Abstract: In this talk we explain the concept of constructive category theory and its implementation in our software project CAP - Categories, algorithms, programming. Furthermore, we show the benefits of CAP’s framework for constructive category theory by demonstrating some applications to homological algebra: diagram chasing via generalized morphisms and computing the purity filtration via spectral sequences.
Abstract: QPA (“Quivers and Path Algebras”) is a GAP package for doing computations related to algebras defined by quivers and relations. A substantial part of QPA’s functionality deals with module categories over such algebras, and certain functors between these categories. The QPA package is currently undergoing a major rewrite where we try to make the code better structured and more consistent. In the new version of QPA, we use the structures provided by CAP to represent categories and functors, replacing our earlier more ad hoc approaches. In this talk, I will give a brief introduction to QPA and show how we are using CAP in QPA.
Abstract: The holy grail of string theory is to prove or disprove that a string vacuum furnishes a description of the physical universe that we are living in. A minimalistic check to tell if a string vacuum could match the experimental results from particle accelerators is to compute the massless spectrum. For a special class of string vacua – the so-called F-theory vacua – I will explain that this eventually leads to cohomologies of coherent sheaves.
The functionality of CAP extends far beyond coherent sheaves. However, the CAP-framework is perfectly suited for a modern implementation of coherent sheaves and their cohomologies. Indeed, much progress has been made on this subject during the last few years. Despite these successes, the required computations test the boundaries of these implementations in terms of computational time and power. To overcome these limitations, we are currently investigating if techniques from machine learning can help. I will conclude this talk with results on this work in progress.
(See pdf for theoretical background)
Abstract: Following Francis Brown, I will explain why questions in number theory (irrationality of zeta values) make it desirable to be able to compute some mixed Tate motives of moduli spaces. Then I will explain an algorithm devised by Clément Dupont in his PhD thesis for that purpose, and its implementation in the package MotivesForBiarrangements which uses CAP.
Abstract: This talk is meant to introduce the Quillen model categories. In short, model categories are used to give an effective construction of the localization of categories, where, similarly to localization of rings, the problem is to convert a class of morphisms, called weak-equivalences, into isomorphisms.
In the case of a model category, the localized category is identified with a homotopy category, a category whose morphism sets consist of equivalence classes of morphisms under a certain homotopy relation which is determined by the model structure.
We demonstrate this by constructing in Gap/CAP the derived category of some categories with finite global projective dimension, for example the category of finite left presentations over a polynomial ring or the category of representations of an acyclic quiver.
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