We prove a strong version of a theorem of Siu and Beauville on morphisms to higher genus curves, and we use it to show that if a variety X in characteristic p lifts to characteristic 0, then any morphism $X\to C$ to a curve of genus $g\ge 2$ can be lifted along. We use this to construct, for every prime p, a smooth projective surface X over ${\overline{\mathbf{F}}}_p$ that cannot be rationally dominated by a smooth proper variety Y that lifts to characteristic 0.

We study the existence of special Lagrangian submanifolds of log Calabi–Yau manifolds equipped with the complete Ricci-flat Kähler metric constructed by Tian and Yau. We prove that if X is a Tian–Yau manifold and if the compact Calabi–Yau manifold at infinity admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians. In complex dimension 2, we prove that if Y is a del Pezzo surface or a rational elliptic surface and $D\in |-K_Y|$ is a smooth divisor with $D^2=d$, then $X=Y\setminus D$ admits a special Lagrangian torus fibration, as conjectured by Strominger–Yau–Zaslow and Auroux. In fact, we show that X admits twin special Lagrangian fibrations, confirming a prediction of Leung and Yau. In the special case that Y is a rational elliptic surface or $Y={\mathbb{P}}^2$, we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kähler rotation, X can be compactified to the complement of a Kodaira type $I_d$ fiber appearing as a singular fiber in a rational elliptic surface $\check{\mathrm{\pi }}:\check{Y}\to {\mathbb{P}}^1$.

We consider hedgehog solutions in the $(3+1)$-dimensional Skyrme model, which is an energy-supercritical problem. We introduce a new strategy to prove global well-posedness for arbitrarily large initial data.

To reduce to resolving Cohen–Macaulay singularities, Faltings initiated the program of “Macaulayfying” a given Noetherian scheme X. For a wide class of X, Kawasaki built the sought-for Cohen–Macaulay modifications, with a crucial drawback that his blowups did not preserve the locus $\mathrm{CM}(X)\subset X$, where X is already Cohen–Macaulay. We extend Kawasaki’s methods to show that every quasi-excellent, Noetherian scheme X has a Cohen–Macaulay $\tilde{X}$ with a proper map $\tilde{X}\to X$ that is an isomorphism over $\mathrm{CM}(X)$. This completes Faltings’s program, reduces the conjectural resolution of singularities to the Cohen–Macaulay case, and implies that every proper, smooth scheme over a number field has a proper, flat, Cohen–Macaulay model over the ring of integers.

By applying the residue method for period integrals, we prove identities between period integrals on cuspidal automorphic representations and period integrals on residual representations for seven spherical varieties. Combining this with the Langlands–Shahidi theory for residues of Eisenstein series, for each case, we prove some relations between the period integrals and certain automorphic L-functions. In some cases, we also study the local multiplicity of the spherical varieties.

We study Kakutani equivalence in the class of unipotent flows acting on finite-volume quotients of semisimple Lie groups. For every such flow, we compute the Kakutani invariant of M. Ratner, the value being explicitly given by the Jordan block structure of the unipotent element generating the flow. This, in particular, answers a question of M. Ratner. Moreover, it follows that the only loosely Kronecker unipotent flows are given by $\left(\begin{array}{cc}1& t\\ 0& 1\end{array}\right)\times \mathrm{id}$ acting on $(\mathrm{SL}(2,\mathbb{R})\times G^{\prime })∕{\mathrm{\Gamma }}^{\prime }$, where ${\mathrm{\Gamma }}^{\prime }$ is an irreducible lattice in $\mathrm{SL}(2,\mathbb{R})\times G^{\prime }$ (with the possibility that $G^{\prime }=\{e\}$).