We consider rational representations of a connected linear algebraic group $\mathbb {G}$ over a field $k$ of positive characteristic $p > 0$. We introduce a natural extension $M \mapsto \Pi (\mathbb {G})_M$ to $\mathbb {G}$-modules of the $\pi$-point support theory for modules $M$ for a finite group scheme $G$ and show that this theory is essentially equivalent to the more ‘intrinsic’ and ‘explicit’ theory $M \mapsto \mathbb {P}\mathfrak{C}(\mathbb {G})_M$ of supports for an algebraic group of exponential type, a theory which uses $1$-parameter subgroups $\mathbb {G}_a \to \mathbb {G}$. We extend our support theory to bounded complexes of $\mathbb {G}$-modules, $C^\bullet \mapsto \Pi (\mathbb {G})_{C^\bullet }$. We introduce the tensor triangulated category $\mathit {StMod}(\mathbb {G})$, the Verdier quotient of the bounded derived category $D^b(\mathit {Mod}(\mathbb {G}))$ by the thick subcategory of mock injective modules. Our support theory satisfies all the ‘standard properties’ for a theory of supports for $\mathit {StMod}(\mathbb {G})$. As an application, we employ $C^\bullet \mapsto \Pi (\mathbb {G})_{C^\bullet }$ to establish the classification of $(r)$-complete, thick tensor ideals of $\mathit {stmod}(\mathbb {G})$ in terms of locally $\mathit {stmod}(\mathbb {G})$-realizable subsets of $\Pi (\mathbb {G})$ and the classification of $(r)$-complete, localizing subcategories of $\mathit {StMod}(\mathbb {G})$ in terms of locally $\mathit {StMod}(\mathbb {G})$-realizable subsets of $\Pi (\mathbb {G})$.

In this paper we investigate the quantity of diagonal quartic surfaces $a_0 X_0^4 + a_1 X_1^4 + a_2 X_2^4 +a_3 X_3^4 = 0$ which have a Brauer–Manin obstruction to the Hasse principle. We are able to find an asymptotic formula for the quantity of such surfaces ordered by height. The proof uses a generalization of a method of Heath-Brown on sums over linked variables. We also show that there exists no uniform formula for a generic generator in this family.

Let $f : X \to S$ be a family of smooth projective algebraic varieties over a smooth connected quasi-projective base $S$, and let $\mathbb {V} = R^{2k} f_{*} \mathbb {Z}(k)$ be the integral variation of Hodge structure coming from degree $2k$ cohomology it induces. Associated to $\mathbb {V}$ one has the so-called Hodge locus $\textrm {HL}(S) \subset S$, which is a countable union of ‘special’ algebraic subvarieties of $S$ parametrizing those fibres of $\mathbb {V}$ possessing extra Hodge tensors (and so, conjecturally, those fibres of $f$ possessing extra algebraic cycles). The special subvarieties belong to a larger class of so-called weakly special subvarieties, which are subvarieties of $S$ maximal for their algebraic monodromy groups. For each positive integer $d$, we give an algorithm to compute the set of all weakly special subvarieties $Z \subset S$ of degree at most $d$ (with the degree taken relative to a choice of projective compactification $S \subset \overline {S}$ and very ample line bundle $\mathcal {L}$ on $\overline {S}$). As a corollary of our algorithm we prove conjectures of Daw–Ren and Daw–Javanpeykar–Kühne on the finiteness of sets of special and weakly special subvarieties of bounded degree.

We give cases in which nearby cycles commute with pushforward from sheaves on the moduli stack of shtukas to a product of curves over a finite field. The proof systematically uses the property that taking nearby cycles of Satake sheaves on the Beilinson–Drinfeld Grassmannian with parahoric reduction is a central functor together with a ‘Zorro's lemma’ argument similar to that of Xue [Smoothness of cohomology sheaves of stacks of shtukas, Preprint (2020), arXiv:2012.12833]. As an application, for automorphic forms at the parahoric level, we characterize the image of tame inertia under the Langlands correspondence in terms of two-sided cells.

