Collaborative Research Unit

Foundations Lab

Research disciplines
  • Natural sciences
    • Order, lattices, ordered algebraic structures
    • Topological groups, Lie groups
    • Calculus of variations and optimal control, optimisation
    • Dynamical systems and ergodic theory
    • Functional analysis
    • Harmonic analysis on Euclidean spaces
    • Integral transforms, operational calculus
    • Measure and integration
    • Operator theory
    • Ordinary differential equations
    • Operations research and mathematical programming
    • Systems theory, control
    • History and biography of mathematical sciences
    • Mathematical logic and foundations
    • Probability theory
    • Statistics
    • Knowledge representation and reasoning
    • Artificial intelligence not elsewhere classified
    • Statistical physics
    • Quantum information, computation and communication
Description
Foundations Lab is a research group within the Faculty of Engineering and Architecture that focuses on basic research, and concentrates on the mathematical foundations of a number of disciplines. The members of this group are responsible for a largepart of the mathematics courses in the engineering programs, which are typically being taught to large groups of 300 to 400 students. They also teach a diversity of more advanced courses with a strong mathematical flavor in later years of the engineering curriculum. The research group is active in a number of fields. A first unit focuses on the study of quantum integrable systems constructed using Clifford algebras. In that context, a crucial role is played by the Dirac operator and its various deformations and extensions. Symmetries of such operators can typically be organized to form remarkable algebraic structures (Lie superalgebras, transvector algebras, Bannai-Ito and Racah algebras, …) and are an important topic of current research.Another theme of interest is the representation theory of Lie (super)algebras and the various orthogonal polynomials and special functions that appear in this context. A second unit is concerned with the mathematical and philosophical foundations of reasoning and decision-making under uncertainty, and has played a pioneering and prominent role in that field for a number of decades. We study robust generalizations of probabilistic models, known as imprecise probability models, which include sets of probabilities, lower expectation operators, sets of desirable gambles, and choice functions. We study their implications in the fields of decision making, financial analysis, AI, statistics, algorithmic randomness, and stochastic processes. For such models, we also develop robust inference methods and algorithms, and have applied them in a number of practical areas, such as Markov chains and Bayesian networks. A third unit investigates the dynamical foundations of information processing. Taking decisions, optimizing, reacting, or interpreting data, are all finally performed by a physical device which often consists of a complex network of interacting components. The basic properties of these devices can play a key role in efficient and accurate computation. For instance, quantum computers promise much faster operation than any classical supercomputer for some tasks; neural networks use a continuous representation of data rather than with digits; conversely, robots compute with digits but must interpret and control real-world signals. We are developing principles to study these devices as dynamical systems from two perspectives: (i) how can we improve accurate, robust operation of such devices, e.g. of quantum computers or robots; and (ii) which fundamental properties of a dynamical system can help solving particular information processing tasks? With this focus, we aim to contribute to Quantum Algorithms, Systems and Control Theory, Quantum Engineering, Robotics, Markov Chains and other related research fields.