Combinatorial Geometry Takes Shape at the CRM

From October 6 to 10, the Centre de Recerca Matemàtica hosted the Research School on Combinatorial Geometries and Geometric Combinatorics – Polytope Week, a meeting point for researchers exploring the rich territory where geometry and combinatorics intersect. The activity was part of the MDM Focused Research Programme on Combinatorial Geometries and Geometric Combinatorics, coordinated by Kolja Knauer, Anna de Mier, Arnau Padrol, Julian Pfeifle, and Vincent Pilaud.

The activity showcased the growing dialogue between geometry and combinatorics, highlighting how shared methods and perspectives can drive advances across both fields.

More than fifty participants from universities across Europe, America, and Asia filled the CRM Auditorium for five days of lectures, problem sessions, and informal discussions. The schedule alternated between theoretical insights and hands-on work, creating a rhythm that blended the rigor of advanced training with the atmosphere of a collaborative workshop.

Carolina Benedetti (Universidad de los Andes) opened the week with her minicourse Perspectives on flags of matroids and positroids, introducing participants to a field that connects combinatorial structures with geometry and algebraic concepts from tropical mathematics. Through examples and open problems, Benedetti guided the audience toward questions still at the frontier of research, including subdivisions of polytopes linked to matroid theory.

In the second minicourse, Moduli polytopes: parametrizing combinatorial structures with polytopes, Raman Sanyal (Goethe-Universität Frankfurt) explored how complex discrete objects, such as permutations, rooted trees, or matroids, can be encoded as vertices of geometric shapes. His lectures revealed the capacity of this geometric viewpoint to understand, compare, and classify combinatorial structures.

Afternoons were devoted to exercise sessions and group work, giving participants the chance to test ideas and develop small collaborative projects. The week also included a session for research presentations and a social dinner that carried discussions well beyond the blackboard.

By the end of the school, what began as a deep dive into the geometry of polytopes had become a broader exchange of perspectives and techniques, linking students and researchers at different stages of their careers. The Polytope Week confirmed the dynamism of the community around combinatorial geometry.