arXiv subjects

Fabian Lenzen

Publications and source records attributed to Fabian Lenzen.

10 recordsLinked to original sources

arXiv2026

Dualities in Multiparameter Persistence

In the theory of persistent homology, a well known duality relates the barcodes of the absolute homology and relative cohomology of a one-parameter simplicial filtration. Motivated by the problem of computing free presentations of the (co)homology of multiparameter Rips filtrations, we give a multiparameter generalization of this duality. Considering two duality functors on multiparameter persistence modules, the pointwise dual $(-)^*$ and the global dual $(-)^\dagger$, we show that $H_q(C)^* \cong H^{N+q}(C^\dagger)$ for chain complexes $C$ of free $N$-parameter persistence modules with acyclic colimit. We give an elementary and accessible proof based on a long exact sequence argument, and also give an alternate proof that casts the result as a special case of multigraded Grothendieck local duality. As a corollary, we recover a simple correspondence between minimal free resolutions of a persistence module $M$ and those of its pointwise dual $M^*$, a result previously obtained by Miller, 2000. These results form the foundation of a state-of-the-art algorithm for computing free resolutions of the homology of Vietoris--Rips bifiltrations, described in a forthcoming paper.

math.AC↗

arXiv2026

Tropical $k$-means clustering for phylogenetic trees

The asymmetric tropical distance is a distance measure on the tropical torus $\mathbb{R}^n/\mathbb{R}\mathbf{1}$ and in particular on the Bergman fan $B(K_N) \subseteq \mathbb{R}^{\binom{N}{2}}/\mathbb{R}\mathbf{1}$ of the complete graphical matroid. In this paper, we define and analyse a clustering algorithm for equidistant phylogenetic trees based on this distance, using the correspondence between $B(K_N)$ and the space of equidistant trees with $N$ leaves.

math.CO↗

arXiv2026

An Empirically Fast Las Vegas Algorithm for Algebraic Shifting

Improved algorithms for computing (partial and full) exterior algebraic shifts of hypergraphs and simplicial complexes are presented. The main benefit is in positive characteristic. Experiments with an implementation in OSCAR with various inputs such as bipartite graphs and triangulations of two and three dimensional manifolds show that the method considerably extends for which simplicial complexes exterior algebraic shifts can be computed in practice.

math.CO↗

arXiv2025

Persistent Cycle Representatives and Generalized Landscapes for Codimension 1 Persistent Homology

For a filtered simplicial complex $K$ embedded in $\mathbb{R}^{d+1}$, the merge tree of the complement of $K$ induces a forest structure on the persistent homology $H_d(K)$ via Alexander duality. We prove that the connected components of $\mathbb{R}^{d+1}\setminus K_r$ correspond to representative cycles for a basis of $H_d(K_r)$ which are volume-optimal. By keeping track of how these representatives evolve with the filtration of $K$, we can equip each interval $I$ in the barcode of $H_d(K)$ with a sequence of canonical representative cycles. We develop and implement an efficient algorithm to compute the progression of cycles in time $\mathcal{O}((\#K)^2)$. We apply functionals to these representatives, such as path length, enclosed volume, or total curvature. This way, we obtain a real-valued function for each interval, which captures geometric information about~$K$. Deriving from this construction, we introduce the \emph{generalized persistence landscapes}. Using the constant one-function as the functional, this construction gives back the standard persistence landscapes. Generalized landscapes can distinguish point clouds with similar persistent homology but distinct shape, which we demonstrate by concrete examples.

math.AT↗

arXiv2025

Computing Flat-Injective Presentations of Multiparameter Persistence Modules

A flat-injective presentation of a multiparameter persistence module $M$ characterizes $M$ as the image of a morphism from a flat to an injective persistence module. Like flat or injective presentations, flat-injective presentations can be easily represented by a single graded matrix, completely describe the persistence module up to isomorphism, and can be used as starting point to compute other invariants of it,such as the rank invariant, persistence images, and others. If all homology modules of a bounded chain complex $F_\bullet$ of flat $n$-parameter modules are finite dimensional,it is known that $F_\bullet$ and its shifted image $νF_\bullet[n]$ under the Nakayama functor are quasi-isomorphic, where $νF_\bullet[n]$ is a complex of injective modules. We give an explicit construction of a quasi-isomorphism $ϕ_\bullet\colon F_\bullet \to νF_\bullet[n]$,based on the boundary morphisms of $F_\bullet$. If $F_\bullet$ is a flat resolution of a finite dimensional persistence module $M$,then the degree-zero part $ϕ_0\colon F_0 \to νF_n$ is a flat-injective resolution of $M$. From our construction of $ϕ$, we obtain a method to compute a matrix representing $ϕ_0$from the matrices representing the resolution $F_\bullet$. A Julia package implementing this method is available.

