Proceedings book of international workshop on theory of submanifolds

Istanbul Technical University, Turkey 02-04 June 2016Differential geometry is proving to be an increasingly powerful tool that improves its ties to other branches of mathematics such as analysis, topology, algebra, PDEs, and so on, as well as to theoretical physics research. The growing number of publications in the field of submanifolds was probably the main reason to organize the ”International Workshop on Theory of Submanifolds”, which took place at Istanbul Technical University, Turkey, from June 2 to June 4, 2016. One of the main features of the conference is the originality of its topic, being the only one focussing particularly on submanifold theory in the last few years. This is remarkable since submanifold theory is a very broad and omnipresent topic, going from surface theory in three-space, with applications in engineering and computer vision for example, to very abstract settings with high dimension and codimension, some of them appearing in modern physical theories. This volume, containing the proceedings of the above mentioned workshop, provides very recent results mainly on the theory of submanifolds, which the reader would be interested in getting acquainted with. The book is divided into three parts, each of them having a distinct editor. The first part contains surveys on submanifolds with certain properties, in particular on surfaces. Part two is the biggest one, and is devoted to the theory of submanifolds. The last part extends the main subject of the workshop toward some related topics, such as some geometrical structures which are extended from a manifold to the whole space containing the manifold (e.g. the total space of its cotangent bundle). The experience of the contributors to the Proceedings is illustrated by their publications in this field and the freshness of this conference was given mainly by the presence of many young mathematicians. The workshop was very successful, despite the critical period of this conference, where many participants had to cancel their participation for reasons beyond their control. All articles included here passed the usual referee process. Our warm thanks go to all those who contributed to this book by their work, to all participants of the workshop, to the referees of the Proceedings, to the host institution for organizing the conference and last but not least to our sponsors.İT

Local and global gestalt laws: A neurally based spectral approach

A mathematical model of figure-ground articulation is presented, taking into account both local and global gestalt laws. The model is compatible with the functional architecture of the primary visual cortex (V1). Particularly the local gestalt law of good continuity is described by means of suitable connectivity kernels, that are derived from Lie group theory and are neurally implemented in long range connectivity in V1. Different kernels are compatible with the geometric structure of cortical connectivity and they are derived as the fundamental solutions of the Fokker Planck, the Sub-Riemannian Laplacian and the isotropic Laplacian equations. The kernels are used to construct matrices of connectivity among the features present in a visual stimulus. Global gestalt constraints are then introduced in terms of spectral analysis of the connectivity matrix, showing that this processing can be cortically implemented in V1 by mean field neural equations. This analysis performs grouping of local features and individuates perceptual units with the highest saliency. Numerical simulations are performed and results are obtained applying the technique to a number of stimuli.Comment: submitted to Neural Computatio