Related papers: Representation theory and homological stability
This paper presents a mathematically rigorous framework for brain-inspired representation learning founded on the interplay between persistent topological structures and cohomological flows. Neural computation is reformulated as the…
Persistence diagrams are efficient descriptors of the topology of a point cloud. As they do not naturally belong to a Hilbert space, standard statistical methods cannot be directly applied to them. Instead, feature maps (or representations)…
Graph convolutional neural networks (GCNNs) have emerged as powerful tools for analyzing graph-structured data, achieving remarkable success across diverse applications. However, the theoretical understanding of the stability of these…
We prove that some of the classical homological stability results for configuration spaces of points in manifolds can be lifted to motivic cohomology.
According to Letourmy and Vendramin, a representation of a skew brace is a pair of representations on the same vector space, one for the additive group and the other for the multiplicative group, that satisfies a certain compatibility…
Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multi-set of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly…
A fundamental problem from invariant theory is to describe the endomorphism algebra of multilinear functions on a representation V invariant under the action of a group G. According to Weyl's classic, a first main (later: fundamental)…
We give applications of equivariant Gromov--Hausdorff convergence in various contexts. Namely, using equivariant Gromov--Hausdorff convergence, we prove a stability result in the setting of compact finite dimensional Alexandrov spaces.…
The symmetric group acts on the power set and also on the set of square free polynomials. These two related representations are analyzed from the stability point of view. An application is given for the action of the symmetric group on the…
Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a…
Polygon representation learning is essential for diverse applications, encompassing tasks such as shape coding, building pattern classification, and geographic question answering. While recent years have seen considerable advancements in…
We study the representation theory of the increasing monoid. Our results provide a fairly comprehensive picture of the representation category: for example, we describe the Grothendieck group (including the effective cone), classify…
We prove that the dual rational homotopy groups of the configuration spaces of a 1-connected manifold of dimension at least 3 are uniformly representation stable in the sense of Church, and that their derived dual integral homotopy groups…
Coboundary expansion (with $\mathbb{F}_2$ coefficients), and variations on it, have been the focus of intensive research in the last two decades. It was used to study random complexes, property testing, and above all Gromov's topological…
In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections…
Associated to every group with a weak spherical Tits system of rank n+1 with an appropriate rank n subgroup, we construct a relative spectral sequence involving group homology of Levi subgroups of both groups. Using the fact that such Levi…
Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of…
Persistent topological properties of an image serve as an additional descriptor providing an insight that might not be discovered by traditional neural networks. The existing research in this area focuses primarily on efficiently…
This paper is based on my talk at ICM on recent progress in a number of classical problems of linear algebra and representation theory, based on new approach, originated from geometry of stable bundles and geometric invariant theory.
A class of peridynamic material models known as constitutive correspondence models provide a bridge between classical continuum mechanics and peridynamics. These models are useful because they allow well-established local constitutive…