Related papers: Representations of Reeb spaces via simplified grap…

As a higher dimensional version of the theory of Morse functions, there have been various studies of smooth manifolds using generic smooth maps. As fundamental results, in these studies, they have found that inverse images of such maps…

Classes of branched surfaces extend the classes of surfaces or 2-dimensional manifolds satisfying suitable properties and defined in various manners. Reeb spaces of smooth maps of suitable classes into surfaces whose codimensions are…

We consider the Reeb graph of a thickening of points sampled from an unknown space. Our main contribution is a framework to transfer reconstruction results similar to the well-known work of Niyogi, Smale, and Weinberger to the setting of…

We introduce topological invariants of semi-decompositions (e.g. filtrations, semi-group actions, multi-valued dynamical systems, combinatorial dynamical systems) on a topological space to analyze semi-decompositions from a dynamical…

In our paper, we introduce special-generic-like maps or SGL maps as smooth maps and study their several algebraic topological and differential topological properties. The new class generalize the class of so-called special generic maps.…

A fold map is a smooth map at each singular point of which it is represented as the product map of a Morse function and the identity map on an open ball. A special generic map is a fold map such that the Morse function can be taken as a…

In this paper, we develop the theory of Sobolev spaces on locally finite graphs, including completeness, reflexivity, separability, and Sobolev inequalities. Since there is no exact concept of dimension on graphs, classical methods that…

We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space $\mathbb{X}$ equipped with a continuous function $f: \mathbb{X} \rightarrow \mathbb{R}$. We first give a categorification of the…

Constructing Morse functions and their higher dimensional versions or fold maps is fundamental, important and challenging in investigating the topologies and the differentiable structures of differentiable manifolds via Morse functions,…

We study function spaces that are related to square-integrable, irreducible, unitary representations of several low-dimensional nilpotent Lie groups. These are new examples of coorbit theory and yield new families of function spaces on…

We prove a Reeb sphere theorem for finite simple graphs. The result bridges two different definitions of spheres in graph theory. We also reformulate Morse conditions in terms of the center manifolds, the level surface graphs {f=f(x)} in…

Morse functions are important objects and tools in understanding topologies of manifolds since the 20th century. Their classification has been natural and difficult problems, and surprisingly, this is recently developing. Since the 2010's,…

As a topic of mathematics, "arrangements", systems of hyperplanes, circles, and general (regular) submanifolds, attract us strongly. We present a natural elementary study of arrangements of circles. It is also a kind of new studies. Our…

This paper is a contribution to the study of the relations between special functions, Lie algebras and rigged Hilbert spaces. The discrete indices and continuous variables of special functions are in correspondence with the representations…

This note deals with quasi-states on the two-dimensional torus. Quasi-states are certain quasi-linear functionals (introduced by Aarnes) on the space of continuous functions. Grubb constructed a quasi-state on the torus, which is invariant…

Riemannian Geometry, Topology and Dynamics permit to introduce partially defined holomorphic functions on the variety of representations of the fundamental group of a manifold. The functions we consider are the complex valued Ray-Singer…

We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology, and introduce new invariants to study these equivalence classes. These new invariants are as simple, but more discerning…

The present paper finds new necessary and sufficient conditions for $6$-dimensional closed and simply-connected manifolds of certain classes to admit special generic maps into certain Euclidean spaces. The class of special generic maps…

As Reeb's theorem shows, Morse functions with exactly two singular points on closed manifolds are very simple and important. They characterize spheres whose dimensions are not $4$ topologically and the $4$-dimensional unit sphere. Special…

We construct a correspondence between epimorphisms $\varphi \colon \pi_1(M) \to F_r$ from the fundamental group of a compact manifold $M$ onto the free group of rank $r$, and systems of $r$ framed non-separating hypersurfaces in $M$, which…