Related papers: Morse shellings and compatible discrete Morse func…
Any watershed, when defined on a stack on a normal pseudomanifold of dimension d, is a pure (d -- 1)-subcomplex that satisfies a drop-of-water principle. In this paper, we introduce Morse stacks, a class of functions that are equivalent to…
We construct a smooth Riemannian metric on any 3-manifold with the property that there are genus zero embedded minimal surfaces of arbitrarily high Morse index.
If A is a finite-dimensional symmetric algebra, then it is well-known that the only silting complexes in $\mathrm{K^b}(\mathrm{proj}A)$ are the tilting complexes. In this note we investigate to what extent the same can be said for weakly…
We develop a theory of simple pentagonal subdivision of quadrilateral tilings, on orientable as well as non-orientable surfaces. Then we apply the theory to answer questions related to pentagonal tilings of surfaces, especially those…
In this note, we give a proof of the famous theorem of M. Morse dealing with the cancellation of a pair of non-degenerate critical points of a smooth function. Our proof consists of a reduction to the one-dimensional case where the question…
For a smooth, closed $n$-manifold $M$, we define an upper semi-continuous integer-valued complexity function on $H^1(M;{\mathbb R})$ using Morse theory. This measures how far an integral class is from being a fiber of a fibration. The fact…
In this paper we consider non-singular Morse-Smale flows on closed orientable 3-manifolds, under the assumption that among the periodic orbits of the flow there is only one saddle orbit and it is twisted. It is found that any manifold…
As a toy model of chiral interactions in crowded spaces, we consider sphinx tilings in finite regions of the triangular lattice. The sphinx tiles, hexiamonds composed of six equilateral triangles in the shape of a stylized sphinx, come in…
We study the weighted partition function for lozenge tilings, with weights given by multivariate rational functions originally defined by Morales, Pak and Panova (2019) in the context of the factorial Schur functions. We prove that this…
We present an algorithm for computing the main topological characteristics of three-dimensional bodies. The algorithm is based on a discretization of Morse theory and uses discrete analogs of smooth functions with only nondegenerate (Morse)…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…
The class of special generic maps contains Morse functions with exactly two singular points, characterizing spheres topologically which are not $4$-dimensional and the $4$-dimensional unit sphere. This class is for higher dimensional…
The Thurston spine consists of the subset of Teichm\"uller space at which the set of shortest curves, the systoles, cuts the surface into polygons. The systole function is a topological Morse function on Teichm\"uller space. This paper…
Let $M$ be a smooth closed orientable surface. Let $F$ be the space of Morse functions on $M$, and $\mathbb{F}^1$ the space of framed Morse functions, both endowed with $C^\infty$-topology. The space $\mathbb{F}^0$ of special framed Morse…
We study here slopes of periodicity of tilings. A tiling is of slope if it is periodic along direction but has no other direction of periodicity. We characterize in this paper the set of slopes we can achieve with tilings, and prove they…
By computing all cyclotomic points on some algebraic varieties, we get an independent and efficient way to find all rational $a^3b$-monotiles for the sphere, thereby completing the classification of edge-to-edge monohedral quadrilateral…
Neeman shows that the completion of a triangulated category with respect to a good metric yields a triangulated category. We compute completions of discrete cluster categories with respect to metrics induced by internal t-structures. In…
We show that compact Riemannian three-manifolds with negative sectional curvature possess closed minimal surfaces of arbitrarily high Morse index.
We investigate the geometric constraints imposed by low Morse index on minimal surfaces with Y-singularities, focusing on the classification of those with Morse index one. Our rigidity result establishes a partial uniqueness theorem,…
We list special graphs of degree 4 with at most 3 vertices (atoms from the theory of integrable hamiltonian systems) which could be represented by a union of closed geodesics on the one of the following surfaces with metric of constant…