Related papers: Affine open subsets in A^3 without the cancellatio…

We investigate the geometry of closed, orientable, hyperbolic $3$-manifolds whose fundamental groups are $k$-free for a given integer $k\ge 3$. We show that any such manifold $M$ contains a point $P$ of $M$ with the following property: If…

We complete the classification of algebraic monoid structures on the affine 3-space. The result is based on a reduction of the general case to that of commutative monoids. We also study various algebraic properties of all monoids appearing…

We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau 3-folds;…

We consider open, oriented 3-manifolds which are infinite connected sums of closed 3-manifolds. We introduce some topological invariants for these manifolds and obtain a classification in the case where there are only finitely many summands…

We show that the generic fiber of a family of smooth $\mathbb{A}^{1}$-ruled affine surfaces always carries an $\mathbb{A}^{1}$-fibration, possibly after a finite extension of the base. In the particular case where the general fibers of the…

The author proves that there is an open non empty set of metrics on any 3-manifold such that there exists a family of stably embedded minimal 2-spheres whose area is unbounded. This generalizes the work of T. Colding and W. Minicozzi who…

We construct new families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of form $Z\times A^1$, where $Z$ is a quasiprojective variety. The affine cones over such a fourfold admit effective…

We show that if $f(u)\in \mathbb{Z}[u]$ is a monic cubic polynomial, then for all but finitely many $n\in \mathbb{Z}$ the affine cubic surface $f(u_{1})+f(u_{2})+f(u_{3})=n \subset \mathbb{A}^{3}_{\mathbb{Z}}$ has no integral Brauer-Manin…

We give a classification of noncommutative algebraic monoid structures on normal affine varieties such that the group of invertible elements of the monoid is connected, solvable, and has a one-dimensional unipotent radical. We describe the…

We give a class of examples of reducible (d-semistable) threefolds of CY type with two irreducible components for which (it is reasonably easy to prove that) no family of admissible genus zero stable maps sweeps out a surface, yet such…

We prove that under some assumptions on how points escape to infinity in the universal cover, homeomorphisms of hyperbolic 3-manifolds are forced to have several invariant sets (in particular, they cannot be minimal). For this, we use some…

We construct algebraic families of exotic affine 3-spheres, that is, smooth affine threefolds diffeomorphic to a non-degenerate smooth complex affine quadric of dimension 3 but non algebraically isomorphic to it. We show in particular that…

We construct families of smooth affine surfaces with pairwise non isomorphic A 1-cylinders but whose A 2-cylinders are all isomorphic. These arise as complements of cuspidal hyperplane sections of smooth projective cubic surfaces.

Let $(M,g)$ be an asymptotically flat $3$-manifold containing no closed embedded minimal surfaces. We prove that for every point $p\in M$ there exists a complete properly embedded minimal plane in $M$ containing $p$.

For $\Gamma_1$-structures on 3-manifolds, we give a very simple proof of Thurston's regularization theorem, first proved in \cite{thurston}, without using Mather's homology equivalence. Moreover, in the co-orientable case, the resulting…

It is well-known that there is a faithful representation of braid groups on automorphism groups of free groups, and it is also well-known that free groups are bi-orderable. We investigate which n-strand braids give rise to automorphisms…

Principal outer automorphisms were introduced in Algom-Kfir-Kapovich-Pfaff to emulate principal pseudo-Anosov surface homeomorphisms, i.e. those whose attracting and repelling invariant foliations have only 3-pronged singularities. It is…

This is an introduction to noncommutative geometry, from an affine viewpoint, that is, by using coordinates. The spaces $\mathbb R^N,\mathbb C^N$ have no free analogues in the operator algebra sense, but the corresponding unit spheres…

In this paper, we study non-commutative projective schemes whose associated non-commutative graded algebras are finite over their centers. We study their moduli spaces of stable sheaves, and construct a symmetric obstruction theory in the…

We show that every convex polyhedron admits a simple edge unfolding after an affine transformation. In particular there exists no combinatorial obstruction to a positive resolution of Durer's unfoldability problem, which answers a question…