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We give a brief survey on local holomorphic dynamics in higher dimensions. The main novelty of this note is that we will organize the material by the "level" of local invariants rather than the type of maps.
This is a survey article, to appear in the Proceedings of the 2018 International Congress of Mathematicians. (Revised, with added and updated references.)
We introduce a notion of discrete topological complexity in the setting of simplicial complexes, using only the combinatorial structure of the complex by means of the concept of contiguous simplicial maps. We study the links of this new…
Harmonicity of holomorphic maps between various subclasses of almost contact metric manifolds is discussed. Consequently, some new results are obtained. Also some known results are recovered, some of them are generalized and some of them…
The aim of this paper is to present a simple way to generate proper monomial rational maps between generalized balls and via the relations between generalized balls and bounded symmetric domains of type I, we suggest new examples of proper…
In recent years, significant advances have been made in the design and analysis of fully dynamic algorithms. However, these theoretical results have received very little attention from the practical perspective. Few of the algorithms are…
We study analysis over infinite dimensional manifolds consisted by sequences of almost K\"ahler manifolds. In particular we develop moduli theory of pseudo holomorphic curves into the spaces with high symmetry. As applications, we study…
This expository paper is concerned with the properties of proper holomorphic mappings between domains in complex affine spaces. We discuss some of the main geometric methods of this theory, such as the Reflection Principle, the scaling…
We give a description of complex geodesics and we study the structure of stationary discs in some non-convex domains for which complex geodesics are not unique.
This paper uses differential spaces to obtain some new results in integrable Hamiltonian systems
Estimating the coefficient functionals on various classes of holomorphic functions traditionally forms an important field of geometric complex analysis and its mathematical and physical applications. These coefficients reflect fundamental…
We study the regularity results of holomorphic correspondences. As an application, we combine it with certain recently developed methods to obtain the extension theorem for proper holomorphic mappings between domains with real analytic…
The paper is devoted to the study of homeomoephisms with finite distortion on the plane with use of the modulus techniques.
These are lecture notes from my talks at the "Current Developments in Mathematics" conference (Harvard, 2006). They cover a variety of topics involving symplectic cohomology. In particular, a discussion of (algorithmic) classification…
We consider links of complex isolated hypersurface singularities in $\mathbb{C}^{n+1}$ and study differentiable maps defined by restricting holomorphic functions to the links. We give an explicit example in which such a restriction gives a…
A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete…
The convenient setting for smooth mappings, holomorphic mappings, and real analytic mappings in infinite dimension is sketched. Infinite dimensional manifolds are discussed with special emphasis on smooth partitions of unity and tangent…
Synthetic algebraic geometry uses homotopy type theory extended with three axioms to develop algebraic geometry internal to a higher version of the Zariski topos. In this article we make no essential use of the higher structure and use…
We classify proper holomorphic mappings between generalized pseudoellipsoids of different dimensions. Those domains are parametrized by the exponents. The relations among them are also obtained. Main tool is the orthogonal decomposition of…
We give a necessary and sufficient condition for the existence of nondegenerate holomorphic mappings between pseudoellipsoidal real hypersurfaces, and provide an explicit parametrization for the collection of all such mappings (in the…