Open PhD Positions

There are currently open PhD positions at the Research Training Group "Cohomological Methods in Geometry" in Freiburg. If you are interested to do your PhD in differential geometry (in particular, gauge theory and/or Riemannian manifolds with special holonomy groups), feel free to send an informal query per e-mail. For further information, have a look at open positions at the RTG.

Research projects

Gauge theory

The aim of this project is a construction of invariants of compact G2-manifolds via gauge-theoretic methods. A program for constructing such invariants was pioneered by Donaldson and Thomas. Roughly speaking, one would like to define an invariant of G2-manifolds which would remain constant along isotopies of G2-metrics by counting certain G2-instantons. One of the main problems of this approach is that the number of G2-instantons does not need to be constant along isotopies of G2-metrics. Conjecturally, one should be able to obtain an invariant of a G2-manifold that remains constant along isotopies of G2-structures by counting G2-instantons together with certain Seiberg-Witten monopoles on associative submanifolds. Some details of this approach can be found for example in Haydys' paper G2 instantons and the Seiberg-Witten monopoles.

Selected publications:

A. Haydys. Topology of the blow-up set for the Seiberg-Witten equation with multiple spinors. arXiv:1607.01763
A. Haydys, Th.Walpuski. A compactness theorem for the Seiberg-Witten equation with multiple spinors in dimension three. Geom. Funct. Anal. (GAFA), 25(6):1799-1821, 2015; arXiv:1406.5683
A. Haydys. Gauge theory, calibrated geometry and harmonic spinors, J. Lond. Math. Soc. (2), 86(2):482-498, 2012. arXiv:0902.3738

HyperKähler manifolds and related geometries

A mid-term goal of this project is to construct examples of hyperKähler metrics admitting the structure of a complex Lagrangian fibration via the so called c-map construction, which takes an (affine) special Kähler manifold as input and gives a hyperKähler manifold as output. The resulting Riemannian metric is sometimes called semi-flat. The main challenge concerned with the c-map construction is to extend semi-flat metrics over the singularities after a suitable modification.

Since a complete special Kähler metric is necessarily flat, a singular special Kähler metric is a relevant input for the c-map construction. Haydys' research in the framework of this project has been concerned so far with a description of isolated singularities of affine special Kähler structures.

Selected publications:

A. Haydys and B. Xu. Special Kähler structures, cubic differentials and hyperbolic metrics, arXiv:1807.08550
M. Callies and A. Haydys. Local models of isolated singularities for affine special Kähler structures in dimension two, IMRN, DOI:10.1093/imrn/rny165, arXiv:1711.09118
A. Haydys. Isolated singularities of affine special Kaehler metrics in two dimensions. Commun. Math. Phys., 340(3):1231-1237, 2015; arXiv:1505.00462

Some Recorded Talks