Cancellation of a critical pair in discrete Morse theory and its effect on (co)boundary operators

Abstract Given a squarefree monomial ideal I of a polynomial ring Q , we show that if the minimal free resolution $$\mathbb {F}$$ F of Q / I admits the structure of a differential graded (dg) algebra, then so does any “pruning” of $$\mathbb {F}$$ F . In the language of combinatorics, this says that if $$Q/\mathcal {F}(\Delta )$$ Q / F ( Δ ) , the quotient of the ambient polynomial ring by the facet ideal $$\mathcal {F}(\Delta )$$ F ( Δ ) of a simplicial complex $$\Delta $$ Δ , is minimally resolved by a dg algebra, then so is the quotient by the facet ideal of each facet-induced subcomplex of $$\Delta $$ Δ (over the smaller polynomial ring). Along with techniques from discrete Morse theory and homological algebra, this allows us to give complete classifications of the trees and cycles G with Q / I ( G ) minimally resolved by a dg algebra in terms of the length of the longest path in G , where I ( G ) is the edge ideal of G .

Abstract From the perspective of Morse theory, it is natural to investigate gradient flow trajectories between critical points. In this short note, we explore the minimal hypersurface analogue of this phenomenon and present examples that suggest additional topological and variational obstructions to the existence of connecting mean curvature flows.

In this paper, we prove a universality result for the limiting distribution of persistence diagrams arising from geometric filtrations over random point processes. Specifically, we consider the distribution of the ratio of persistence values (death/birth), and show that for fixed dimension, homological degree and filtration type (Čech or Vietoris-Rips), the limiting distribution is independent of the underlying point process distribution, i.e., universal. In proving this result, we present a novel general framework for universality in scale-invariant functionals on point processes. Finally, we also provide a number of new results related to Morse theory in random geometric complexes, which may be of an independent interest.

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Topological reconstruction of sampled surfaces via Morse theory

In this paper, we study Spin*(8)-Higgs bundles over compact Riemann surfaces, extending the work of Bradlow, García-Prada, and Gothen on SO*(8). The group Spin*(8) is exceptional among classical real forms, as its complexification Spin(8,C) admits triality, an outer automorphism of order 3, but triality does not preserve the real form Spin*(8). We establish the Toledo bound |τ|≤4(g−1) for semistable Spin*(8)-Higgs bundles and characterize maximal bundles through rigidity theorems. We prove that the moduli space of maximal bundles fibers over the SO*(8) moduli space with discrete fibers parametrized by spin structures, and has a dimension of 15(g−1), one less than expected. Using Morse theory, we establish connectedness of moduli spaces for τ=0 and maximal |τ|. Via the non-abelian Hodge correspondence, our results yield connectedness theorems for character varieties of surface group representations into Spin*(8). We analyze how triality determines the decomposition of the isotropy representation despite not acting on the real form.

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, where each Morse function on a simplicial complex $K$ defines a sequence of elementary internal collapses. This reduction guarantees the existence of a CW-complex that is homotopy equivalent to $K$ , with cells corresponding to the critical simplices of the Morse function. However, this approach lacks an explicit combinatorial description of the attaching maps, which limits the reconstruction of the homotopy type of $K$ . By restricting discrete Morse functions to those induced by total orders on the vertices, we develop a strong discrete Morse theory , generalizing the strong collapses introduced by Barmak and Minian. We show that, in this setting, the resulting reduced CW-complex is regular, enabling us to recover its homotopy type combinatorially. We also provide an algorithm to compute this reduction and apply it to obtain efficient structures for complexes in the library of triangulations by Benedetti and Lutz.

In this paper, we investigate the existence and the multiplicity of solutions to the elliptic equation with Dirichlet boundary value condition$ \begin{cases}-\Delta u = f(u), &amp; \text{in} \ \Omega, \\ u = 0, &amp; \text{on} \ \partial \Omega, \end{cases} $where $ \Omega\subset\mathbb{R}^{N} $ is an open, bounded, smooth domain. We show that under suitable conditions, the elliptic equation with superlinear terms possesses at least nine nontrivial solutions. Our approach combines the sub-supersolution method, the mountain pass theorem in order intervals, bifurcation theory, degree theory, Morse theory, and the invariance of descending flow method.

Mapping cone and Morse theory

This paper investigates the existence of nontrivial solutions for the Schrödinger-Bopp-Podolsky system with an indefinite potential function V( x ) . The indefiniteness of V prevents the direct application of standard variational methods such as the mountain pass theorem, as the associated Schrödinger operator −Δ+V possesses a finite-dimensional negative space, leading to a loss of coercivity and the standard linking structure in the energy functional. By employing a reduction method to handle the coupling with the Bopp-Podolsky equation and applying Morse theory combined with critical groups at infinity, we establish the existence of at least one nontrivial solution under appropriate assumptions on the indefinite potential V . and the nonlinearity g .

Abstract The existence of acyclic complete matchings on the face poset of a regular CW complex implies that the underlying topological space of the CW complex is contractible by discrete Morse theory. In this paper, we construct explicitly acyclic complete matchings on any nontrivial Bruhat interval based on any reflection order on the Coxeter group . We then apply this combinatorial result to regular CW complexes arising from the theory of total positivity. As an application, we show that the totally nonnegative Springer fibers are contractible. This verifies a conjecture of Lusztig. As another application, we show that the totally nonnegative fibers of the natural projection from full flag varieties to partial flag varieties are contractible. This leads to a much simplified proof of the regularity property on totally nonnegative partial flag varieties compared to the proofs by Galashin et al. and Bao and He.

