Related papers: A short course on Tensor algebra and analysis, wit…
These lectures given to graduate students in theoretical particle physics, provide an introduction to the ``inner workings'' of computer algebra systems. Computer algebra has become an indispensable tool for precision calculations in…
Tensor networks provide extremely powerful tools for the study of complex classical and quantum many-body problems. Over the last two decades, the increment in the number of techniques and applications has been relentless, and especially…
These are notes from the lectures I gave at the Oberwolfach seminar `Tensor Triangular Geometry and Interactions' which was held in October 2025. The aim of these notes is to give an introduction to tensor triangular geometry, for both…
The main concepts of the theory of tensors are presented. The emphasis is on the basic notions of tensor algebra and practical skills in culculations involving the Kronecker delta and Levi-Civita symbol. Sixty routine exercises are…
This course, intended for undergraduates familiar with elementary calculus and linear algebra, introduces the extension of differential calculus to functions on more general vector spaces, such as functions that take as input a matrix and…
This is the first part of the lecture notes that grew out of the special course given during the 2021-2022 academic year. In these lecture notes we present an approach to the fundamental structures of differential geometry that uses the…
This book is an introduction to the nascent field of Fourier analysis on polytopes, and cones. There is a rapidly growing number of applications of these methods, so it is appropriate to invite students, as well as professionals, to the…
These lecture notes present a computation driven pathway from classical complex analysis to the theory of compact Riemann surfaces and their connections to algebraic geometry. The exposition follows a compute first then abstract philosophy,…
This is a set of lecture notes on the operator algebraic approach to 2-dimensional conformal field theory. Representation theoretic aspects and connections to vertex operator algebras are emphasized. No knowledge on operator algebras or…
The book covers basics of noncommutative geometry and its applications in topology, algebraic geometry and number theory. A brief survey of main parts of noncommutative geometry with historical remarks, bibliography and a list of exercises…
MSc thesis of the author offering an introduction to the operator algebraic approach to noncommutative geometry, with a treatment of some more advanced elements such as the noncommutative geometry of quantum groups, fuzzy physics, and…
Scaling problems have a rich and diverse history, and thereby have found numerous applications in several fields of science and engineering. For instance, the matrix scaling problem has had applications ranging from theoretical computer…
This article is an expository account of the theory of twisted commutative algebras, which simply put, can be thought of as a theory for handling commutative algebras with large groups of linear symmetries. Examples include the coordinate…
It is often difficult to learn new mathematics semantically and syntactically, even when there are similarities in the words and meaning when discussed aloud. The goal of this document is to facilitate learning through explanations and…
This is the first paper in a series of eight where in the first three we develop a systematic approach to the geometric algebras of multivectors and extensors, followed by five papers where those algebraic concepts are used in a novel…
There have been several modifications of how basic calculus has been taught, but very few of these modifications have considered the computational tools available at our disposal. Here, we present a few tools that are easy to develop and…
We review the notion of shape tensor of an embedded manifold, which efficiently combines intrinsic and extrinsic geometry, and allows for intuitive understanding of some basic concepts of classical differential geometry, such as parallel…
This set of lecture notes constitutes the free textbook project I initiated towards the end of Summer 2015, while preparing for the Fall 2015 Analytical Methods in Physics course I taught to upper level undergraduates at the University of…
In this expository note, I present some of the key features of the lattice of torsion classes of a finite-dimensional algebra, focussing in particular on its complete semidistributivity and consequences thereof. This is intended to serve as…
We introduce a cohomology set for groups defined by algebraic difference equations and show that it classifies torsors under the group action. This allows us to compute all torsors for large classes of groups. We also develop some tools for…