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Masuda (2008) provided the characterization of real Bott manifolds in terms of three operations on upper triangular matrices. We provide a combinatorial characterization of real Bott manifolds up to diffeomorphism in terms of operations on…

Calculus and geometry are ubiquitous in the theoretical modelling of scientific phenomena, but have historically been very challenging to apply directly to real data as statistics. Diffusion geometry is a new theory that reformulates…

We give a method to resolve 4-dimensional symplectic orbifolds making use of techniques from complex geometry and gluing of symplectic forms. We provide some examples to which the resolution method applies.

It is important to have fast and effective methods for simplifying 3-manifold triangulations without losing any topological information. In theory this is difficult: we might need to make a triangulation super-exponentially more complex…

In this article, we introduce a fixed parameter tractable algorithm for computing the Turaev-Viro invariants TV(4,q), using the dimension of the first homology group of the manifold as parameter. This is, to our knowledge, the first…

We describe an efficient algorithm to compute a pseudotriangulation of a finite planar family of pairwise disjoint convex bodies presented by its chirotope. The design of the algorithm relies on a deepening of the theory of visibility…

It is proved that isomorphisms between algebras of smooth functions on Hausdorff smooth manifolds are implemented by diffeomorphisms. It is not required that manifolds are second countable nor paracompact. This solves a problem stated by A.…

Convex optimization is a well-established research area with applications in almost all fields. Over the decades, multiple approaches have been proposed to solve convex programs. The development of interior-point methods allowed solving a…

We consider the class of profinite diffeological spaces, that is, diffeological spaces which diffeologies are deduced by pull-back of diffeologies on finite-dimensional manifolds through a system of projection mappings. This class includes…

We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three…

This dissertation contains a comprehensive study of the topology of 2-manifolds and a complementary analysis of the work done by Edwin E. Moise, L. V. Ahlfors and Ian Richards. Our aim is to study the well known classification of surfaces.…

We propose a new method for constructing partially hyperbolic diffeomorphisms on closed manifolds. As a demonstration of the method we show that there are simply connected closed manifolds that support partially hyperbolic diffeomorphisms.

We provide an alternative, simpler proof of the existence of thick triangulations for noncompact $\mathcal{C}^1$ manifolds. Moreover, this proof is simpler than the original one given in \cite{pe}, since it mainly uses tools of elementary…

In this note we demonstrate that a number of case-heavy combinatorial proofs in the mathematical phylogenetics literature can be proven more compactly using computational support. We use these techniques to also prove several new…

It is proved that the number of deformation types of complex structures on a fixed oriented smooth four-manifold can be arbitrarily large. The considered examples are locally simple abelian covers of rational surfaces.

We show examples of pairs of smooth, compact, homeomorphic 4-manifolds, whose diffeomorphism types are distinguished by the topology of the singular sets of smooth stable maps defined on them. In this distinction we rely on results from…

New invariants for 2-dimensional cell complexes are defined, which can be interpreted as curvature bounds. These invariants are proved to be rational and computable in a companion article. This document is a survey that collects theorems…

This article investigates a new gauge theoretic approach to Einstein's equations in dimension 4. Whilst aspects of the formalism are already explained in various places in the mathematics and physics literature, our first goal is to give a…

Basic facts and definitions of conformal moduli of rings and quadrilaterals are recalled. Some computational methods are reviewed. For the case of quadrilaterals with polygonal sides, some recent results are given. Some numerical…

We describe an algorithm to subdivide automatically a given set of PL n-manifolds (via coloured triangulations or, equivalently, via crystallizations) into classes whose elements are PL-homeomorphic. The algorithm, implemented in the case…