Related papers: $E_8$-plumbings and exotic contact structures on s…

We prove that if a contact 3-manifold admits an open book decomposition of genus 0, a certain intersection pattern cannot appear in the homology of any of its minimal symplectic fillings, and moreover, fillings cannot contain symplectic…

In this paper we study topological properties of stable Hamiltonian structures. In particular, we prove the following results in dimension three: The space of stable Hamiltonian structures modulo homotopy is discrete; there exist stable…

In this note we show that a closed oriented contact manifold is obtained from the standard contact sphere of the same dimension by contact surgeries on isotropic and coisotropic spheres. In addition, we observe that all closed oriented…

Let A be an affine variety inside a complex N dimensional vector space which has an isolated singularity at the origin. The intersection of A with a very small sphere turns out to be a contact manifold called the link of A. Any contact…

We prove a contact non-squeezing phenomenon on homotopy spheres that are fillable by Liouville domains with infinite dimensional symplectic homology: there exists a smoothly embedded ball in such a sphere that cannot be made arbitrarily…

It is known that the long line supports $2^{\aleph_1}$ many non-diffeomorphic differential structures. We show that the long plane supports a similar number of exotic differential structures, ie structures which are not merely diffeomorphic…

In this paper, it is proved that under dynamically convex condition, there exist at least $[\frac{n+1}{2}]$ closed Reeb orbits on a closed contact type hypersurface in $T^*S^n$ enclosing the zero section and bounding a simply connected…

Let C be the contact structure naturally induced on the lens space L(p,q) by the standard contact structure on the three--sphere. We obtain a complete classification of the symplectic fillings of (L(p,q),C) up to orientation-preserving…

It is known that the $(2k-1)$-sphere has at most $2^{O(n^k \log n)}$ combinatorially distinct triangulations with $n$ vertices, for every $k\ge 2$. Here we construct at least $2^{\Omega(n^k)}$ such triangulations, improving on the previous…

We introduce a procedure for gluing Weinstein domains along Weinstein subdomains. By gluing along flexible subdomains, we show that any finite collection of high-dimensional Weinstein domains with the same topology are Weinstein subdomains…

We give a construction of contact homology in the sense of Eliashberg--Givental--Hofer. Specifically, we construct coherent virtual fundamental cycles on the relevant compactified moduli spaces of pseudo-holomorphic curves.

In this paper we prove the presence of an embedded plastikstufe implies overtwistedness of the contact structure in any dimension. Moreover, we show in dimension 5 that the presence of an embedded bordered Legendrian open book (bLob) also…

We obtain necessary conditions for the existence of special K\"ahler structures with isolated singularities on compact Riemann surfaces. We prove that these conditions are also sufficient in the case of the Riemann sphere and, moreover, we…

In this paper we develop a method for studying tight contact structures on lens spaces. We then derive uniqueness and non-existence statements for tight contact structures with certain (half) Euler classes on lens spaces. We also prove that…

One of the emerging problems in algebraic geometry is to characterize the affine $n$-space $\mathbb{A}^n$ among smooth affine schemes up to $\mathbb{A}^1$-contractibility. Recent efforts show that this characterization holds in dimensions…

Continuing the investigations of Harborth (1974) and the author (2002) we study the following two rather basic problems on sphere packings. Recall that the contact graph of an arbitrary finite packing of unit balls (i.e., of an arbitrary…

We prove that a compact smooth 4-manifold admits generalized complex structures of odd type if and only if it has a transversely holomorphic 2-foliation. Consequently, there exist generalized complex structures of odd type on a circle…

We construct 2^{\Omega(n^{5/4})} combinatorial types of triangulated 3-spheres on n vertices. Since by a result of Goodman and Pollack (1986) there are no more than 2^{O(n log n)} combinatorial types of simplicial 4-polytopes, this proves…

We show that the novel figure eight singularity in a pseudoholomorphic quilt can be continuously removed when composition of Lagrangian correspondences is cleanly immersed. The proof of this result requires a collection of width-independent…

We show some characterizations of hyperspheres in the $(n+1)$-dimensional Euclidean space ${\Bbb E}^{n+1}$ with intrinsic and extrinsic properties such as the $n$-dimensional area of the sections cut off by hyperplanes, the…