Related papers: Cascades and perturbed Morse-Bott functions

We prove that the topological flow category $\mathcal{M}$ arising from a Morse-Smale pair $(f,\xi)$ on a smooth closed manifold $X$ is equivalent, as an $\infty$-category, to Lurie's $\infty$-category $\mathrm{Sing}_A(X)$ of exit paths in…

Various curve complexes with vertices representing multicurves on a surface $S$ have been defined, for example [3], [4] and [8]. The homology curve complex $\mathcal{HC}(S,\alpha)$ defined in [7] is one such complex, with vertices…

One of the main aims of this article is to give the complete classification of critical metrics of the volume functional on a compact manifold $M$ with boundary $\partial M$ and with harmonic Weyl tensor, which improves the corresponding…

We relate the topology of the Morse boundary of a group to geometric and algorithmic properties of the group. In particular, we show that a group has $\sigma$-compact Morse boundary if and only if it is Morse local-to-global. We also…

The purpose of this work is to develop a version of Forman's discrete Morse theory for simplicial complexes, based on internal strong collapses. Classical discrete Morse theory can be viewed as a generalization of Whitehead's collapses,…

We will use the tools developed in [Rie24] to give a Morse-theoretic description of a string topology product on the homology of the space of paths in a manifold Y with endpoints in a submanifold X and a module structure on this homology…

We show that self-similar sets arising from iterated function systems that satisfy the Moran open-set condition, a canonical class of fractal sets, are `equi-homogeneous'. This is a regularity property that, roughly speaking, means that at…

We introduce the notion of a two-scale branching function associated with an arbitrary metric space, which encodes the lower and upper box dimensions as well as the Assouad spectrum. If the metric space is quasi-doubling, this function is…

We consider a class of infinite weighted metric trees obtained as perturbations of self-similar regular trees. Possible definitions of the boundary traces of functions in the Sobolev space on such a structure are discussed by using…

We introduce the notion of measurable bounded cohomology for measured groupoids, extending continuous bounded cohomology of locally compact groups. We show that the measurable bounded cohomology of the semidirect groupoid associated to a…

We introduce a notion of Morse shellings (and tilings) on finite simplicial complexes which extends the classical one and its relation to discrete Morse theory.Skeletons and barycentric subdivisions of Morse shellable (or tileable)…

We show that Busemann functions on a smooth, non-compact, complete, boundaryless, connected Riemannian manifold are viscosity solutions with respect to the Hamilton-Jacobi equation determined by the Riemannian metric and consequently they…

We show that the Morse index of a closed minimal hypersurface in a four-dimensional Riemannian manifold cannot be bound in terms of the volume and the topological invariants of the hypersurface itself by presenting a method for constructing…

We compare the higher analytic torsion of Bismut and Lott of a fibre bundle p: M -> B equipped with a flat vector bundle F -> M and a fibre-wise Morse function h on M with a higher torsion T that is constructed in terms of a families…

It is well known that the cohomology groups of a closed manifold $M$ can be reconstructed using the gradient dynamical of a Morse-Smale function $f\colon M\to \R$. A direct result of this construction are Morse inequalities that provide…

Given a manifold $M$, some closed $\beta\in\Omega^1(M)$ and a map $f\in C^\infty(M)$, a $\beta$-critical point is some $x\in M$ such that $d_\beta f_{x}=0$ for the Lichnerowicz derivative $d_\beta$. In this paper, we will give a lower bound…

Here we show a procedure to generate an analytical structure producing a cascade that scales as the energy spectrum in isotropic homogeneous turbulence. We obtain a function that unveils a non-self-similar fractal at the origin of the…

We develop a version of discrete Morse theory for finite regular CW complexes equipped with an auxiliary stratification. The key construction is the halo of a cell, which contains all those faces in the boundary that enter closed…

We compute the Hausdorff, upper box and packing dimensions for certain inhomogeneous Moran set constructions. These constructions are beyond the classical theory of iterated function systems, as different nonlinear contraction…

Consider a family $\mathcal F$ of $C_{2r+1}$-free graphs, where $r\geq 2$. Suppose that each graph in $\mathcal F$ has minimum degree linear in its number of vertices. Thomassen showed that such a family has bounded chromatic number, or,…