arXiv research

New proof for discrete Morse theory using combinatorial construction.

We derive a discrete analogue of Morse-Bott theory on CW complexes and use this discrete Morse-Bott function to do some Conley theory analysis. It turns out that our discrete Morse-Bott theory is indeed a generalization of Forman's discrete Morse theory.

Paper studies gradient fields from discrete Morse functions for watershed-cut computation.

Discrete Morse functions induce shellings with critical tiles corresponding to function's critical faces.

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) simplicial complexes are Morse shellable (or tileable). Moreover, every triangulated clo…

After surveying classical notions of PL topology of the Seventies, we clarify the relation between Morse theory and its discretization by Forman. We show that PL handles theory and discrete Morse theory are equivalent, in the sense that every discrete Morse vector on some PL triangulation is also a PL handle vector, an…

In the case of smooth manifolds, we use Forman's discrete Morse theory to realize combinatorially any Thom-Smale complex coming from a smooth Morse function by a couple triangulation-discrete Morse function. As an application, we prove that any Euler structure on a smooth oriented closed 3-manifold has a particular rea…

1) We introduce random discrete Morse theory as a computational scheme to measure the complicatedness of a triangulation. The idea is to try to quantify the frequence of discrete Morse matchings with a certain number of critical cells. Our measure will depend on the topology of the space, but also on how nicely the spa…

Local-to-global principle for Morse actions on symmetric spaces.

It is proved that every discrete Morse function in the sense of Forman on a finite regular CW complex can be represented by a polyhedral Morse function in the sense of Banchoff on an appropriate embedding in Euclidean space of the barycentric subdivision of the CW complex; such a representation preserves critical point…

Extends Morse-Forman theory to vector-valued functions for multiparameter persistence.

In bounding the homology of a manifold, Forman's Discrete Morse theory recovers the full precision of classical Morse theory: Given a PL triangulation of a manifold that admits a Morse function with c_i critical points of index i, we show that some subdivision of the triangulation admits a boundary-critical discrete Mo…

The paper uses MDM theory to analyze multifiltering functions on simplicial complexes.

Given a finite set of points in $\mathbb R^n$ and a radius parameter, we study the Čech, Delaunay-Čech, Delaunay (or Alpha), and Wrap complexes in the light of generalized discrete Morse theory. Establishing the Čech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, we prove that the four…

In this paper, we study the discrete Morse flow for the Ricci flow on football, which is the 2-sphere with removed north and south poles and with the metric $g_0$ of constant scalar curvature, and and for Porous media equation on a bounded regular domain in the plane. We show that with a suitable assumption about $g(0)…

The paper characterizes discrete Morse functions on knot diagrams and generalizes a clock theorem.

We present extremal constructions connected with the property of simplicial collapsibility. (1) For each $d \ge 2$, there are collapsible (and shellable) simplicial $d$-complexes with only one free face. Also, there are non-evasive $d$-complexes with only two free faces. (Both results are optimal in all dimensions.) (2…

New method connects curvature and Persistent Homology for networks.

We study the geometry and dynamics of discrete infinite covolume subgroups of higher rank semisimple Lie groups. We introduce and prove the equivalence of several conditions, capturing "rank one behavior'' of discrete subgroups of higher rank Lie groups. They are direct generalizations of rank one equivalents to convex…

The study identifies all possible vector field structures on specific 2D shapes.

Discrete Morse theory simplifies Khovanov homology calculations.

New connection found between shape reconstruction methods and persistent homology.

The counting function on the natural numbers defines a discrete Morse-Smale complex with a cohomology for which topological quantities like Morse indices, Betti numbers or counting functions for critical points of Morse index are explicitly given in number theoretical terms. The Euler characteristic of the Morse filtra…

The paper develops algorithms and topological invariants for distinguishing dynamic systems.

Morse inequalities for noncompact manifolds with group action.

Floer constructs homology from flow lines in generalized dynamical systems and combinatorial vector fields.

Optimal Morse matchings reveal essential structures of cell complexes which lead to powerful tools to study discrete geometrical objects, in particular discrete 3-manifolds. However, such matchings are known to be NP-hard to compute on 3-manifolds, through a reduction to the erasability problem. Here, we refine the stu…

Contractible Vietoris-Rips complexes for integer n proved using discrete Morse theory.

Discrete Morse theory emerged as an essential tool for computational geometry and topology. Its core structures are discrete gradient fields, defined as acyclic matchings on a complex $C$, from which topological and geometrical informations of $C$ can be efficiently computed, in particular its homology or Morse-Smale d…

We study Morse representations of discrete subgroups in higher rank semi-simple Lie groups defined by M. Kapovich, B. Leeb and J. Porti. We show that, if a sequence of Morse representations $ρ_n : Γ\rightarrow G$ is (strongly) unbounded in the character variety, the group must have a very particular structure.

New methods merge discrete gradient fields from patches to correct errors.

Study Morse complexity of manifolds and homology classes, proving bounds and implications.

In 1967, Chillingworth proved that all convex simplicial 3-balls are collapsible. Using the classical notion of tightness, we generalize this to arbitrary manifolds: We show that all tight simplicial 3-manifolds admit some perfect discrete Morse function. We also strengthen Chillingworth's theorem by proving that all c…

Study on Gauss map of anisotropic minimal surfaces with Morse index estimates.

New technique connects graph matching complexes to Morse theory for better topology understanding.

New spectral sequences derived from shellable tilings.

We formalize an equivariant version of Bestvina-Brady discrete Morse theory, and apply it to Vietoris-Rips complexes in order to exhibit finite universal spaces for proper actions for all asymptotically CAT(0) groups.

We show how to construct homology bases for certain CW complexes in terms of discrete Morse theory and cellular homology. We apply this technique to study certain subcomplexes of the half cube polytope studied in previous works. This involves constructing explicit complete acyclic Morse matchings on the face lattice of…

Product of shellable complexes yields shellable triangulations under tameness conditions.

We prove a Morse Lemma for coarsely regular quasigeodesics in nonpositively curved symmetric spaces and euclidean buildings X. The main application is a simpler coarse geometric characterization of Morse subgroups of the isometry groups Isom(X) as undistorted subgroups which are coarsely uniformly regular. We show furt…

We prove several combinatorial results on path algebras over discrete structures related to directed graphs. These results are motivated by Morse theory on a manifold with boundary and, more generally, by Floer theory on a configuration space with boundary. Their purpose is to organize cobordism relationships among mod…

The paper explores heat flow and constants on graphs, proving properties and proposing new concepts.

In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of one-dimensional lat…

We introduce a novel combinatorial method to study $Q^{**}$-transformations of group presentations or, equivalently, 3-deformations of CW-complexes of dimension 2. Our procedure is based on a refinement of discrete Morse theory that gives a Whitehead simple homotopy equivalence from a regular CW-complex to the simplifi…

Morse neural networks improve uncertainty quantification and detection.

The aim of this paper is to develop a refinement of Forman's discrete Morse theory. To an acyclic partial matching $μ$ on a finite regular CW complex $X$, Forman introduced a discrete analogue of gradient flows. Although Forman's gradient flow has been proved to be useful in practical computations of homology groups, i…

Researchers use discrete Morse theory to improve the topology of matching complexes of complete graphs.

Completeness of the eigenfunctions of a quantum mechanical system is crucial for its probability interpretation. By using the method of contour integral we give properly normalized eigenfunctions for both discrete and continuum spectrum of the Morse potential, and explicitly prove the completeness relation. As an appli…