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. Adv. Math., 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

Special Kähler and related geometries

Special Kähler structures appear in geometry as natural structures on certain important moduli spaces and as the structure of the base of a holomorphic Lagrangian fibration. These structures also appear in physics, for example in the low energy limit of N = 2 supersymmetric Yang-Mills theory in dimension four.

Since a complete special Kähler metric is necessarily flat, a singular special Kähler metric is a relevant object of study. From the integrable system point of view, singularities of special Kähler structures correspond to singular fibers of the corresponding holomorphic Lagrangian fibration. From the moduli space point of view, singularities correspond to the "boundary points", i.e., degenerations of the structures under consideration. The main goal of this project is to understand singularities of special Kähler structures in a systematic way.

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