    01/10/2000 - 30/09/2002

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      Fundamental categorical and applied research in approximationtheory in a non-metric setting. 01/10/1999 - 31/12/2000

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        01/10/1998 - 30/09/2001

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          01/10/1998 - 30/09/2000

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            Generic optimalisation. 01/01/1998 - 31/12/2001

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              01/10/1997 - 30/09/1999

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                01/01/1997 - 31/12/1997

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                  01/01/1997 - 31/10/1997

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                    Analysis of queueing systems with correlated and bursty input. 01/10/1996 - 30/09/1998

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                    Future telecommunications systems, such as ATM networks, have to carry traffic with specific characteristics originating from the properties of the types of services that are offered (video, voice, data). The stochastic models describing this traffic, should incorporate the most important characteristics (i.e. those with the highest impact on the system performance), in particular the correlation between rivals and the burstiness of the arrivals. The aim of this research is to find efficient algorithms to derive the important performance measures of queueing systems with correlated and bursty input asymptotic behavior and distribution of the buffer occupation, waiting time distribution, etc...)

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                      01/10/1996 - 30/09/1998

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                        01/10/1996 - 30/09/1998

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                          Algebraic and topological categories. 01/05/1996 - 30/11/1999

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                          Algebra, Analysis, Topology, Geometry and Theoretical Computer Science make each of them use of specific category-theoretical techniques. The aim of the project is to investigate the interaction between these specific methods and to test them on their reciprocal applicability.

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                          • Promotor: Lowen Bob
                          • Co-promotor: Verschoren Alain

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                            Fundamental methods and techniques in mathematics. 01/01/1996 - 31/12/2010

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                            General Mathematical research with particular emphasis on interdisciplinary aspects.

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                            • Promotor: Lowen Bob
                            • Promotor: Verschoren Alain

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                            01/10/1995 - 30/09/1997

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                              Development of approximation theory in a non-metric setting. 01/07/1995 - 31/12/1996

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                              Techniques will be developed to calculate exactly or approximately the so-called limitfunctions in approach spaces: 1) set-up of approximation theory in the setting of locally convex spaces and afterwards in more general situations; 2) generalization and refinement of approximative methods in the theory of hyperspaces on metric spaces and multifunctions; 3) refinement of important limit theses in probability theory.

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                                01/10/1994 - 30/09/1996

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                                  01/10/1994 - 30/09/1996

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                                    Topological and algebraic categories. 01/04/1994 - 31/03/1996

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                                    Algebra, mathematical analysis, topology, geometry and theoretical computer science make, each in its own way, use of specific category-theoretical techniques. The aim of this project is to investigate the interaction between these specific methods and to test them as regards their reciprocal applicability.

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                                    • Promotor: Lowen Bob
                                    • Co-promotor: Verschoren Alain

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                                      Vision. 01/10/1993 - 31/12/1997

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                                        01/10/1992 - 30/09/1993

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