Related papers: The tangential map and associated integrable equat…

The McMillan map is a well-known example of a rational integrable system for one particle in a two-dimensional phase space. An elegant recent paper presented a generalization of the McMillan map to an $N$-body system, for particles moving…

Turaev's shadow can be seen locally as the Stein factorization of a stable map. In this paper, we define the notion of stable map complexity for a compact orientable 3-manifold bounded by (possibly empty) tori counting, with some weights,…

In this short article, we determine the bigness of the tangent bundle $T_X$ of the projective bundle $X=\mathbb{P}_C(E)$ associated to a vector bundle $E$ on a smooth projective curve $C$.

We show that through a point of an affine variety there always exists a smooth plane curve inside the ambient affine space, which has the multiplicity of intersection with the variety at least 3. This result has an application to the study…

An identity map $(M,g)\longrightarrow(M,g)$ is a harmonic from a Riemannian manifold $(M,g)$ onto itself. In this paper, we study the harmonicity of identity maps $(M,g)\longrightarrow(M,g-df\otimes df)$ and $(M,g-df\otimes…

We study projective surfaces in $\mathbb{P}^3$ which can be written as Hadamard product of two curves. We show that quadratic surfaces which are Hadamard product of two lines are smooth and tangent to all coordinate planes, and such…

We develop a theory of conically smooth stratified spaces and their smooth moduli, including a notion of classifying maps for tangential structures. We characterize continuous space-valued sheaves on these conically smooth stratified spaces…

A Schnyder wood is an orientation and coloring of the edges of a planar map satisfying a simple local property. We propose a generalization of Schnyder woods to graphs embedded on the torus with application to graph drawing. We prove…

We consider some planar triangular maps. These maps preserve certain fibration of the plane. We assume that there exists an invariant attracting fiber and we study the limit dynamics of those points in the basin of attraction of this…

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components, and $\mathcal{C}(N)$ be the complex of curves of $N$. Suppose that $g + n \leq 3$ or $g + n \geq 5$. If $\lambda : \mathcal{C}(N) \rightarrow…

These notes summarize two different connections between two discrete integrable systems, the $A_d$ $T$-system and its infinite-rank analog, the octahedron relation, and the pentagram map and its various generalizations.

This paper describes how to define and work with differential equations in the abstract setting of tangent categories. The key notion is that of a curve object which is, for differential geometry, the structural analogue of a natural number…

Let $X$ be a smooth projective variety. Define a stable map $f:C\to X$ to be "eventually smoothable" if there is an embedding $X\hookrightarrow\mathbb{P}^N$ such that $(C,f)$ occurs as the limit of a $1$-parameter family of stable maps to…

We describe how the loop group maps corresponding to special submanifolds associated to integrable systems may be thought of as certain Grassmann submanifolds of infinite dimensional homogeneous spaces. In general, the associated families…

We study the reflectional symmetry of a surface in the Euclidean 3-dimensional space with a cross-cap singularity with respect to planes. This symmetry is picked up by the singularities of folding maps on the cross-cap. We give a list of…

This article is devoted to the study of mappings defined in the region on the plane. Under certain conditions, the upper estimate of the distortion of the modulus of families of paths is obtained. Similarly, the upper estimate of the…

A framed surface is a smooth surface in the Euclidean space with a moving frame. By using the moving frame, we can define Bertrand framed surfaces as the same idea as Bertrand framed curves. Then we find the caustics and involutes as…

In this paper, we are concerned with the existence of invariant curves of the planar twist mappings where the perturbations are almost periodic. As an application, the existence of almost periodic solutions and the boundedness of all…

Curvature and torsion of linear transports along paths in, respectively, vector bundles and the tangent bundle to a differentiable manifold are defined and certain their properties are derived.

Let M be a smooth strictly convex closed surface in space and denote by H the set of points x in the exterior of M such that all the tangent segments from x to M have equal lengths. In this note we prove that if H is either a closed surface…