Such a fervent and vigorous activity can be easily explained: tensors are ubiquitous in describing physical quantities and laws, in modelling systems, and in representing constraints. Thus, more and more often so, even theoretical results about tensors can have significant practical consequences.
One of the aims of this workshop is to bring together experts from different branches of mathematics and engineering in which tensors play a crucial role.
One other significant aim is to introduce young researchers to the art of building bridges between different research fields using tensors as the leading example.
The workshop is a scientific activity of the project DISMA - Dipartimento di Eccellenza 2018-2022
Workshop web-site
This approach has several advantages, from highly efficient numerical algorithms that seem to be very stable also in long term or asymptotic behaviour, to a deeper understanding of the integrable nature of the smooth theory being discretized.
The type of discretization to be discussed depends on the integrable nature of the smooth theory to be discretized: in partial differential equations this relates to the fact that the pde under investigation can be transformed into a (possibly infinite dimensional) system of odes; in differential geometry it depends on the fact that the surfaces under consideration admit a sufficiently rich "transformation theory", i.e., procedures to generate new surfaces (solutions of a pde) from given ones. Examples are surfaces of constant mean curvature (soap bubbles) or the sinh-Gordon equation.
A key issue in creating a discrete theory analogous to a smooth one is to find the "correct" definitions for discrete analogues of smooth objects or quantities. Darboux-Baecklund type transformations allow for a nearly algorithmic approach to integrable discretization: permutability can be used to generate discrete or semi-discrete nets with the desired properties.
The lectures will give an introduction to basic ideas of integrable discretization, an outline of some key methodology and of the problems to be solved on the way; to elucidate the presented concepts some particular classes of smooth/discrete surfaces will be discussed in detail in the second part of the lectures. then, students will prepare presentations on smooth and discrete geometries in pairs/groups, based on classical references and recent research articles.
Lecturers
Udo Hertrich-Jeromin, Vienna University of Technology
Mason Pember, Vienna University of Technology
Gudrun Szewieczek, Vienna University of Technology
The course is a scientific activity of the project DISMA - Dipartimento di Eccellenza 2018-2022