Related papers: Recent progress in the geometry of Q-acyclic surfa…
In this paper we give a streamlined overview of some of the recent constructions provided with K.-H. Neeb, G. \'Olafsson and collaborators for a new geometric approach to Algebraic Quantum Field Theory (AQFT). Motivations, fundamental…
This paper will appear in the Proceedings of the 1995 Santa Cruz Summer Institute. The paper is a survey of recent developments in the theory of toric varieties, including new constructions of toric varieties and relations to symplectic…
The correspondence between 2-parameter families of oriented lines in ${\Bbb{R}}^3$ and surfaces in $T{\Bbb{P}}^1$ is studied, and the geometric properties of the lines are related to the complex geometry of the surface. Congruences…
A correspondence between different $Pin$-type structures on a compact surface and quadratic (linear) forms on its homology is constructed. Addition of structures is defined and expressed in terms of these quadratic forms.
A study of symplectic forms associated with two dimensional quantum planes and the quantum sphere in a three dimensional orthogonal quantum plane is provided. The associated Hamiltonian vector fields and Poissonian algebraic relations are…
We consider some integral-geometric quantities that have recently arisen in harmonic analysis and elsewhere, derive some sharp geometric inequalities relating them, and place them in a wider context.
We review recent results concerning closed G$_2$-structures on seven-dimensional manifolds. In particular, we discuss the construction of examples and some related problems.
This paper provides an overview of modern digital geometry and topology through mathematical principles, algorithms, and measurements. It also covers recent developments in the applications of digital geometry and topology including image…
We survey old and new results on the existence of moduli spaces of semistable coherent sheaves both in algebraic and in complex geometry.
One of the themes in algebraic geometry is the study of the relation between the ``topology'' of a smooth projective variety and a (``general'') hyperplane section. Recent results of Nori produce cohomological evidence for a conjecture that…
The embeddings of complex plane projective curves in the plane are a cornerstone of the topological study of algebraic varieties. In this work, we deal with the local and global aspects of these embeddings, with a special attention to its…
We generalize results by Wakabayashi and Orevkov about rational cuspidal curves on the projective plane to that on $\mathbb{Q}$-homology projective planes. It turns out that the result is exactly the same as the projective plane case under…
Results on $8$-dimensional topological planes are scattered in the literature. It is the aim of the present paper to give a survey of these geometries, in particular of information obtained after the appearance of the treatise Compact…
This paper constructs a Riemann surface associated to the icosahedron and discusses the geodesics associated to a flat metric on this surface. Because of the icosahedral symmetry, this is a distinguished special case of the example treated…
This work deals with relations between a bounded cohomological invariant and the geometry of Hermitian symmetric spaces of noncompact type. The invariant, obtained from the K\"ahler class, is used to define and characterize a special class…
This paper is devoted to the study of the global properties of harmonically immersed Riemann surfaces in $\mathbb{R}^3.$ We focus on the geometry of complete harmonic immersions with quasiconformal Gauss map, and in particular, of those…
We give an up-to-date overview of geometric and topological properties of cosymplectic and coKaehler manifolds. We also mention some of their applications to time-dependent mechanics.
We describe a part of the recent developments in the theory of separately holomorphic mappings between complex analytic spaces. Our description focuses on works using the technique of holomorphic discs.
A general scheme for determining and studying integrable deformations of algebraic curves is presented. The method is illustrated with the analysis of the hyperelliptic case. An associated multi-Hamiltonian hierarchy of systems of…
Recent developments in Seiberg-Witten theory and relations with Complex Geometry.