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We present the theory of rank-metric codes with respect to the 3-tensors that generate them. We define the generator tensor and the parity check tensor of a matrix code, and describe the properties of a code through these objects. We define…

The emergence of a diverging length scale in many-body systems at a quantum phase transition implies that total entanglement has to reach its maximum there. In order to fully characterize this, one has to consider multipartite entanglement…

The theory of tensor categories has found applications across various fields, including representation theory, quantum field theory (conformal in 2 dimensions, and topological in 3 and 4 dimensions), quantum invariants of low-dimensional…

In a triangulated category T with a pair of triangulated subcategories X and Y, one may consider the subcategory of extensions X*Y. We give conditions for X*Y to be triangulated and use them to provide tools for constructing stable…

We define and study a certain relative tensor product of subfactors over a modular tensor category. This gives a relative tensor product of two completely rational heterotic full local conformal nets with trivial superselection structures…

A quadratic algebra is a homogeneous algebra generated by its elements of degree 1. Manin has endowed the category of quadratic algebras with two tensor products. These structures have been adapted to operads by Ginsburg and Kapranov.…

3D generative modeling is accelerating as the technology allowing the capture of geometric data is developing. However, the acquired data is often inconsistent, resulting in unregistered meshes or point clouds. Many generative learning…

This paper has been withdrawn and replaced by arXiv:1309.5035. In this paper we describe some examples of so called spherical functors between triangulated categories, which generalize the notion of a spherical object. We also give…

Deep generative models learn the data distribution, which is concentrated on a low-dimensional manifold. The geometric analysis of distribution transformation provides a better understanding of data structure and enables a variety of…

Additive categories play a fundamental role in mathematics and related disciplines. Given an additive category equipped with a biadditive functor, one can construct its category of extensions, which encodes important structural information.…

Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. A notion of proper class in an extriangulated category is defined in this paper. Let…

These are the notes from lectures I gave at the Oberwolfach Seminar "Tensor Triangular Geometry and Interactions" which was held in October 2025. The aim of these notes is twofold: We develop notions of support for triangulated categories,…

A fundamental question in learning to classify 3D shapes is how to treat the data in a way that would allow us to construct efficient and accurate geometric processing and analysis procedures. Here, we restrict ourselves to networks that…

The algebra of invariants of several 3 x 3 matrices under the action of the orthogonal group by simultaneous conjugation is considered over a field of characteristic different from two. The maximal degree of elements of minimal system of…

This is the first paper in a general program to automate skein theoretic arguments. In this paper, we study skein theoretic invariants of planar trivalent graphs. Equivalently, we classify trivalent categories, which are nondegenerate…

Tensors are often studied by introducing preorders such as restriction and degeneration: the former describes transformations of the tensors by local linear maps on its tensor factors; the latter describes transformations where the local…

This paper addresses the challenge of object-centric layout generation under spatial constraints, seen in multiple domains including floorplan design process. The design process typically involves specifying a set of spatial constraints…

For a tensor triangulated category which is well generated in the sense of Neeman, it is shown that the collection of Bousfield classes forms a set. This set has a natural structure of a complete lattice which is then studied, using the…

Recent text-to-3D models can render high-quality assets, yet they still stumble on objects with complex attributes. The key obstacles are: (1) existing text-to-3D approaches typically lift text-to-image models to extract semantics via text…

We investigate the problem when the tensor functor by a bimodule yields a singular equivalence. It turns out that this problem is equivalent to the one when the Hom functor given by the same bimodule induces a triangle equivalence between…