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At the Institute of Mathematics, a position as

**PhD Student in Mathematics - Differential Geometry**

is to be filled in the research group of Prof. Dr. Thomas Wannerer starting October 1, 2024.

Scientific background of the PhD project:

In its most basic form, the isoperimetric problem asks: Among all planar regions of fixed perimeter, which have the largest possible area? That circular discs are the solution was known already in ancient Greece. However, mathematically rigorous proofs did not appear until much later, in the 19th century. The theory of mixed volumes of convex bodies, developed by Hermann Minkowski at the beginning of the 20th century, provides a remarkable framework for investigating all kinds of questions about the volume of convex bodies, including the isoperimetric problem. A fundamental result in this area and a cornerstone of convex geometry is the Alexandrov-Fenchel inequality. It directly implies the isoperimetric inequality and has many applications in convex geometry and other areas.

Recently, a new perspective on the Alexandrov-Fenchel inequality has emerged from the groundbreaking work of Fields medalist June Huh and his collaborators. A philosophy underlying their work is that certain inequalities point to the existence of a “Kähler package” in the background. This is an algebraic structure that was observed to occur in algebra, geometry, and combinatorics, and encompasses Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. This perspective led to the conjecture that a certain algebra arising in convex geometry, the algebra of smooth valuations, should satisfy the properties of the Kähler package. Recently, this conjecture has been confirmed using methods from convex geometry, differential geometry, and spectral theory.

The prospective PhD student will conduct research aimed at extending and deepening our understanding of this recent result.

More information and application instructions can be found in the official announcement:

https://jobs.uni-jena.de/jobposting/355158256364402616c9ef450b1426427aecb65c0

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