# Scientic objectives

## Beyond Permutahedra and Associahedra: Geometry, Combinatorics, Algebra, and Probability

The permutahedron and the associahedron are two classical objects that encode the mathematical structure of permutations and associations of an n-element set. Over the years, they have inspired a vast amount of research due to their significant appearance in many different contexts, and have led to numerous connections to diverse fields in mathematics, computer science and physics. Many relevant questions about these objects have been completely solved, and the impact of the solutions onto various problems in related fields are well understood. However, the permutahedron and the associahedron can be viewed as particular examples of much more general families of mathematical objects. In these general contexts, many new open problems and connections to other fields arise, opening up new horizons and directions of research.

The present project lies at the interplay between theoretical computer science and pure mathematics. It approaches a selection of questions and open problems that go beyond the study of the permutahedron and the associahedron, with focus on four different areas:

- Combinatorics: combinatorial properties, bijections, and enumeration of related objects.
- Discrete Geometry: geometric structure and construction methods.
- Algortihmics: graph properties and complexity of the shortest path distance problem.
- Algebra and Probability: new insights into the combinatorial foundations of free probability and its relations with combinatorial Hopf algebras.

In order to make substantial progress, we have brought together a team of experts from these disparate areas who will work together on problems arising from the intersection/ interplay of these fields. Our approach includes the use of computer exploration and the development of free open-source software.

##### Pipe dreams, Tamari lattices, and binary trees

The theory of Schubert polynomials is an active area of research that was initiated by A. Lascoux and M.-P. Schützenberger in connection to the study of cohomology classes of Schubert cycles in flag varieties. A combinatorial understanding of Schubert polynomials was discovered by S. Fomin and A. Kirillov and …

Read More##### Cumulants, free probability and Hopf algebras

Free Probability was introduced by Voiculescu 40 years ago in order to solve problems in von Neumann algebras. It has since developed into a whole new field of mathematics with intimate connections to established fields like classical probability, combinatorics and analysis, in particular random matrices , noncrossing partitions and operator algebras …

Read More##### Realization Spaces

The realization space of a combinatorial polytope is the set of all geometric polytopes that realize it. Mnëv's celebrated universality theorem states that realization spaces of polytopes can be arbitrarily complicated (stably equivalent to any primary basic semialgebraic set \(S\)). An outstanding improvement by Richter-Gebert shows that this holds already …

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