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Looking for a geometric framework to study plectic Heegner points, we define a collection of abelian varieties – called plectic Jacobians—using the middle-degree cohomology of quaternionic Shimura varieties (QSVs). The construction is inspired by the definition of Griffiths’ intermediate Jacobians and rests on Nekovář–Scholl’s notion of plectic Hodge structures. Moreover, we construct exotic Abel–Jacobi maps sending certain zero cycles on QSVs to plectic Jacobians.

We establish an implication between two long-standing open problems in complex dynamics. The roots of the $n$th Gleason polynomial $G_{n}\in{\mathbb{Q}}[c]$ comprise the $0$-dimensional moduli space of quadratic polynomials with an $n$-periodic critical point. $\operatorname{Per}_{n}(0)$ is the $1$-dimensional moduli space of quadratic rational maps on ${\mathbb{P}}^{1}$ with an $n$-periodic critical point. We show that if $G_{n}$ is irreducible over ${\mathbb{Q}}$, then $\operatorname{Per}_{n}(0)$ is irreducible over ${\mathbb{C}}$. To do this, we exhibit a ${\mathbb{Q}}$-rational smooth point on a projective completion of $\operatorname{Per}_{n}(0)$, using the admissible covers completion of a Hurwitz space. In contrast, the Uniform Boundedness Conjecture in arithmetic dynamics would imply that for sufficiently large $n$, $\operatorname{Per}_{n}(0)$ itself has no ${\mathbb{Q}}$-rational points.

In this paper, we determine the number of general points through which a Brill–Noether curve of fixed degree and genus in any projective space can be passed.

Building on work of Gerstenhaber, we show that the space of integrable derivations on an Artin algebra forms a Lie algebra, and a restricted Lie algebra if contains a field of characteristic . We deduce that the space of integrable classes in forms a (restricted) Lie algebra that is invariant under derived equivalences, and under stable equivalences of Morita type between self‐injective algebras. We also provide negative answers to questions about integrable derivations posed by Linckelmann and by Farkas, Geiss and Marcos. Along the way, we compute the first Hochschild cohomology of the group algebra of any symmetric group.

We find the scaling limits of a general class of boundary-to-boundary connection probabilities and multiple interfaces in the critical planar FK-Ising model, thus verifying predictions from the physics literature. We also discuss conjectural formulas using Coulomb gas integrals for the corresponding quantities in general critical planar random-cluster models with cluster-weight$${q \in [1,4)}$$$q\in [1,4)$. Thus far, proofs for convergence, including ours, rely on discrete complex analysis techniques and are beyond reach for other values ofqthan the FK-Ising model ($$q=2$$$q=2$). Given the convergence of interfaces, the conjectural formulas for other values ofqcould be verified similarly with relatively minor technical work. The limit interfaces are variants of$$\text {SLE}_\kappa $$${\text{SLE}}_{\kappa }$curves (with$$\kappa = 16/3$$$\kappa =16/3$for$$q=2$$$q=2$). Their partition functions, that give the connection probabilities, also satisfy properties predicted for correlation functions in conformal field theory (CFT), expected to describe scaling limits of critical random-cluster models. We verify these properties for all$$q \in [1,4)$$$q\in [1,4)$, thus providing further evidence of the expected CFT description of these models.

A simple polytopePis calledB-rigidif its combinatorial type is determined by the cohomology ring of the moment-angle manifold$\mathcal {Z}_P$overP. We show that any tensor product decomposition of this cohomology ring is geometrically realized by a product decomposition of the moment-angle manifold up to equivariant diffeomorphism. As an application, we find thatB-rigid polytopes are closed under products, generalizing some recent results in the toric topology literature. Algebraically, our proof establishes that the Koszul homology of a Gorenstein Stanley–Reisner ring admits a nontrivial tensor product decomposition if and only if the underlying simplicial complex decomposes as a join of full subcomplexes.

We consider manifold-knot pairs$(Y,K)$, whereYis a homology 3-sphere that bounds a homology 4-ball. We show that the minimum genus of a PL surface$\Sigma $in a homology ballX, such that$\partial (X, \Sigma ) = (Y, K)$can be arbitrarily large. Equivalently, the minimum genus of a surface cobordism in a homology cobordism from$(Y, K)$to any knot in$S^3$can be arbitrarily large. The proof relies on Heegaard Floer homology.

Assume that a ground‐based vehicle moves in a room with walls or other planar surfaces. Can the vehicle reconstruct the positions of the walls from the echoes of a single sound event? We assume that the vehicle carries some microphones and that a loudspeaker is either also mounted on the vehicle or placed at a fixed location in the room. We prove that the reconstruction is almost always possible if (1) no echoes are received from floors, ceilings, or sloping walls and the vehicle carries at least three noncollinear microphones, or if (2) walls of any inclination may occur, the loudspeaker is fixed in the room and there are four noncoplanar microphones. The difficulty lies in the echo‐matching problem: How to determine which echoes come from the same wall. We solve this by using a Cayley–Menger determinant. Our proofs use methods from computational commutative algebra.

For any subset$Z \subseteq {\mathbb {Q}}$, consider the set$S_Z$of subfields$L\subseteq {\overline {\mathbb {Q}}}$which contain a co-infinite subset$C \subseteq L$that is universally definable inLsuch that$C \cap {\mathbb {Q}}=Z$. Placing a natural topology on the set${\operatorname {Sub}({\overline {\mathbb {Q}}})}$of subfields of${\overline {\mathbb {Q}}}$, we show that ifZis not thin in${\mathbb {Q}}$, then$S_Z$is meager in${\operatorname {Sub}({\overline {\mathbb {Q}}})}$. Here,thinandmeagerboth mean “small”, in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fieldsLhave the property that the ring of algebraic integers$\mathcal {O}_L$is universally definable inL. The main tools are Hilbert’s Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every$\exists $-definable subset of an algebraic extension of${\mathbb Q}$is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.

Let$\mathrm{E}/\mathbb{Q}$\mathrm{E}/\mathbb{Q}be an elliptic curve and 𝑝 a prime of supersingular reduction for$\mathrm{E}$\mathrm{E}.Consider a quadratic extension$L/{\mathbb{Q}}_p$L/\mathbb{Q}_{p}and the corresponding anticyclotomic${\mathbb{Z}}_p$\mathbb{Z}_{p}-extension$L_{\mathrm{\infty }}/L$L_{\infty}/L.We analyze the structure of the points$\mathrm{E}(L_{\mathrm{\infty }})$\mathrm{E}(L_{\infty})and describe two global implications of our results.