Using min-max theory, we show that in any closed Riemannian manifold of dimension at least $3$ and at most $7$, there exist infinitely many smoothly embedded closed minimal hypersurfaces. It proves a conjecture of S.-T. Yau. This paper builds on the methods developed by F. C. Marques and A. Neves.

Let $F$ be a CM number field. We prove modularity lifting theorems for regular $n$-dimensional Galois representations over $F$ without any self-duality condition. We deduce that all elliptic curves $E$ over $F$ are potentially modular, and furthermore satisfy the Sato--Tate conjecture. As an application of a different sort, we also prove the Ramanujan Conjecture for weight zero cuspidal automorphic representations for $\mathrm{GL}_2(\mathbb{A}_F)$.

It is known that the Brauer--Manin obstruction to the Hasse principle is vacuous for smooth Fano hypersurfaces of dimension at least $3$ over any number field. Moreover, for such varieties it follows from a general conjecture of Colliot-Thélène that the Brauer--Manin obstruction to the Hasse principle should be the only one, so that the Hasse principle is expected to hold. Working over the field of rational numbers and ordering Fano hypersurfaces of fixed degree and dimension by height, we prove that almost every such hypersurface satisfies the Hasse principle provided that the dimension is at least $3$. This proves a conjecture of Poonen and Voloch in every case except for cubic surfaces.

With respect to a $C^\infty$ metric which is close to the standard Euclidean metric on $\mathbb{R}^{N+1+\ell}$, where $N\ge 7$ and $\ell\ge 1$ are given, we construct a class of embedded $(N+\ell)$-dimensional hypersurfaces (without boundary) which are minimal and strictly stable, and which have singular set equal to an arbitrary preassigned closed subset $K\subset \{0\}\times \mathbb{R}^{\ell}$. Thus the question is settled, with a strong affirmative, as to whether there can be " gaps" or even fractional dimensional parts in the singular set. Such questions, for both stable and unstable minimal submanifolds, remain open in all dimensions in the case of real analytic metrics and, in particular, for the standard Euclidean metric.

The construction used here involves the analysis of solutions $u$ of the symmetric minimal surface equation on domains $\Omega\subset\mathbb{R}^{n}$ whose symmetric graphs (i.e., $\{(x,\xi)\in\Omega\times \mathbb{R}^{m}: |\xi| = u(x)\}$) lie on one side of a cylindrical minimal cone including, in particular, a Liouville type theorem for complete solutions (i.e., the case $\Omega=\mathbb{R}^{n}$).

A fundamental theorem of P. Deligne (2002) states that a pre-Tannakian category over an algebraically closed field of characteristic zero admits a fiber functor to the category of supervector spaces (i.e., is the representation category of an affine proalgebraic supergroup) if and only if it has moderate growth (i.e., the lengths of tensor powers of an object grow at most exponentially). In this paper we prove a characteristic $p$ version of this theorem. Namely, we show that a pre-Tannakian category over an algebraically closed field of characteristic $p>0$ admits a fiber functor into the Verlinde category $\mathrm{Ver}_p$ (i.e., is the representation category of an affine group scheme in $\mathrm{Ver}_p$) if and only if it has moderate growth and is Frobenius exact. This implies that Frobenius exact pre-Tannakian categories of moderate growth admit a well behaved notion of Frobenius-Perron dimension.

It follows that any semisimple pre-Tannakian category of moderategrowth has a fiber functor to $\mathrm{Ver}_p$ (so in particular Deligne's theorem holds on the nose for semisimple pre-Tannakian categories in characteristics $2$,$3$). This settles a conjecture of the third author from 2015.

In particular, this result applies to semisimplifications of categories of modular representations of finite groups (or, more generally, affine group schemes), which gives new applications to classical modular representation theory. For example, it allows us to characterize, for a modular representation $V$, the possible growth rates of the number of indecomposable summands in $V^{\otimes n}$of dimension prime to $p$.