One fundamental consequence of a scheme $X$ being proper is that the functor classifying maps from $X$ to any other suitably nice scheme or algebraic stack is representable by an algebraic stack. This result has been generalized by replacing $X$ with a proper algebraic stack. We show, however, that it also holds when $X$ is replaced by many examples of algebraic stacks which are not proper, including many global quotient stacks. This leads us to revisit the definition of properness for stacks. We introduce the notion of a formally proper morphism of stacks and study its properties. We develop methods for establishing formal properness in a large class of examples. Along the way, we prove strong $h$-descent results which hold in the setting of derived algebraic geometry but not in classical algebraic geometry. Our main applications are algebraicity results for mapping stacks and the stack of coherent sheaves on a flat and formally proper stack.

We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs, and apply this invariant to various problems in algebraic and symplectic geometry. On the algebro-geometric side, we introduce a new method based on symplectic flexibility and mirror symmetry to bound the Rouquier dimension of derived categories of coherent sheaves on certain complex algebraic varieties and stacks. These bounds are sharp in dimension at most $3$. As an application, we resolve a well-known conjecture of Orlov for new classes of examples (e.g. toric $3$-folds, certain log Calabi–Yau surfaces). We also discuss applications to non-commutative motives on partially wrapped Fukaya categories. On the symplectic side, we study various quantitative questions including the following. (1) Given a Weinstein manifold, what is the minimal number of intersection points between the skeleton and its image under a generic compactly supported Hamiltonian diffeomorphism? (2) What is the minimal number of critical points of a Lefschetz fibration on a Liouville manifold with Weinstein fibers? We give lower bounds for these quantities which are to the best of the authors’ knowledge the first to go beyond the basic flexible/rigid dichotomy.

We present a quantitative isolation property of the lifts of properly immersed geodesic planes in the frame bundle of a geometrically finite hyperbolic $3$-manifold. Our estimates are polynomials in the tight areas and Bowen–Margulis–Sullivan densities of geodesic planes, with degree given by the modified critical exponents.

We initiate the study of outer automorphism groups of special groups $G$, in the Haglund–Wise sense. We show that $\operatorname {Out}(G)$ is infinite if and only if $G$ splits over a co-abelian subgroup of a centraliser and there exists an infinite-order ‘generalised Dehn twist’. Similarly, the coarse-median preserving subgroup $\operatorname {Out}_{\rm cmp}(G)$ is infinite if and only if $G$ splits over an actual centraliser and there exists an infinite-order coarse-median-preserving generalised Dehn twist. The proof is based on constructing and analysing non-small, stable $G$-actions on $\mathbb {R}$-trees whose arc-stabilisers are centralisers or closely related subgroups. Interestingly, tripod-stabilisers can be arbitrary centralisers, and thus are large subgroups of $G$. As a result of independent interest, we determine when generalised Dehn twists associated to splittings of $G$ preserve the coarse median structure.

In this paper, we give an explicit computable algorithm for the Zelevinsky–Aubert duals of irreducible representations of $p$-adic symplectic and odd special orthogonal groups. To do this, we establish explicit formulas for certain derivatives and socles. We also give a combinatorial criterion for the irreducibility of certain parabolically induced representations.

Let $\psi : \mathbb {N} \to [0,1/2]$ be given. The Duffin–Schaeffer conjecture, recently resolved by Koukoulopoulos and Maynard, asserts that for almost all reals $\alpha$ there are infinitely many coprime solutions $(p,q)$ to the inequality $|\alpha - p/q| < \psi (q)/q$, provided that the series $\sum _{q=1}^\infty \varphi (q) \psi (q) / q$ is divergent. In the present paper, we establish a quantitative version of this result, by showing that for almost all $\alpha$ the number of coprime solutions $(p,q)$, subject to $q \leq Q$, is of asymptotic order $\sum _{q=1}^Q 2 \varphi (q) \psi (q) / q$. The proof relies on the method of GCD graphs as invented by Koukoulopoulos and Maynard, together with a refined overlap estimate from sieve theory, and number-theoretic input on the ‘anatomy of integers’. The key phenomenon is that the system of approximation sets exhibits ‘asymptotic independence on average’ as the total mass of the set system increases.

