Related papers: Hedgehogs in higher dimensions and their applicati…

We study a homological epimorphism of proper dg algebras and show that if the source is a finite dimensional piecewise hereditary algebra, then the target is derived equivalent to some piecewise hereditary algebra.

We investigate the $C^0$-topology of the group of symplectic diffeomorphisms of positive symplectic rational surfaces. For all but a few exceptions, we prove that the group of Hamiltonian diffeomorphisms forms a connected component in the…

Given a closed oriented surface $\Sigma$ of genus greater than 0, we construct a map $\mathcal{F}$ from the higher-dimensional Heegaard Floer homology of the cotangent fibers of $T^*\Sigma$ to the Hecke algebra associated to $\Sigma$ and…

Suppose that $X$ and $Y$ are surfaces of finite topological type, where $X$ has genus $g\geq 6$ and $Y$ has genus at most $2g-1$; in addition, suppose that $Y$ is not closed if it has genus $2g-1$. Our main result asserts that every…

In the study of holomorphic maps, the term "rigidity" refers to certain types of results that give us very specific information about a general class of holomorphic maps owing to the geometry of their domains or target spaces. Under this…

We study the orbit behavior of a four dimensional smooth symplectic diffeomorphism $f$ near a homoclinic orbit $\Gamma$ to an 1-elliptic fixed point under some natural genericity assumptions. 1-elliptic fixed point has two real eigenvalues…

It is shown that the geometry of locally homogeneous multisymplectic manifolds (that is, smooth manifolds equipped with a closed nondegenerate form of degree > 1, which is locally homogeneous of degree k with respect to a local Euler field)…

We study conjugacy classes of germs of non-flat diffeomorphisms of the real line fixing the origin. Based on the work of Takens and Yoccoz, we establish results that are sharp in terms of differentiability classes and order of tangency to…

We show that any conservative partially hyperbolic diffeomorphism homotopic to the identity is accessible unless the fundamental group of its ambient 3-manifold is virtually solvable. As a consequence, such diffeomorphisms are ergodic,…

For a pointed topological space $X$, we use an inductive construction of a simplicial resolution of $X$ by wedges of spheres to construct a "higher homotopy structure" for $X$ (in terms of chain complexes of spaces). This structure is then…

We analyse divergent diagrams of \(k\)-fold map-germs on \((\mathbb{C}^n,0)\), for $k, n \geq 2$, associated with reflections, adapting to the complex setting the theory of folds associated with involutions on \((\mathbb{R}^n,0)\). In the…

Representations of $C^*$-algebras are realized on section spaces of holomorphic homogeneous vector bundles. The corresponding section spaces are investigated by means of a new notion of reproducing kernel, suitable for dealing with…

We study groups of formal or germs of analytic diffeomorphisms in several complex variables. Such groups are related to the study of the transverse structure and dynamics of Holomorphic foliations, via the notion of holonomy group of a leaf…

For any smooth compact manifold $W$ of dimension at least two we prove that the classifying spaces of its group of diffeomorphisms which fix a set of $k$ points or $k$ embedded disks (up to permutation) satisfy homology stability. The same…

The purpose of this article is towards systematically characterizing (holomorphic) retracts of domains of holomorphy; to begin with, bounded balanced pseudoconvex domains $B \subset \mathbb{C}^N$. Specifically, we show that every retract of…

Quasi-invariant curves are used in the study of hedgehog dynamics. Denjoy-Yoccoz lemma is the preliminary step for Yoccoz's complex renormalization techniques for the study of linearization of analytic circle diffeomorphisms. We give a…

We introduce a new extension in symbolic dynamics on two sets of alphabets, called the zip shift space. In finite case, it represents a finite-to-1 local homeomorphism called zip shift map. Such extension, offers a conjugacy between some…

Employing Morse theory for the global control of monodromy and the method of analytic discs for local extension, we establish a version of the global Hartogs extension theorem in a singular setting: for every domain D of an (n-1)-complete…

In some previous work, we defined an invariant of genus zero nonabelian Hodge spaces taking the form of a diagram. Here, enriching the diagram by fission data to obtain a refined invariant, the enriched tree, including a partition of the…

Let $g:\mathbb{R}^2\to\mathbb{R}$ be a homogeneous polynomial of degree $p>1$, $G=(-g'_{y}, g'_{x})$ be its Hamiltonian vector field, and $G_t$ be the local flow generated by $G$. Denote by $E(G,O)$ the space of germs of $C^{\infty}$…