https://homalg-project.github.io/capdays-2024/
CAP (shorthand for Categories, Algorithms, Programming) is a software project for algorithmic category theory written in GAP. It facilitates both the realization of specific instances of categories and the implementation of generic categorical algorithms.
The workshop is split into three kinds of activities:
Introductory talks about algorithmic category theory (ALCT), CAP as a dialect of ALCT, and CompilerForCAP.
An explicit live show-case implementation of a category constructor and an explicit categorical tower.
Exercise sessions implementing categorical algorithms in CAP. Participants are also invited to implement their own work in CAP and we will be happy to provide help.
The workshop aims at mathematicians, physicists, and computer scientists who want to learn about applications of algorithmic category theory and how CAP can be used to structure implementations in a categorical way.
Our project has two intertwined goals: Compute the graded parts of Orlik-Solomon algebras of geometric lattices together with its algebra structure and generalise the Lehrer-Solomon conjecture, possibly altering its statement, to other classes of arrangements and nonrepresentable matroids.
For the first goal, a major strategy will be the exploitation of equivariant structures under (subgroups of) the automorphism group G of the underlying lattice of the Orlik-Solomon algebra.
For the second goal, reasonable classes of arrangements that allow both for an inductive process and have nontrivial automorphism groups are the class of inductively free arrangements and the larger class of free arrangements. Among these classes are the class of Coxeter arrangements and the larger classes of ideal subarrangements of Weyl arrangements and of crystallographic arrangements, all of which include arrangements with relatively large automorphism groups.
The need to combine both strategies naturally leads us to the use of the category-theoretic language, both as a theoretical framework and as an algorithmic machinery.
The results will be made accessible via an online database.
]]>The Berkeley Seminar
Various constructions of categories have a universal property
expressing the freeness/initiality of the construction within a
specific categorical doctrine. Expressed in an algorithmic framework,
it turns out that this universal property is in a certain sense a
doctrine-specific ur-algorithm
from which various known categorical
constructions/algorithms (including spectral sequences of bicomplexes)
can be derived in a purely computational way. This can be viewed as a
categorical version of the Curry-Howard correspondence to extract
programs from proofs.
https://web.northeastern.edu/martsinkovsky/p/Conferences/M%C3%A1laga2023/FM-23.html
Organizers
https://web.northeastern.edu/martsinkovsky/p/Conferences/Ghent2023/AA-23.html
]]>The workshop aims at bringing together experts from Mathematics and Physics to discuss the latest developments and future directions in unraveling the Mathematical Structures in Feynman Integrals. Topics will include among others Structures of Feynman integrals, Integral reduction, Applications from algebraic geometry, Finite fields and rational reconstruction, Differential equations, etc.
The program will feature dedicated talks, but will also leave ample time for discussions among workshop participants.
February 11, 2023 marks the 100th anniversary of the death of the mathematician Wilhelm Killing, born in Burbach, only 20km from Siegen. Killing formulated a research program that is still relevant today and has significantly influenced mathematical research for a century. The 100th anniversary of Killing’s death is an appropriate opportunity to honour his merits with an inaugural talk by Prof. Wolfgang Hein.
]]>Computational and algorithmic methods
Chevalley proved that the image of an algebraic morphism between algebraic varieties is a constructible set. Examples are orbits of algebraic group actions. A constructible set in a topological space is a finite union of locally closed sets and a locally closed set is the difference of two closed subsets. Simple examples show that even if the source and target of the morphism are affine varieties the image may neither be affine nor quasi-affine. In this talk, I will present a Gröbner-basis-based algorithm that computes the constructible image of a morphism of affine spaces, along with some applications.
]]>