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We construct a geometric system from which the Hall algebra can be recovered. This system inherently satisfies higher associativity conditions and thus leads to a categorification of the Hall algebra. We then suggest how to use this…

In this paper we extend the Lie theory of integration in two different ways. First we consider a finite dimensional Lie algebra of vector fields and discuss the most general conditions under which the integral curves of one of the fields…

This paper presents a complete Pascal interpolation scheme for use in the plane geometry mapping applied in association with numerical methods. The geometry of a domain element is approximated by a complete Pascal polynomial. The…

We propose a general multi-class visual recognition model, termed the Classifier Graph, which aims to generalize and integrate ideas from many of today's successful hierarchical recognition approaches. Our graph-based model has the…

We present a graph-based deep learning framework for predicting the magnetic properties of quasi-one-dimensional Ising spin systems. The lattice geometry is encoded as a graph and processed by a graph neural network (GNN) followed by fully…

In the present paper, we revisit the geometry of smooth plane quartics and their bitangents from several perspectives. First, we study in detail the weak combinatorics of arrangements of bitangents associated with highly symmetric quartic…

Automated recognition and simplification of fillet features in CAD models is critical for CAE analysis, yet it remains an open challenge. Traditional rule-based methods lack robustness, while existing deep learning models suffer from poor…

This paper generalizes the bordered-algebraic knot invariant introduced in an earlier paper, giving an invariant now with more algebraic structure. It also introduces signs to define these invariants with integral coefficients. We describe…

A. Girand has constructed an explicit two-parameter family of flat connections over the complex projective plane $\mathbb{P}^2$. These connections have dihedral monodromy and their polar locus is a prescribed quintic composed of a conic and…

We study polynomial group convolutional neural networks (PGCNNs) for an arbitrary finite group $G$. In particular, we introduce a new mathematical framework for PGCNNs using the language of graded group algebras. This framework yields two…

The use of complexified quaternions and $i$-complex geometry in formulating the Dirac equation allows us to give interesting geometric interpretations hidden in the conventional matrix-based approach.

We give a geometric approach to the relation between the irreducible components of the characteristic varieties of local systems on a plane curve arrangement complement and the associated pencils of plane curves discovered recently by M.…

Analogously as in classical algebraic geometry, linear pencils of tropical plane curves are parameterized by tropical lines in a coefficient space. A special example of such a linear pencil is the set of tropical plane curves with an…

We review three vector encodings of Bayesian network structures. The first one has recently been applied by Jaakkola 2010, the other two use special integral vectors formerly introduced, called imsets [Studeny 2005, Studeny 2010]. The…

We present a framework for characterizing injectivity of classes of maps (on cosets of a linear subspace) by injectivity of classes of matrices. Using our formalism, we characterize injectivity of several classes of maps, including…

In generalized complex geometry, we revisit linear subspaces and submanifolds that have an induced generalized complex structure. We give an expression of the induced structure that allows us to deduce a smoothness criteria, we dualize the…

Simplicial surfaces describe the incidence relations between vertices, edges and faces of triangulated 2-dimensional manifolds in a purely combinatorial way. By considering only the incidences of edges and faces, simplicial surfaces are…

We automatically verify the crucial steps in the original proof of correctness of an algorithm which, given a geometric graph satisfying certain additional properties removes edges in a systematic way for producing a connected graph in…

We describe an algorithm that recognizes some (perhaps all) intrinsically knotted (IK) graphs, and can help find knotless embeddings for graphs that are not IK. The algorithm, implemented as a Mathematica program, has already been used by…

The scattering transform is a multilayered wavelet-based deep learning architecture that acts as a model of convolutional neural networks. Recently, several works have introduced generalizations of the scattering transform for non-Euclidean…