Narrow Results

It is shown how actions corresponding to antisymmetric non-Abelian tensorial gauge field theories of (p+1)-dimensional diffeomorphisms yield p-brane actions associated with their (p+1)-dimensional worldvolume evolution. We conclude with a discussion of how to obtain p-brane actions from the large N limit of covariant matrix models based on generalized hypermatrices. A deformation quantization of Nambu classical mechanics furnishing Nambu quantum mechanics by constructing the n-ary noncommutative product of n functions f₁ •f₂ • ⋯ •f_n, the n-ary version of the Moyal bracket, and the analog of the Weyl–Wigner–Groenowold–Moyal map among operators and c-functions remains an open problem. A solution to this problem will reveal important relations between the physics of p-branes and matrix models based on generalized hypermatrices in the large N limit.

General covariance is the cornerstone of Einstein’s general relativity (GR) and implies that any two metrics related by diffeomorphisms are physically equivalent. There are, however, many examples pointing to the fact that this strict statement of general covariance needs refinement. There are a very special (measure-zero) subset of diffeomorphisms, the residual diffeomorphisms, to which one can associate well-defined conserved charges. This would hence render these diffeomorphic geometries physically distinct. We discuss that these symmetries may be appropriately called “symplectic symmetries”. Existence of residual diffeomorphisms and symplectic symmetries can be a quite general feature and not limited to the examples discussed so far in the literature. We propose that, in the context of black holes, these diffeomorphic, but distinct, geometries may be viewed as “symplectic soft hair” on black holes. We comment on how this may remedy black hole microstate problem, which in this context are dubbed as “horizon fluffs”.

Following a line of reasoning suggested by Eliashberg, we prove Cerf's theorem that any diffeomorphism of the 3-sphere extends over the 4-ball. To this end we develop a moduli-theoretic version of Eliashberg's filling-with-holomorphic-discs method.

Over the last four decades, group actions on manifolds have deserved much attention by people coming from different fields, as for instance group theory, low-dimensional topology, foliation theory, functional analysis, and dynamical systems. This text focuses on actions on 1-manifolds. We present a (non exhaustive) list of very concrete open questions in the field, each of which is discussed in some detail and complemented with a large list of references, so that a clear panorama on the subject arises from the lecture.

In content-based image retrieval there is a need to reduce the gap between the high- level semantics of visual objects and the low-level features such as color, texture and shape descriptors extracted from them. In this paper, we develop an image retrieval system that bridges the gap between low-level visual features and high- level semantics and extracts a similarity measure directly from the data itself using machine learning. We proposed a fuzzy color histogram for color features and Bayesian estimation for texture features and Lie descriptors for shape features. We have used unsupervised Kohonen's Self-Organizing Maps (SOM) technique to train the images and our own indexing scheme with reference system based on R-tree SOM.