相关论文: Immersions and translation structures I: The space…

A translation structure on a surface is an atlas of charts to the plane so that the transition functions are translations. We allow our surfaces to be non-compact and infinite genus. We endow the space of all pointed surfaces equipped with…

In this paper we introduce a new kind of topological space, called 'structured space', which locally resembles various kinds of algebraic structures. This can be useful, for instance, to locally study a space that cannot be globally endowed…

In this paper, we prove a theorem about embedding of some partially ordered topological spaces in topological hyperspaces equipped with Fell topology. Then we give some examples to show that the map defining the embedding may not be…

Translation surfaces can be defined in an elementary way via polygons, and arise naturally in in the study of various basic dynamical systems. They can also be defined as Abelian differentials on Riemann surfaces, and have moduli spaces…

The study of topological information of spatial objects has for a long time been a focus of research in disciplines like computational geometry, spatial reasoning, cognitive science, and robotics. While the majority of these researches…

Modulated symmetries are internal symmetries that are not invariant under spacetime symmetry actions. We propose a general way to describe the lattice translation modulated symmetries in 1+1D, including the non-invertible ones, via the…

In this paper, we introduce a generalized piecewise translation map on the Euclidean space. We provide a special case when this map is always of finite type. For a finite type map in this case, we form conjectures on the semi-continuity of…

To understand an aperiodic tiling (or a quasicrystal modeled on an aperiodic tiling), we construct a space of similar tilings, on which the group of translations acts naturally. This space is then an (abstract) dynamical system. Dynamical…

Persistence modules are a central algebraic object arising in topological data analysis. The notion of interleaving provides a natural way to measure distances between persistence modules. We consider various classes of persistence modules,…

Starting from filters over the set of indices, we introduce structures in a product of sets where the coordinate sets have the given structures.

We characterize subgroups of the mapping class group that stabilize a Teichmueller disk in terms of ellipses and strips that are immersed in the associated translation surface. In particular, we show that the space of immersed…

We give an explicit simple construction for classifying spaces of maps obtained as hyperplane projections of immersions. We prove structure theorems for these classifying spaces.

After introducing some motivations for this survey, we describe a formalism to parametrize a wide class of algebraic structures occurring naturally in various problems of topology, geometry and mathematical physics. This allows us to define…

Differentials on Riemann surfaces correspond to translation surfaces with conical singularities, and affine transformations acting on them preserve the orders of these singularities. This viewpoint allows the moduli spaces of differentials…

A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules. We…

Finite translation surfaces can be classified by the order of their singularities. When generalizing to infinite translation surfaces, however, the notion of order of a singularity is no longer well-defined and has to be replaced by new…

Let $S$ be a closed orientable surface of genus at least two. We introduce a bordification of the moduli space $\mathcal{PT}(S)$ of complex projective structures, with a boundary consisting of projective classes of half-translation…

We discuss selected topics on the topology of moduli spaces of curves and maps, emphasizing their relation with Gromov--Witten theory and integrable systems.

In this paper, we are interested in flat metric structures with conical singularities on surfaces which are obtained by deforming translation surface structures. The moduli space of such flat metric structures can be viewed as some…

Let M be a closed hyperbolic three manifold. We construct closed surfaces which map by immersions into M so that for each one the corresponding mapping on the universal covering spaces is an embedding, or, in other words, the corresponding…