Workshop "Singularities and Applications, Victor Goryunov 60"

Victor Goryunov is a leading figure in Singularity Theory, whose contributions to the subject are fundamental. He has published several books and a vast variety of papers in singularity theory, Vassiliev type invariants, and Legendrian knots. Many of his papers in Lagrange and Legendiran geometry are now considered to be classical in the subject. In particular he has developed stability theory of maps between Lagrangian and Legendrian varieties and studied local properties of their singularities. In addition Victor Goryunov has worked on complex crystallographic groups, and he has calculated Tjurina and Milnor numbers of matrix singularities. Victor Goryunov's great enthusiasm, his amiability and his belief in scientific progress have helped to maintain a strong research network of mathematicians all over the world. Originally from Arnol'd's school in Moscow, he has worked in Russia, Denmark, US and UK. He is an active collaborator with former Soviet Union mathematicians and those from the West. Victor Goryunov was among the principal organizers of a semester program in the Newton Institute on singularity theory. He has been an organizer of many international conferences. For many years he has been a principal organizer of 'Singularity Days' joining researchers of Liverpool, Warwick, and Valencia (funded by LMS).

The proposed workshop in singularity theory and its applications will attract major figures from, at least, Europe including Scandinavia and Israel, North and South America, Russia and Japan.

The immediate impact of a workshop of this kind will certainly be academic impact, that is the dissemination of new ideas through lectures, informal discussions and a rapidly produced follow-up document giving background and promising directions for progress, including explicit problems (this to have wide circulation among the international community).

Two members of the SINGULAR team from Kaiserslauten have agreed to present recent developments in the software; this will give participants an opportunity to find out what the software can do for them, and of course especially to talk to the team members about their own computational needs.

This kind of impact is hard to measure precisely; however we will ask participants to acknowledge the workshop as appropriate in future publications, especially when the workshop resulted in a new collaboration, besides contributing to the document mentioned above.

Here is an outline of the external impact of singularity theory in various disciplines. The main goal of many problems of singularity theory is to understand the dependence on parameters of certain structures from analysis, geometry, physics, or from some other science. Thus singularity theory has influenced the development of many areas of mathematics and physics: it has a direct impact within other scientific disciplines. There are many examples; here are a few which are relevant to the material of the workshop.

Singularity theory connects the investigation of optical caustics with simple Lie algebras and regular polyhedral theory, while also relating hyperbolic PDE wavefronts to knot theory and the theory of the shape of solids to commutative algebra.

One of the most famous modern applications of singularity theory was the introduction by Vassiliev of finite type invariants in knot theory. Further investigations led to many other topological applications of his ideas.

Frobenius manifolds, the basics of which were introduced by Saito and which are finding nowadays numerous applications in theoretical physics.

Singularities of distance functions (in various geometries) have led to a whole area of application to computer vision, via medial and skeletal structures, and through reconstruction of scenes from images by moving cameras. The singularities of linkage configurations have application to robotics. Recent developments include stochastic geometry and bio imaging, space-time modeling and statistical methods within bioinformatics.

There is a significant input from singularity theory to contemporary control theory, and thereby into areas where control theory is used such as mathematical economics. This includes the complete local classifications of generic second order linear PDEs on the plane and generic singularities of slow motion of relaxation type equations with one fast and two slow variables; classification of generic relative minima singularities for low dimensional parameters; and results on the structural stability of both local and non-local controllability of generic control systems on orientable two-dimensional manifolds.

Configuration spaces of tensegrities are considered as particular example of stratified spaces arising in singularity theory. Currently there are various applications of the theory of tesegrities in art and free-form structure buildings (e.g. in 2015 a tree-like tensegrity sculpture, or "TensegriTree" was erected in the out-door teaching area on the University of Kent Campus).