Geometry

Differential Geometry (Real and Complex), Algebraic Geometry, Simplectic Geometry and Topology are closely connected. For example, a complex (immersed) projective variety is a special case of a Kähler variety, which in turn is a special case of both a Riemannian and a symplectic variety. Moreover, a Riemannian or Kähler structure induces strong constraints on the topology of a variety. There are multiple and deep implications between these areas of Geometry (and Topology). To give just a few examples, the search for special metrics on complex varieties is linked to issues of algebro-geometric stability. On the other hand, every analytic Riemannian variety in the real sense admits an essentially canonical complexification, which is a Stein variety (a concept in which symplectic and complex analysis properties converge) and whose properties, which can be investigated by methods of complex analysis and symplectic geometry, reflect the properties of the starting Riemannian variety. Or again, the geodesic flow of a Riemannian metric is a special case of Hamiltonian flow on a symplectic variety. The research activity in the various areas of Geometry in our Department is summarised here.

Algebraic Geometry

Topics:

  • Geometry of toric varieties. Geometry and topology of Mori Dream Spaces. Geometric transitions and their analytical equivalence. Aspects of Mirror Symmetry and Homological Mirror Symmetry
  • Vector fibres on algebraic varieties. Forms spaces of vector fibres on algebraic projective curves
  • Motivic theories. Group actions on patterns

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Riemannian Geometry

Topics:

  • Geometry of subvarieties with constant medium curvature and solitons of geometric flows
  • Geometry of Riemannian varieties with border

Participants:

Geometric Analysis

Topics:

  • (p-)harmonic maps between Riemannian varieties and related Dirichlet problems
  • Global integral inequalities on Riemannian varieties and metric rigidity
  • Qualitative analysis of solutions of elliptic PDEs of geometric origin, possibly nonlinear, on Riemannian varieties
  • Monodromy of the Schwarz equation on Riemann Surfaces

Participants:

Complex Differential Geometry

Topics:

  • Szegő core and applications in complex geometry and CR, immersion theorems
  • Asymptotically locally Euclidean or complex hyperbolic Kählerian metrics

Participants:

Simplectic And Complex Geometry

Topics:

  • Asymptotic aspects in geometric quantization of symplectic varieties
  • Grauert tubes on analytic Riemannian varieties and Szegő and Poisson cores
  • Symplectic quotients of CR varieties (fibres in circles) associated with Hamiltonian actions and their geometric properties
  • Quantization for CR varieties and Berezin-Toeplitz operators
  • Complex non-integrable compatible structures on symplectic varieties

Participants:

Low-Dimensional Topology

Topics:

  • Realisation of geometric structures on surfaces
  • Geometry and topology of forms spaces of geometric structures
  • Dynamics of class group mapping on spaces of representations of a surface group in Lie groups

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