Related papers: A Very Elementary Introduction to Sheaves
The purpose of this paper is to develop an efficient computational model for Abelian categories of coherent sheaves over certain classes of varieties. These categories are naturally described as Serre quotient categories. Hence, our…
The main aim of the present work is to arrive at a mathematical theory close to the historically original conception of generalized functions, i.e. set theoretical functions defined on, and with values in, a suitable ring of scalars and…
This article serves as an introduction to the study of networks of social systems. First, we introduce the reader to key mathematical tools to study social networks, including mathematical representations of networks and essential…
This text is devoted to the theory of varieties, which provides an important tool, based in universal algebra, for the classification of regular languages. In the introductory section, we present a number of examples that illustrate and…
This is a written-up version of eight introductory lectures to the Hodge theory of projective manifolds. The table of contents should be self-explanatory. The only exception is section 8 where I discuss, in a simple example, a technique for…
These notes aim to give a first introduction to intersection cohomology and perverse sheaves with applications to representation theory or quantum groups in mind.
What is a time-varying graph, a time-varying topological space, or, more generally, a mathematical structure that evolves over time? In this work, we lay the foundations for a general theory of temporal data by introducing categories of…
We give a modern introduction to the moduli of sheaves. After reviewing the classical theory, we give a catalogue of results from the last decade. We then consider a more "symmetric" formulation of the theory by working with gerbes from the…
Network science is the field dedicated to the investigation and analysis of complex systems via their representations as networks. We normally model such networks as graphs: sets of nodes connected by sets of edges and a number of node and…
Pyknotic objects are (hyper)sheaves on the site of compacta. These provide a convenient way to do algebra and homotopy theory with additional topological information present. This appears, for example, when trying to contemplate the derived…
This paper provides a preparatory introduction to sheaves and topoi, written as a conceptual continuation of the author's earlier introduction to torsors and as preparatory background for the author's arXiv paper \emph{Grothendieck…
We propose a categorical framework to reason about scientific explanations: descriptions of a phenomenon meant to translate it into simpler terms, or into a context that has been already understood. Our motivating examples come from systems…
This article summarises a Web-book on "Complexity" that was developed to introduce undergraduate students to interesting complex systems in the biological, physical and social sciences, and the common tools, principles and concepts used for…
They run our lives, if you believe the hype in the news, but there is no precise definition of "algorithms" which is generally accepted by the mathematicians, logicians and computer scientists who create and study them. My main aims here…
Data analysis and machine learning have become an integrative part of the modern scientific methodology, offering automated procedures for the prediction of a phenomenon based on past observations, unraveling underlying patterns in data and…
This paper is the first in a series of papers in which we define and study a category of "sheaves of $\mathcal Z$-modules on the set of alcoves" that carries important information on the category of representations of semisimple Lie…
This survey is meant to provide an introduction to the fundamental theorem of linear algebra and the theories behind them. Our goal is to give a rigorous introduction to the readers with prior exposure to linear algebra. Specifically, we…
We introduce a sheaf-theoretic characterization of task solvability in general distributed computing models, unifying distinct approaches to message-passing models. We establish cellular sheaves as a natural mathematical framework for…
The authors are doing the readers of Statistical Science a true service with a well-written and up-to-date overview of boosting that originated with the seminal algorithms of Freund and Schapire. Equally, we are grateful for high-level…
In [1] we introduced the concept of structured space, which is a topological space that locally resembles some algebraic structures. In [2] we proceeded the study of these spaces, developing two cohomology theories. The aim of this paper is…