Related papers: Tensor categories: A selective guided tour
These are lecture notes of a mini-course given by the first author in Moscow in July 2019, taken by the second author and then edited and expanded by the first author. They were also a basis of the lectures given by the first author at the…
During the last years, low-rank tensor approximation has been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. This survey…
The groundbreaking performance of deep neural networks (NNs) promoted a surge of interest in providing a mathematical basis to deep learning theory. Low-rank tensor decompositions are specially befitting for this task due to their close…
These course note first provide an introduction to secondary characteristic classes and differential cohomology. They continue with a presentation of a stable homotopy theoretic approach to the theory of differential extensions of…
We introduce the notions of categorical integrals and categorical cointegrals of a finite tensor category $\mathcal{C}$ by using a certain adjunction between $\mathcal{C}$ and its Drinfeld center $\mathcal{Z}(\mathcal{C})$. These notions…
A group-category is an additively semisimple category with a monoidal product structure in which the simple objects are invertible. For example in the category of representations of a group, 1-dimensional representations are the invertible…
This paper introduces a formalism that aims to describe the intricacies of quantum computation by establishing a connection with the mathematical foundations of tensor theory and multilinear maps. The focus is on providing a comprehensive…
Homological algebra is often understood as the translator between the world of topology and algebra. However, this branch of mathematics is worth studying by itself, given that it provides fascinating perspectives about other disciplines,…
This paper gives an introduction to the homotopy theory of quasi-categories. Weak equivalences between quasi-categories are characterized as maps which induce equivalences on a naturally defined system of groupoids. These groupoids…
This short introductory category theory textbook is for readers with relatively little mathematical background (e.g. the first half of an undergraduate mathematics degree). At its heart is the concept of a universal property, important…
We shall discuss how the notions of multicategories and their linear representations are related with tensor categories. When one focuses on the ones arizing from planar diagrams, it particularly implies that there is a natural one-to-one…
These are lecture notes for a 4h mini-course held in Toulouse, May 9-12th, at the thematic school on "Quantum topology and geometry". The goal of these lectures is to (a) explain some incarnations, in the last ten years, of the idea of…
Extending tensor models at the field theoretical level, tensor field theories are nonlocal quantum field theories with Feynman graphs identified with simplicial complexes. They become relevant for addressing quantum topology and geometry in…
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…
Are introduced six examples of non-braidable tensor categories which are extensions of the category Comod(H), for H a super-group algebra; and two examples of braided categories where the only possible braiding is the trivial braiding.
A tensor space is a vector space equipped with a finite collection of multilinear forms. The length of a tensor space is its length as a representation of its symmetry group. Infinite dimension tensor spaces of finite length are special,…
These are the notes for a minicourse held in Odessa (2016) and Belo Horizonte (2017). My aim was to provide a short introduction to basic notions of category theory and representation theory of finite-dimensional algebras. We learnt the…
We show that once-extended anomalous 3-dimensional topological quantum field theories valued in the 2-category of k-linear categories are in canonical bijection with modular tensor categories equipped with a square root of the global…
These notes contain an introduction to the theory of complex semisimple quantum groups. Our main aim is to discuss the classification of irreducible Harish-Chandra modules for these quantum groups, following Joseph and Letzter. Along the…
These lecture notes are intended as an introduction to several notions of tensor rank and their connections to the asymptotic complexity of matrix multiplication. The latter is studied with the exponent of matrix multiplication, which will…