Given an irreducible projective variety X, the covering gonality of X is the least gonality of an irreducible curve $E\subset X$ passing through a general point of X. In this paper, we study the covering gonality of the k-fold symmetric product $C^{(k)}$ of a smooth complex projective curve C of genus $g\geq k+1$. It follows from a previous work of the first author that the covering gonality of the second symmetric product of C equals the gonality of C. Using a similar approach, we prove the same for the $3$-fold and the $4$-fold symmetric product of C.

A crucial point in the proof is the study of the Cayley–Bacharach condition on Grassmannians. In particular, we describe the geometry of linear subspaces of $\mathbb {P}^n$ satisfying this condition, and we prove a result bounding the dimension of their linear span.

We consider stochastic differential equations (SDEs) driven by a fractional Brownian motion with a drift coefficient that is allowed to be arbitrarily close to criticality in a scaling sense. We develop a comprehensive solution theory that includes strong existence, path-by-path uniqueness, existence of a solution flow of diffeomorphisms, Malliavin differentiability and $\rho $-irregularity. As a consequence, we can also treat McKean-Vlasov, transport and continuity equations.

We consider generalised Dirac-Schrödinger operators, consisting of a self-adjoint elliptic first-order differential operator $\mathcal {D}$ with a skew-adjoint ‘potential’ given by a (suitable) family of unbounded operators. The index of such an operator represents the pairing (Kasparov product) of the K-theory class of the potential with the K-homology class of $\mathcal {D}$. Our main result in this paper is a generalisation of the Callias Theorem: the index of the Dirac-Schrödinger operator can be computed on a suitable compact hypersurface. Our theorem simultaneously generalises (and is inspired by) the well-known result that the spectral flow of a path of relatively compact perturbations depends only on the endpoints.

In [15], using methods from ergodic theory, a longstanding conjecture of Erdős (see [5, Page 305]) about sumsets in large subsets of the natural numbers was resolved. In this paper, we extend this result to several important classes of amenable groups, including all finitely generated virtually nilpotent groups and all abelian groups $(G,+)$ with the property that the subgroup $2G := \{g+g : g\in G\}$ has finite index. We prove that in any group G from the above classes, any $A\subset G$ with positive upper Banach density contains a shifted product set of the form $\{tb_ib_j\colon i<j\}$, for some infinite sequence $(b_n)_{n\in \mathbb {N}}$ and some $t\in G$. In fact, we show this result for all amenable groups that posses a property which we call square absolute continuity. Our results provide answers to several questions and conjectures posed in [13].

We study linearizability of actions of finite groups on singular cubic threefolds, using cohomological tools, intermediate Jacobians, Burnside invariants, and the equivariant Minimal Model Program.

We consider faithful actions of simple algebraic groups on self-dual irreducible modules and on the associated varieties of totally singular subspaces, under the assumption that the dimension of the group is at least as large as the dimension of the variety. We prove that in all but a finite list of cases, there is a dense open subset where the stabilizer of any point is conjugate to a fixed subgroup, called the generic stabilizer. We use these results to determine whether there exists a dense orbit. This in turn lets us complete the answer to the problem of determining all pairs of maximal connected subgroups of a classical group with a dense double coset.

Let $(W,S)$ be a Coxeter system, and write $S=\{s_i:i\in I\}$, where I is a finite index set. Fix a nonempty convex subset $\mathscr {L}$ of W. If W is of type A, then $\mathscr {L}$ is the set of linear extensions of a poset, and there are important Bender–Knuth involutions $\mathrm {BK}_i\colon \mathscr {L}\to \mathscr {L}$ indexed by elements of I. For arbitrary W and for each $i\in I$, we introduce an operator $\tau _i\colon W\to W$ (depending on $\mathscr {L}$) that we call a noninvertible Bender–Knuth toggle; this operator restricts to an involution on $\mathscr {L}$ that coincides with $\mathrm {BK}_i$ in type A. Given a Coxeter element $c=s_{i_n}\cdots s_{i_1}$, we consider the operator $\mathrm {Pro}_c=\tau _{i_n}\cdots \tau _{i_1}$. We say W is futuristic if for every nonempty finite convex set $\mathscr {L}$, every Coxeter element c and every $u\in W$, there exists an integer $K\geq 0$ such that $\mathrm {Pro}_c^K(u)\in \mathscr {L}$. We prove that finite Coxeter groups, right-angled Coxeter groups, rank-3 Coxeter groups, affine Coxeter groups of types $\widetilde A$ and $\widetilde C$, and Coxeter groups whose Coxeter graphs are complete are all futuristic. When W is finite, we actually prove that if $s_{i_N}\cdots s_{i_1}$ is a reduced expression for the long element of W, then $\tau _{i_N}\cdots \tau _{i_1}(W)=\mathscr {L}$; this allows us to determine the smallest integer $\mathrm {M}(c)$ such that $\mathrm {Pro}_c^{{\mathrm {M}}(c)}(W)=\mathscr {L}$ for all $\mathscr {L}$. We also exhibit infinitely many non-futuristic Coxeter groups, including all irreducible affine Coxeter groups that are not of type $\widetilde A$, $\widetilde C$, or $\widetilde G_2$.

