Polytopes, Matroids, and the Friends You Make: Martina Juhnke on Two Months at the CRM

The Centre de Recerca Matemàtica recently hosted a research programme on Combinatorial Geometries and Geometric Combinatorics, focusing on the overlap between polytopes and matroids. Martina Juhnke, a member of the scientific committee, reflects on how this programme prioritized collaboration, allowing postdocs and students to build the professional networks and broad expertise required in a rapidly moving field.

Martina Juhnke still has friends from her first summer school. She meets them at conferences, years later, and the connection holds. It is a detail that anchors her view of this autumn at the Centre de Recerca Matemàtica (CRM); two months of research, collaboration, and the social architecture of mathematics.

From October through November, the CRM hosted an intensive research programme on Combinatorial Geometries and Geometric Combinatorics. Juhnke, a professor of discrete mathematics at the University of Osnabrück and member of the scientific committee, came to Barcelona for the closing conference. What she saw confirmed a long-held belief: research centres are not just venues for solving problems. They are where young mathematicians learn to work together, take on responsibility, and build the relationships that shape careers.

“The goal was to bring them together,” Juhnke says, describing the design. “To have time, to do research together, to have collaboration time, to get to know new projects, to have contact with senior researchers, and also just have a great time.” The structure was deliberate. After preliminary sessions on polytopes and oriented matroids, a two-week research school introduced the field’s core topics through advanced courses. Then came five weeks of collaborative research projects and seminars. The programme closed with a conference where participants presented their progress.

For the postdocs who led research groups, it was a chance to take responsibility for a team, a factor when applying for faculty positions. For PhD and master’s students, it was an immersion in how mathematics actually gets done: not alone in an office, but in conversation, at a blackboard, or over coffee.

“If someone comes who just wants to work alone, then maybe he’s at the wrong place.”

Juhnke is frank about the selection criteria. Scientific qualification mattered, but so did a willingness to collaborate. “If someone comes who just wants to work alone, then maybe they’re at the wrong place.” Gender balance and diversity were considered throughout for participants, visitors, and speakers. The mathematical focus reflected a shift Juhnke sees across combinatorics. The programme’s two branches, geometric combinatorics, where polytopes play a central role, and combinatorial geometries, which deal with matroids, used to feel more distinct.

“Basically, a matroid is a polytope with some specific properties.” The boundaries have blurred. Polytopes appear in optimization, algebra, statistics, theoretical physics, and topology. Juhnke notes a saying often attributed to the field: “Polytopes want to be everywhere.” Matroids generate combinatorial objects and give rise to geometry. “I don’t think that one can separate polytopes and matroids anymore,” Juhnke says. “Synergies just arise naturally.”

This naturalness showed up in the research projects. Topics had to be accessible but also current. The result was a programme responsive to where the field is moving. And the field is moving fast. Juhnke describes it as “a quickly emerging field,” which she sees as an opportunity for young mathematicians. There are plenty of open problems, and as they are solved, new directions open. But this requires a shift in how researchers prepare.

“It’s not enough to be a specialist in just one area,” she says. “It’s not enough to just know something about polytopes. You should also know a lot about matroids.” She cites work in theoretical physics where solving problems requires knowledge across multiple areas. “The younger generation really have to become broad and have broad knowledge.” This is where research centres like the Centre de Recerca Matemàtica become necessary. These spaces provide time to focus without teaching obligations and offer a community to researchers from smaller universities who might otherwise work in isolation.

Before the interview ended, Juhnke added: “I would like to congratulate the organizers, and also all the participants in this programme, for the huge success of this programme, because I think they really achieved a lot. I think everybody can just be proud of themselves.”

It loops back to the friends from that first summer school. The theorems and proofs matter, but so does the structure that supports them: the postdocs learning to lead and the students finding collaborators. If polytopes want to be everywhere, it seems mathematicians do too; at least the ones who have learned that the work goes better when you are not doing it alone.