Related papers: Bigraded Betti numbers and Generalized Persistence…
Torsion and Betti numbers for knots are special cases of more general invariants associated to a finitely generated group G and epimorphism from G to the integers. The sequence of Betti numbers is always periodic; under mild hypotheses, the…
This paper considers an important inverse problem in topological data analysis (TDA): How many different barcodes produce the same Betti curve? Equivalently, given a function $\beta\colon [n]=\{1<\cdots< n\} \to \mathbb{Z}_{\geq 0}$, how…
We compute Betti numbers of the moduli spaces of arbitrary rank stable sheaves on ruled surfaces. Our result generalizes the formula of Goettsche for rank one sheaves and the formula of Yoshioka for rank two sheaves. It also confirms the…
We generalise the finite biquandle colouring invariant to a polynomial invariant based on labelling a knot diagram with a finite birack that reduces to the biquandle colouring invariant in that case. The polynomial is an invariant of a…
The following article is an application of commutative algebra to the study of multiparameter persistent homology in topological data analysis. In particular, the theory of finite free resolutions of modules over polynomial rings is applied…
Let $\Lambda$ be a basic finite dimensional algebra over an algebraically closed field, presented as a path algebra modulo relations; further, assume that $\Lambda$ is graded by lengths of paths. The paper addresses the classifiability, via…
We show that a purely algebraic structure, a two-dimensional scattering diagram, describes a large part of the wall-crossing behavior of moduli spaces of Bridgeland semistable objects in the derived category of coherent sheaves on…
We present a generalization of Transformers to any-order permutation invariant data (sets, graphs, and hypergraphs). We begin by observing that Transformers generalize DeepSets, or first-order (set-input) permutation invariant MLPs. Then,…
We show that a persistence module (for a totally ordered indexing set) consisting of finite-dimensional vector spaces is a direct sum of interval modules. The result extends to persistence modules with the descending chain condition on…
Symmetry plays a fundamental role in understanding natural phenomena and mathematical structures. This work develops a comprehensive theory for studying the persistent symmetries and degree of asymmetry of finite point configurations over…
Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isomorphism, the representations of $\Lambda$ with fixed…
The (gradient-based) bilevel programming framework is widely used in hyperparameter optimization and has achieved excellent performance empirically. Previous theoretical work mainly focuses on its optimization properties, while leaving the…
Proximities are at the heart of almost all machine learning methods. If the input data are given as numerical vectors of equal lengths, euclidean distance, or a Hilbertian inner product is frequently used in modeling algorithms. In a more…
Chinese remainder theorem (CRT) is widely applied in cryptography, coding theory, and signal processing. It has been extended to the multidimensional CRT (MD-CRT), which reconstructs an integer vector from its vector remainders modulo…
In this work, we are interested in characterizing typical (generic) dimensional properties of invariant measures associated with the full-shift system, $T$, in a product space whose alphabet is a perfect and separable metric space (thus,…
Multidimensional persistence has been proposed to study the persistence of topological features in data indexed by multiple parameters. In this work, we further explore its algebraic complications from the point of view of higher…
A foundational principle in the study of modules over standard graded polynomial rings is that geometric positivity conditions imply vanishing of Betti numbers. The main goal of this paper is to determine the extent to which this principle…
A number of machine learning tasks entail a high degree of invariance: the data distribution does not change if we act on the data with a certain group of transformations. For instance, labels of images are invariant under translations of…
Numerical invariants of a minimal free resolution of a module $M$ over a regular local ring $(R,\n)$ can be studied by taking advantage of the rich literature on the graded case. The key is to fix suitable $\n$-stable filtrations ${\mathbb…
Multiparameter persistence is a natural extension of the well-known persistent homology, which has attracted a lot of interest. However, there are major theoretical obstacles preventing the full development of this promising theory. In this…