We show that the simple group ${\mathrm{PSL}}_2\left({\mathbb{F}}_p\right)$ occurs as the Galois group of an extension of the rationals for all primes $p\ge 5$. We obtain our Galois extensions by studying the Galois action on the second étale cohomology groups of a specific elliptic surface.

We prove the transformation formula of Donaldson–Thomas (DT) invariants counting two-dimensional torsion sheaves on Calabi–Yau 3-folds under flops. The error term is described by the Dedekind eta function and the Jacobi theta function, and our result gives evidence of a 3-fold version of the Vafa–Witten S-duality conjecture. As an application, we prove a blow-up formula of DT-type invariants on the total spaces of canonical line bundles on smooth projective surfaces. It gives an analogue of the similar blow-up formula in the original S-duality conjecture by Yoshioka, Li and Qin, and Göttsche.

We study noncompact surfaces evolving by mean curvature flow. Without any symmetry assumptions, we prove that any solution that is $C^3$-close at some time to a standard neck will develop a neckpinch singularity in finite time, will become asymptotically rotationally symmetric in a space-time neighborhood of its singular set, and will have a unique tangent flow.

Generalized Baxter’s relations on the transfer matrices (also known as Baxter’s $TQ$ relations) are constructed and proved for an arbitrary untwisted quantum affine algebra. Moreover, we interpret them as relations in the Grothendieck ring of the category $\mathcal{O}$, introduced by Hernandez and Jimbo, involving infinite-dimensional representations, which we call here “prefundamental.” We define the transfer matrices associated to the prefundamental representations and prove that their eigenvalues on any finite-dimensional representation are polynomials up to a universal factor. These polynomials are the analogues of the celebrated Baxter polynomials. Combining these two results, we express the spectra of the transfer matrices in the general quantum integrable systems associated to an arbitrary untwisted quantum affine algebra in terms of our generalized Baxter polynomials. This proves a conjecture postulated by Frenkel and Reshetikhin in 1998. We also obtain generalized Bethe ansatz equations for all untwisted quantum affine algebras.