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Geometry

The study of geometry can be traced back to antiquity, whereby questions can be motivated by concrete application problems, for example from physics, or the pure gain of knowledge. The individual branches of geometry differ in which objects are studied and which methods are used.

Algebraic geometry studies curves, surfaces and higher dimensional algebraic varieties defined by polynomials. A simple example is the circle defined by the equation x²+y² =1, but more complicated equations describe algebraic varieties that still hold many secrets. The methods are mainly based on the rich arsenal of modern algebra, but complex analysis, number theory or topology are also used when needed.At the department, research is mainly done on the geometry of linear systems and the classification of algebraic surfaces.

In spectral geometry, the goal is to extract information about the eigenvalues of elliptic differential operators (especially the Laplace operator and the Dirac operator) from the geometry (such as the curvature properties or the behavior at infinity) of a manifold. This question is often summarized as "Can one hear the shape of a drum?" (Marc Kac, 1966) - the frequencies of the sounds produced by the drum are precisely the eigenvalues of the Dirichlet problem for the Laplace operator. In this form, the question has now been answered negatively, but it is still possible to obtain an astonishing amount of information about the "drum" (the Manng manifold) from its frequency metric. Again, symmetry groups play a significant role in reducing the complexity of an otherwise almost unsolvable problem to one that is still difficult but treatable. Two important research topics of the Analysis and Differential Geometry group here are on the one hand the distribution, more precisely, the asymptotic growth of eigenvalues (Weyl's law), and on the other hand the derivation of geometric estimates of the smallest eigenvalue (energy levels) and the characterization of the equality case.