math.AC↗

arXiv2025

Partial Algebraic Shifting

We study algebraic shifting of uniform hypergraphs and finite simplicial complexes in the exterior algebra with respect to matrices which are not necessarily generic. Several questions raised by Kalai (2002) are addressed. For instance, it turns out that the combinatorial shifting of Erdős$\unicode{x2013}$Ko$\unicode{x2013}$Rado (1961) arises as a special case. Moreover, we identify a sufficient condition for partial shifting to preserve the Betti numbers of a simplicial complex; examples show that this condition is sharp.

math.CO↗

arXiv2025

Inferring a Cell Structure on the Space of Cyclooctane Conformations

The conformation space of cyclooctane, a ringlike organic molecule comprising eight carbon atoms, is a two-dimensional algebraic variety, which has been studied extensively for more than 90 years. We propose a cell structure representing this space, which arises naturally by partitioning the space into subsets of conformations that admit particular symmetries. We do so both for the labeled conformation space, in which the carbon atoms are considered as distinct, and for the actual, unlabeled, conformation space. The proposed cell structure is obtained by identifying subspaces of conformations based on symmetry patterns and studying the geometry and topology of these subsets using methods from dimensionality reduction and topological data analysis. Our findings suggest that, in contrast to the labeled variant, the conformation space of cyclooctane is contractible.

math.GN↗

arXiv2023

Efficient two-parameter persistence computation via cohomology

Clearing is a simple but effective optimization for the standard algorithm of persistent homology (PH), which dramatically improves the speed and scalability of PH computations for Vietoris--Rips filtrations. Due to the quick growth of the boundary matrices of a Vietoris--Rips filtration with increasing dimension, clearing is only effective when used in conjunction with a dual (cohomological) variant of the standard algorithm. This approach has not previously been applied successfully to the computation of two-parameter PH. We introduce a cohomological algorithm for computing minimal free resolutions of two-parameter PH that allows for clearing. To derive our algorithm, we extend the duality principles which underlie the one-parameter approach to the two-parameter setting. We provide an implementation and report experimental run times for function-Rips filtrations. Our method is faster than the current state-of-the-art by a factor of up to 20.

math.AT↗

arXiv2023

Clifford-symmetric polynomials

Based on the NilHecke algebra $\mathsf{NH}_n$, the odd NilHecke algebra developed by Ellis, Khovanov and Lauda and Kang, Kashiwara and Tsuchioka's quiver Hecke superalgebra, we develop the Clifford Hecke superalgebra $\mathsf{NH}\mathfrak{C}_n$ as another super-algebraic analogue of $\mathsf{NH}_n$. We show that there is a notion of symmetric polynomials fitting in this picture, and we prove that these are generated by an appropriate analogue of elementary symmetric polynomials, whose properties we shall discuss in this text.

math.RT↗

arXiv2021

Shuffling functors and spherical twists on $D^\mathrm b(\mathcal O_0)$

For a semisimple complex Lie algebra $\mathfrak g$, the BGG category $\mathcal{O}$ is of particular interest in representation theory. It is known that Irving's shuffling functors $\mathrm{Sh}_{w}$, indexed by elements $w\in W$ of the Weyl group, induce an action of the braid group $B_W$ associated to $W$ on the derived categories $D^\mathrm{b}(\mathcal{O}_λ)$ of blocks of $\mathcal{O}$. We show that for maximal parabolic subalgebras $\mathfrak{p}$ of $\mathfrak{sl}_n$ corresponding to the parabolic subgroup $W_\mathfrak{p}=S_{n-1}\times S_1$ of $S_n$, the derived shuffling functors $\mathbf{L}\mathrm{Sh}{s_i}$ are instances of Seidel and Thomas' spherical twist functors. Namely, we show that certain parabolic indecomposable projectives $P^\mathfrak{p}(w)$ are spherical objects, and the associated twist functors are naturally isomorphic to $\mathbf{L}\mathrm{Sh}{w}[1]$ as auto-equivalences of $D^\mathrm{b}(\mathcal{O}^\mathfrak{p})$. We give an overview of the main properties of the BGG category $\mathcal{O}$, the construction of shuffling and spherical twist functors, and give some examples how to determine images of both. To this end, we employ the equivalence of blocks of $\mathcal{O}$ and the module categories of certain path algebras.

math.RT↗