We consider periodic solutions of the following nonlinear system associated with the fractional Laplacian $$(−∂_{xx})^su(x)+∇F(u(x))=0 \quad {\rm in} \quad \mathbb{R},$$ where $F$ : $\mathbb{R}^2$→$\mathbb{R}$ is a smooth double-well potential. For the case that $F$ is even in its two variables we obtain the existence of more and more periodic solutions with large period, by using Clark’s theorem. For the case that $F$ is only even in its the second variable and the origin is a saddle critical point of $F$, we give two periodic solutions by using Morse theory.

In this paper, we show that a notion of non-degeneracy which allows to develop Morse theory is generically satisfied for a large class of [Formula: see text]-functionals defined on Banach spaces. The main element of novelty with respect to the previous work [L. Asselle and M. Starostka, A note on the Morse homology for a class of functionals in Banach spaces involving the [Formula: see text]-area functional, NoDEA Nonlinear Differential Equations Appl. 31 (2024) 75, doi:10.1007/s00030-024-00962-3.] of the first and third author is that we do not assume the splitting induced by the second differential at a critical point to persist in a neighborhood, provided one can give precise estimates on how much persistence fails. This allows us to enlarge significantly the class of elliptic pde’s for which non-degeneracy holds and Morse homology can be defined. A concrete example is given by equations involving the [Formula: see text]-Laplacian, [Formula: see text]. As a byproduct, we provide a criterion of independent interest to check whether critical points are non-degenerate in the sense above, and give an abstract construction of Morse homology in a Banach setting for functionals satisfying the Cerami condition.

We study the algebraic complexity of Euclidean Distance (ED) minimization from a generic tensor to a variety of rank-one tensors. The ED degree of the Segre-Veronese variety counts the number of complex critical points of this optimization problem. We regard this invariant as a function of inner products. We prove that Frobenius inner product is a local minimum of the ED degree, and conjecture that it is a global minimum. We prove our conjecture in the case of matrices and symmetric binary and 3 × 3 × 3 3\times 3\times 3 tensors. We discuss the above optimization problem for other algebraic varieties, classifying all possible values of the ED degree. Our approach combines tools from Singularity Theory, Morse Theory, and Algebraic Geometry.

A multiparameter filtration, or a multifiltration, may in many cases be seen as the collection of sublevel sets of a vector function, which we call a multifiltering function. The main objective of this paper is to obtain a better understanding of such functions through multiparameter discrete Morse (mdm) theory, which is an extension of Morse-Forman theory to vector-valued functions. Notably, we prove algorithmically that any multifiltering function defined on a simplicial complex can always be approximated by a compatible mdm function. Moreover, we define the Pareto set of a discrete multifiltering function and show that the concept links directly to that of critical simplices of a mdm function. Finally, we experiment with these notions using triangular meshes.

For mathematical programs with complementarity constraints (MPCC), we study the stability properties of their Scholtes regularization. Our goal is to relate nondegenerate C-stationary points of MPCC with nondegenerate Karush-Kuhn-Tucker points of the Scholtes regularization up to their topological type. As it is standard in the framework of Morse theory, the topological types are captured by the C-index and the quadratic index, respectively. It turns out that a change of the topological type for the approximating Karush-Kuhn-Tucker points of the Scholtes regularization and their limiting C-stationary point is possible. In particular, a minimizer of MPCC with zero C-index might be approximated by saddle points of the Scholtes regularization with nonzero quadratic index. In order to bypass this index shift phenomenon, an additional generic condition for nondegenerate C-stationary points of MPCC is identified. It says that nonbiactive multipliers under consideration should not vanish. Then, we uniquely trace nondegenerate Karush-Kuhn-Tucker points of the Scholtes regularization and successively maintain the topological type of their limiting C-stationary point. The main technical issue here is to relate the first-order information of the defining functions, which enters the biactive part of the C-index, with the second-order information, which enters the quadratic index of the Karush-Kuhn-Tucker points.

We give a new, short proof of a result of Virk, that the Vietoris–Rips complex of the group Z n \mathbb {Z}^n with the standard word metric is contractible at large enough scales. This is inspired by a key observation in Virk’s proof, but we use Bestvina–Brady discrete Morse theory to get a very short proof with better bounds. In the course of this, we get a new, general criterion for a metric space to have contractible Vietoris–Rips complexes at large enough scales, which could prove useful in the future.

In this paper, we define Reidemeister torsion and analytic torsion for digraphs by means of Morse theory (abbreviated as M-torsion) of digraphs. We study the structure of Morse complex of digraphs and join of digraphs, illustrate through some typical examples that M-torsions are different from torsions defined by the path complex, and prove that M-torsion of contractible digraphs and cone are all equal to 1.

We study the k-th nearest neighbor distance function from a finite point-set in . We provide a Morse theoretic framework to analyze the sub-level set topology. In particular, we present a simple combinatorial-geometric characterization for critical points and their indices, along with detailed information about the possible changes in homology at the critical levels. We conclude by computing the expected number of critical points for a homogeneous Poisson process. Our results deliver significant insights and tools for the analysis of persistent homology in order-k Delaunay mosaics, and random k-fold coverage.