Translators in \(\mathbb{R}^3\) are solitons of the mean curvature flow for embedded 2-surfaces. In the semigraphical case, where the translators are allowed graphical as well as vertical components, Hoffman-Martín-White classified the surfaces into six types. They conjectured the uniqueness of the objects within two families contained in slabs, the "helicoids" and the "pitchforks," for any given width. We present the proof of the conjecture by combining an arc-counting argument motivated by Morse-Radó theory for translators with a rotational application of the maximum principle. We then discuss applications of this result to the classification of semigraphical translators in \(\mathbb{R}^3\) and their limits, related to the work of Hoffman-Martín-White and Gama-Martín-Møller. This is joint work with F. Martín and M. Sáez.

The moduli space of solutions to Nahm’s equations with values in the Lie algebra of a Lie group G (more generally, self-dual Yang-Mills equations) carries a complete hyperkähler structure, obtained via infinite-dimensional reduction by Hitchin (1987). Kronheimer (1989) proved that this is diffeomorphic to the total space of the cotangent bundle T*Gc of a complex Lie group. Both the hyperkähler structure on the moduli space and the diffeomorphism with T*Gc are proved to exist abstractly; hence, the resulting hyperkähler metric on T*Gc is challenging to describe explicitly even for basic Lie groups. We present joint work with Richard Melrose and Michael Singer obtaining the asymptotics of the metric on the moduli space and the resolution of the critical set of the Nahm vector. We also present an explicit description of the diffeomorphism and induced metric on T* in the case of G=SU(2).