We connect two developments that aim to extend Voevodsky's theory of motives over a field in such a way as to encompass non-$\mathbf {A}^1$-invariant phenomena. One is theory of reciprocity sheaves introduced by Kahn, Saito and Yamazaki. The other is theory of the triangulated category $\operatorname {\mathbf {logDM}}^{{\operatorname {eff}}}$ of logarithmic motives launched by Binda, Park and Østvær. We prove that the Nisnevich cohomology of reciprocity sheaves is representable in $\operatorname {\mathbf {logDM}}^{{\operatorname {eff}}}$.

Let $\{\Lambda ^\infty _t\}$ be an isotopy of Legendrians (possibly singular) in a unit cosphere bundle $S^*M$ that arise as slices of a singular Legendrian $\Lambda _I^\infty \subset S^*M \times T^*I$. Let $\mathcal {C}_t = Sh(M, \Lambda ^\infty _t)$ be the differential graded derived category of constructible sheaves on $M$ with singular support at infinity contained in $\Lambda ^\infty _t$. We prove that if the isotopy of Legendrians embeds into an isotopy of Liouville hypersurfaces, then the family of categories $\{\mathcal {C}_t\}$ is constant in $t$.

We bound from below the complexity of the top Chern class $\lambda _g$ of the Hodge bundle in the Chow ring of the moduli space of curves: no formulas for $\lambda _g$ in terms of classes of degrees 1 and 2 can exist. As a consequence of the Torelli map, the 0-section over the second Voronoi compactification of the moduli of principally polarized abelian varieties also cannot be expressed in terms of classes of degree 1 and 2. Along the way, we establish new cases of Pixton's conjecture for tautological relations. In the log Chow ring of the moduli space of curves, however, we prove $\lambda _g$ lies in the subalgebra generated by logarithmic boundary divisors. The proof is effective and uses Pixton's double ramification cycle formula together with a foundational study of the tautological ring defined by a normal crossings divisor. The results open the door to the search for simpler formulas for $\lambda _g$ on the moduli of curves after log blow-ups.

We prove an extension of the homology version of the Hofer–Zehnder conjecture proved by Shelukhin to the weighted projective spaces which are symplectic orbifolds. In particular, we prove that if the number of fixed points counted with their isotropy order as multiplicity of a non-degenerate Hamiltonian diffeomorphism of such a space is larger than the minimum number possible, then there are infinitely many periodic points.

Let $\mathbb {V}$ be a polarized variation of Hodge structure over a smooth complex quasi-projective variety $S$. In this paper, we give a complete description of the typical Hodge locus for such variations. We prove that it is either empty or equidistributed with respect to a natural differential form, the pull–push form. In particular, it is always analytically dense when the pull–push form does not vanish. When the weight is two, the Hodge numbers are $(q,p,q)$ and the dimension of $S$ is least $rq$, we prove that the typical locus where the Picard rank is at least $r$ is equidistributed in $S$ with respect to the volume form $c_q^r$, where $c_q$ is the $q$th Chern form of the Hodge bundle. We obtain also several equidistribution results of the typical locus in Shimura varieties: a criterion for the density of the typical Hodge loci of a variety in $\mathcal {A}_g$, equidistribution of certain families of CM points and equidistribution of Hecke translates of curves and surfaces in $\mathcal {A}_g$. These results are proved in the much broader context of dynamics on homogeneous spaces of Lie groups which are of independent interest. The pull–push form appears in this greater generality, we provide several tools to determine it, and we compute it in many examples.