Complex Manifolds: Heisenberg Funding for LMU Mathematician

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Mathematics often involves studying shapes – but not just the familiar ones we can see and touch. An important class of these mathematical shapes follows rules that originate from the world of complex numbers. These complex manifolds, as they are known, appear across modern mathematics and even in theoretical physics. Yet despite many decades of research, their deeper structure is still only partially understood.

With his Heisenberg project entitled “Topological methods and the ddbar equation on complex manifolds”, Dr. Jonas Stelzig is tackling the problem by bringing together two very different mathematical perspectives:

On one side is topology and homotopy theory, which study shapes in a highly flexible way. From this perspective, objects can be stretched, bent, or deformed without breaking – as if they were made of rubber. This flexibility allows mathematicians to extract general structural information and classify spaces using powerful algebraic tools.

On the other side is complex geometry, which is much more rigid. This is where shapes must satisfy strict analytical conditions, making them far less deformable. Small changes can fundamentally alter their nature, and many of the techniques that work in topology can no longer be applied here.

Reconciling flexibility and rigidity

A central idea of the project, which is receiving a total of almost 700,000 euros of funding over a period of five years, is to bridge this gap. By developing a new framework – known as pluripotential homotopy theory – the research combines the flexibility of topology with the rigidity of complex geometry.

This synthesis opens up new possibilities. It allows mathematicians to compare different complex shapes more effectively, to understand how they can change or deform, to study their symmetries and to uncover hidden structures that were previously inaccessible.

Moreover, many currently available methods apply only to a particular subclass of complex manifolds, known as Kähler manifolds, while beyond this realm the research is still based to a certain extent on individual examples. By contrast, the methods developed in this proposal can be applied uniformly to all complex manifolds.

“In summary, the project provides new, unifying methods for the study of complex manifolds that strike a balance between flexibility and rigidity,” explains Stelzig.

The Heisenberg Programme of the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) supports outstanding scientists who do not yet hold a professorship but meet all the requirements for appointment to a chair. The programme is intended to promote this next generation of talented scientists.