The Torelli theorem establishes that the Jacobian of a smooth projective curve, together with the polarization provided by the theta divisor, fully characterizes the curve. In the case of nodal curves, there exists a concept known as fine compactified Jacobian. The fine compactified Jacobian of a curve comes with a natural stratification that can be regarded as a poset. Furthermore, this poset is entirely determined by the dual graph of the curve and is referred to as the poset of quasistable divisors on the graph. We present a combinatorial version of the Torelli theorem, which demonstrates that the poset of quasistable divisors of a graph completely determines the biconnected components of the graph (up to contracting separating edges). Moreover, we achieve a natural extension of this theorem to tropical curves.

In this paper, we define compact open subgroups of quasi-split even unitary groups for each even non-negative integer and establish the theory of local newforms for irreducible tempered generic representations with a certain condition on the central characters. To do this, we use the local Gan–Gross–Prasad conjecture, the local Rankin–Selberg integrals and the local theta correspondence.

Chen-Gounelas-Liedtke recently introduced a powerful regeneration technique, a process opposite to specialization, to prove existence results for rational curves on projective $K3$ surfaces. We show that, for projective irreducible holomorphic symplectic manifolds, an analogous regeneration principle holds and provides a very flexible tool to prove existence of uniruled divisors, significantly improving known results.

An empty simplex is a lattice simplex in which vertices are the only lattice points. We show two constructions leading to the first known empty simplices of width larger than their dimension:

• ◦ We introduce cyclotomic simplices and exhaustively compute all the cyclotomic simplices of dimension $10$ and volume up to $2^{31}$. Among them, we find five empty ones of width $11$ and none of larger width.

• ◦ Using circulant matrices of a very specific form, we construct empty simplices of arbitrary dimension d and width growing asymptotically as $d/\operatorname {\mathrm {arcsinh}}(1) \sim 1.1346\,d$.

We develop the theory and algorithms necessary to be able to verify the strong Birch–Swinnerton-Dyer Conjecture for absolutely simple modular abelian varieties over ${\mathbf {Q}}$. We apply our methods to all 28 Atkin–Lehner quotients of $X_0(N)$ of genus $2$, all 97 genus $2$ curves from the LMFDB whose Jacobian is of this type and six further curves originally found by Wang. We are able to verify the strong BSD Conjecture unconditionally and exactly in all these cases; this is the first time that strong BSD has been confirmed for absolutely simple abelian varieties of dimension at least $2$. We also give an example where we verify that the order of the Tate–Shafarevich group is $7^2$ and agrees with the order predicted by the BSD Conjecture.

In our previous paper, we gave a presentation of the torus-equivariant quantum K-theory ring $QK_{H}(Fl_{n+1})$ of the (full) flag manifold $Fl_{n+1}$ of type $A_{n}$ as a quotient of a polynomial ring by an explicit ideal. In this paper, we prove that quantum double Grothendieck polynomials, introduced by Lenart-Maeno, represent the corresponding (opposite) Schubert classes in the quantum K-theory ring $QK_{H}(Fl_{n+1})$ under this presentation. The main ingredient in our proof is an explicit formula expressing the semi-infinite Schubert class associated to the longest element of the finite Weyl group, which is proved by making use of the general Chevalley formula for the torus-equivariant K-group of the semi-infinite flag manifold associated to $SL_{n+1}(\mathbb {C})$.

Let X be a smooth projective variety of dimension $n\geq 2$ and $G\cong \mathbf {Z}^{n-1}$ a free abelian group of automorphisms of X over $\overline {\mathbf {Q}}$. Suppose that G is of positive entropy. We construct a canonical height function $\widehat {h}_G$ associated with G, corresponding to a nef and big $\mathbf {R}$-divisor, satisfying the Northcott property. By characterizing the zero locus of $\widehat {h}_G$, we prove the Kawaguchi–Silverman conjecture for each element of G. As for other applications, we determine the height counting function for non-periodic points and show that X satisfies potential density.

This paper considers the large N limit of Wilson loops for the two-dimensional Euclidean Yang–Mills measure on all orientable compact surfaces of genus larger or equal to $1$, with a structure group given by a classical compact matrix Lie group. Our main theorem shows the convergence of all Wilson loops in probability, given that it holds true on a restricted class of loops, obtained as a modification of geodesic paths. Combined with the result of [20], a corollary is the convergence of all Wilson loops on the torus. Unlike the sphere case, we show that the limiting object is remarkably expressed thanks to the master field on the plane defined in [3, 39], and we conjecture that this phenomenon is also valid for all surfaces of higher genus. We prove that this conjecture holds true whenever it does for the restricted class of loops of the main theorem. Our result on the torus justifies the introduction of an interpolation between free and classical convolution of probability measures, defined with the free unitary Brownian motion but differing from t-freeness of [5] that was defined in terms of the liberation process of Voiculescu [67]. In contrast to [20], our main tool is a fine use of Makeenko–Migdal equations, proving uniqueness of their solution under suitable assumptions, and generalising the arguments of [21, 33].

We consider log Calabi-Yau surfaces $(Y, D)$ with singular boundary. In each deformation type, there is a distinguished surface $(Y_e,D_e)$ such that the mixed Hodge structure on $H_2(Y \setminus D)$ is split. We prove that (1) the action of the automorphism group of $(Y_e,D_e)$ on its nef effective cone admits a rational polyhedral fundamental domain; and (2) the action of the monodromy group on the nef effective cone of a very general surface in the deformation type admits a rational polyhedral fundamental domain. These statements can be viewed as versions of the Morrison cone conjecture for log Calabi–Yau surfaces. In addition, if the number of components of D is no greater than six, we show that the nef cone of $Y_e$ is rational polyhedral and describe it explicitly. This provides infinite series of new examples of Mori Dream Spaces.

Let G be an almost simple group with socle $G_0$. In this paper we prove that whenever $G/G_0$ is abelian, then there exists an abelian subgroup A of G such that $G=AG_0$. We propose a few applications of this structural property of almost simple groups.

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.