International Laboratory of Discrete and Computational Geometry named after Boris Delaunay

Scientific results:

• The Laboratory has solved a number of recognized complex problems in the domain of geometry and topology. We developed an ideology of the use of algebro-topological methods in image processing problems. On the basis of this ideology we managed to apply topological methods to problems of the segmentation and classification of medical images and the generalization of vector cartographic images.

• We have created a new algorithm for generalizing cartographic data preserving their topological properties, new algorithms and methods of the classification and segmentation of images obtained by gastroendoscopy.

• The Laboratory has developed a software complex for use in the modern school and allows not only to perform all the necessary operations for the design of stereometric models but also to automatically form layouts for 3D printing.

• Using asymptotic and numerical methods we have researched the Kolmogorov–Petrovsky–Piskunov equation with delay that describes the propagation of waves of concentration in an active medium. We studied the local properties of solutions, performed a numerical analysis of the propagation of a wave from one or from two initial perturbations. A complex spatially inhomogeneous structure occurring during the propagation and interaction of waves has been found and an explanation of its properties was provided.

• We have reviewed special systems of ordinary differential equations – so-called fully-connected systems of nonlinear oscillators. For this class of systems we suggested new methods that allow to tackle problems of the existence and stability of periodic modes of two-cluster synchronization. On the basis of these methods we found two-cluster synchronization modes for two important applied examples.

• The Laboratory has proposed and substantiated a new definition of an infinite-dimensional torus whose main advantage is that within this definition the torus is an analytic Banach manifold with a Finsler metric. This allows to determine the notion of hyperbolicity for an arbitrary diffeomorphism. We have introduced a new class of diffeomorphisms of the torus and for functions from this class we have developed a criterion of hyperbolicity that is new not only in the infinite-dimensional but also in the finite-dimensional case.

Implemented results of research:

• New algorithms for the classification and segmentation of gastroendoscopy imageautomation of the diagnostics of s have been developed for the automation of  diagnostics of oncological diseases of the gastrointestinal tract.

• The program for editing stereometric models with the capability of 3D printing is aimed at developing spatial intuition in school students.

• New algorithms for cartographic generalization are developed to improve  improve the quality of information representation in modern electronic maps. 

Education and career development:

• Two summer mathematical schools have been organized to provide additional training to undergraduate and postgraduate students and young researchers.
• The Laboratory conducted the International Interactive Exhibition «IMAGINARY with a mathematicians’s eyes» (2013).
• We are actively working with school students. Employees of the Laboratory participate  in organizing and staging exhibition, they conduct classes, academic competitions in mathematics for school students, deliver  popular lectures. 

Organizational and structural changes: 

We have created a computation cluster that is currently used by many divisions of Yaroslavl State University. 

Collaborations:

• Institute of Science and Technology Austria (Austria), University of Texas at Brownsville (USA), Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow State University (Russia): joint scientific conferences and publications.
• Free University of Brussels (Belgium): joint research and publications.