## 9. Other problems

This is the section for problems which (a) were not multi-part and (b) I had fewer notes on.-
### Simplify the fibres of the Hodge bundle.

#### Problem 9.1.

Seek notion of genericity for complex structures simplifying the stratification of the singular locus of the Hitchin fibration (for $SL_2(\mathbb{C})$). -
### Relation to Hilbert schemes of points.

#### Problem 9.2.

Better understand the relationship between the moduli of Higgs bundles on $C$ and the Hilbert scheme of points $\text{Hilb}^{\text{pt}}(T^*C)$. -
### Compactifications of the moduli of Higgs bundles.

#### Problem 9.3.

Compare compactifications of the moduli space of Higgs bundles for $GL_n(\mathbb{C})$ of Hausel, Schmitt, and MSWW-F. -
### Deformations from Hochschild homology.

#### Problem 9.4.

Give a geometric realisation of objects given by deforming a Hitchin pair by a non-zero class in $HH_0(C)$ (c.f. Keller). -
### Branes in the moduli of Higgs bundles.

#### Problem 9.5.

Review techniques for constructing Lagrangians in hyperkahler spaces, see which can be applied to the moduli space of Higgs bundles, and explicitly translate/apply those techniques. -
### $tt^*$-equations and Higgs bundles.

#### Problem 9.6.

Make explicit the relationship between the $tt^*$-equations and Higgs bundles in a specific example.

Cite this as: *AimPL: Singular geometry and Higgs bundles in string theory, available at http://aimpl.org/singularhiggs.
*