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# Welcome to Algetiq.eu

This website is dedicated to applications of discrete geometry and non-linear
algebra in information theory. It is maintained as part of the Marie Skłodowska-Curie
Actions grant no. 101110545 ["*Algetiq: Algebraic theory of information quantities*"](https://doi.org/10.3030/101110545).

Its purpose is to collect research notes and expository articles to link
relevant ideas and computational tools from discrete mathematics and algebra
to information-theoretic problems. We hope to stimulate interest from the
applied algebra side and also advance the robustness of computational
methodologies available to tackle conjectures in information theory.
As the project advances, we will also collect special-purpose mathematical
software and research data.

The central object of interest is the **entropy region**. Since many questions
in information theory can be answered by optimizing a linear functional over
this region, we are interested in the geometric features of the entropy region.
The following objects and algorithms form key topics of the project:
* **Information inequalities** are the linear inequalities delimiting the
  boundary of the entropy region. They are a main research interest in
  information theory.
* **Conditional independence** relations give a coarse combinatorial
  decomposition of those parts of the boundary of the entropy region
  which are inhabited by entropy vectors.
* **Gröbner bases** and **cylindrical algebraic decomposition** are exact
  symbolic algorithms which make it possible to process the varieties and
  semialgebraic sets defined by conditional independence relations on a
  computer. These costly exact methods are complemented by numerical
  approaches such as **homotopy continuation**.
* **Matroids and polymatroids** appear as the underlying structures of
  entropy vectors: every point in the entropy region is a polymatroid and
  many of its extreme rays are (connected) matroids.
* Discrete **polyhedral operations** such as convolution and various
  **extension properties** inspired by information-theoretic coding lemmas
  have been used to reveal additional local polyhedral structures of the
  entropy region which lead to **conditional information inequalities**.
* (Poly)matroid representations such as by **subspace arrangements** or
  **algebraic varieties** over finite fields lead to (asymptotically)
  entropic points. This leads to questions bordering coding theory and
  number theory.