Let C be a curve defined over a number field K and write g for the genus of C and J for the Jacobian of C. Let $n \ge 2$. We say that an algebraic point $P \in C(\overline {K})$ has degree n if the extension $K(P)/K$ has degree n. By the Galois group of P we mean the Galois group of the Galois closure of $K(P)/K$ which we identify as a transitive subgroup of $S_n$. We say that P is primitive if its Galois group is primitive as a subgroup of $S_n$. We prove the following ‘single source’ theorem for primitive points. Suppose $g>(n-1)^2$ if $n \ge 3$ and $g \ge 3$ if $n=2$. Suppose that either J is simple or that $J(K)$ is finite. Suppose C has infinitely many primitive degree n points. Then there is a degree n morphism $\varphi : C \rightarrow \mathbb {P}^1$ such that all but finitely many primitive degree n points correspond to fibres $\varphi ^{-1}(\alpha )$ with $\alpha \in \mathbb {P}^1(K)$.

We prove, moreover, under the same hypotheses, that if C has infinitely many degree n points with Galois group $S_n$ or $A_n$, then C has only finitely many degree n points of any other primitive Galois group.

We investigate properties of closed approximate subgroups of locally compact groups, with a particular interest for approximate lattices (i.e., those approximate subgroups that are discrete and have finite co-volume).

We prove an approximate subgroup version of Cartan’s closed-subgroup theorem and study some applications. We give a structure theorem for closed approximate subgroups of amenable groups in the spirit of the Breuillard–Green–Tao theorem. We then prove two results concerning approximate lattices: we extend to amenable groups a structure theorem for mathematical quasi-crystals due to Meyer; we prove results concerning intersections of radicals of Lie groups and discrete approximate subgroups generalising theorems due to Auslander, Bieberbach and Mostow. As an underlying theme, we exploit the notion of good models of approximate subgroups that stems from the work of Hrushovski, and Breuillard, Green and Tao. We show how one can draw information about a given approximate subgroup from a good model, when it exists.

We show that if $\Gamma $ is a point group of $\mathbb {R}^{k+1}$ of order two for some $k\geq 2$ and $\mathcal {S}$ is a k-pseudomanifold which has a free automorphism of order two, then either $\mathcal {S}$ has a $\Gamma $-symmetric infinitesimally rigid realisation in ${\mathbb R}^{k+1}$ or $k=2$ and $\Gamma $ is a half-turn rotation group. This verifies a conjecture made by Klee, Nevo, Novik and Zheng for the case when $\Gamma $ is a point-inversion group. Our result implies that Stanley’s lower bound theorem for centrally symmetric polytopes extends to pseudomanifolds with a free simplicial automorphism of order 2, thus verifying (the inequality part of) another conjecture of Klee, Nevo, Novik and Zheng. Both results actually apply to a much larger class of simplicial complexes – namely, the circuits of the simplicial matroid. The proof of our rigidity result adapts earlier ideas of Fogelsanger to the setting of symmetric simplicial complexes.

We express the total space of a principal circle bundle over a connected sum of two manifolds in terms of the total spaces of circle bundles over each summand, provided certain conditions hold. We then apply this result to provide sufficient conditions for the existence of free circle and torus actions on connected sums of products of spheres and obtain a topological classification of closed, simply connected manifolds with a free cohomogeneity-four torus action. As a corollary, we obtain infinitely many manifolds with Riemannian metrics of positive Ricci curvature and isometric torus actions.

We exhibit large families of K3 surfaces with real multiplication, both abstractly, using lattice theory, the Torelli theorem and the surjectivity of the period map, as well as explicitly, using dihedral covers and isogenies.

We consider the extremes of the logarithm of the characteristic polynomial of matrices from the C$\beta $E ensemble. We prove convergence in distribution of the centered maxima (of the real and imaginary parts) toward the sum of a Gumbel variable and another independent variable, which we characterize as the total mass of a ‘derivative martingale’. We also provide a description of the landscape near extrema points.