We extend most of the results of generic vanishing theory to bundles of holomorphic forms and rank-one local systems, and more generally to certain coherent sheaves of Hodge-theoretic origin associated with irregular varieties. Our main tools are Saito’s mixed Hodge modules, the Fourier–Mukai transform for $\mathscr{D}$-modules on abelian varieties introduced by Laumon and Rothstein, and Simpson’s harmonic theory for flat bundles. In the process, we also discover two natural categories of perverse coherent sheaves.

We study the mixing properties of progressions $(x, xg, x{g}^{2} )$, $(x, xg, x{g}^{2} , x{g}^{3} )$ of length three and four in a model class of finite nonabelian groups, namely the special linear groups ${\mathrm{SL} }_{d} (F)$ over a finite field $F$, with $d$ bounded. For length three progressions $(x, xg, x{g}^{2} )$, we establish a strong mixing property (with an error term that decays polynomially in the order $\vert F\vert $ of $F$), which among other things counts the number of such progressions in any given dense subset $A$ of ${\mathrm{SL} }_{d} (F)$, answering a question of Gowers for this class of groups. For length four progressions $(x, xg, x{g}^{2} , x{g}^{3} )$, we establish a partial result in the $d= 2$ case if the shift $g$ is restricted to be diagonalizable over $F$, although in this case we do not recover polynomial bounds in the error term. Our methods include the use of the Cauchy–Schwarz inequality, the abelian Fourier transform, the Lang–Weil bound for the number of points in an algebraic variety over a finite field, some algebraic geometry, and (in the case of length four progressions) the multidimensional Szemerédi theorem.

We construct a hyperbolic group with a hyperbolic subgroup for which inclusion does not induce a continuous map of the boundaries.

Hassett and Tschinkel gave counterexamples to the integral Hodge conjecture among 3-folds over a number field. We work out their method in detail, showing that essentially all known counterexamples to the integral Hodge conjecture over the complex numbers can be made to work over a number field.

Since the foundational work of Chenciner and Montgomery in 2000 there has been a great deal of interest in choreographic solutions of the $n$-body problem: periodic motions where the $n$ bodies all follow one another at regular intervals along a closed path. The principal approach combines variational methods with symmetry properties. In this paper, we give a systematic treatment of the symmetry aspect. In the first part, we classify all possible symmetry groups of planar $n$-body collision-free choreographies. These symmetry groups fall into two infinite families and, if $n$ is odd, three exceptional groups. In the second part, we develop the equivariant fundamental group and use it to determine the topology of the space of loops with a given symmetry, which we show is related to certain cosets of the pure braid group in the full braid group, and to centralizers of elements of the corresponding coset. In particular, we refine the symmetry classification by classifying the connected components of the set of loops with any given symmetry. This leads to the existence of many new choreographies in $n$-body systems governed by a strong force potential.