We prove the weight elimination direction of the Serre weight conjectures as formulated by Herzig for forms of $U\left(n\right)$ which are compact at infinity and split at places dividing $p$ in generic situations. That is, we show that all modular weights for a mod $p$ Galois representation are contained in the set predicted by Herzig. Under some additional hypotheses, we also show modularity of all the “obvious” weights.

The topological classification of even the simplest Morse–Smale diffeomorphisms on $3$-manifolds does not fit into the concept of singling out a skeleton consisting of stable and unstable manifolds of periodic orbits. The reason for this lies primarily in the possibility of “wild” behavior of separatrices of saddle points. Another difference between Morse–Smale diffeomorphisms in dimension $3$ and their surface analogues lies in the variety of heteroclinic intersections: a connected component of such an intersection may not be only a point, as in the $2$-dimensional case, but also a curve, compact or noncompact. The problem of topological classification of Morse–Smale cascades on $3$-manifolds either without heteroclinic points (gradient-like cascades) or without heteroclinic curves was solved in a series of papers from 2000 to 2016 by C. Bonatti, V. Grines, F. Laudenbach, V. Medvedev, E. Pecou, and O. Pochinka. The present article is devoted to completing the topological classification of the set $MS\left(M^3\right)$ of orientation-preserving Morse–Smale diffeomorphisms $f$ on a smooth closed orientable $3$-manifold $M^3$. The complete topological invariant for a diffeomorphism $f\in MS\left(M^3\right)$ is the equivalence class of its scheme $S_f$ which contains information on the periodic data and the topology of embedding of $2$-dimensional invariant manifolds of the saddle periodic points of $f$ into the ambient manifold.

A combination of direct and inverse Fourier transforms on the unitary group $U\left(N\right)$ identifies normalized characters with probability measures on $N$-tuples of integers. We develop the $N\to \infty$ version of this correspondence by matching the asymptotics of partial derivatives at the identity of logarithm of characters with the law of large numbers and the central limit theorem for global behavior of corresponding random $N$-tuples.

As one application we study fluctuations of the height function of random domino and lozenge tilings of a rich class of domains. In this direction we prove the Kenyon–Okounkov conjecture (which predicts asymptotic Gaussianity and the exact form of the covariance) for a family of non-simply-connected polygons.

Another application is a central limit theorem for the $U\left(N\right)$ quantum random walk with random initial data.