In this paper we show that the rank of the normal function function of the genus $g$ Ceresa cycle over the moduli space of curves has the maximal rank possible, $3g-3$ , provided that $g\ge 3$. In genus 3 we show that the Green–Griffiths invariant of this normal function is a Teichmüller modular form of weight $(4,0,-1)$ and use this to show that the rank of the Ceresa normal function is exactly 1 along the hyperelliptic locus. We also introduce new techniques and tools for studying the behaviour of normal functions along and transverse to boundary divisors. These include the introduction of residual normal functions and the use of global monodromy arguments to compute them.

We construct universal G-zips on good reductions of the Pappas-Rapoport splitting models for PEL-type Shimura varieties. We study the induced Ekedahl-Oort stratification, which sheds new light on the mod p geometry of splitting models. Building on the work of Lan on arithmetic compactifications of splitting models, we further extend these constructions to smooth toroidal compactifications. Combined with the work of Goldring-Koskivirta on group theoretical Hasse invariants, we get an application to Galois representations associated to torsion classes in coherent cohomology in the ramified setting.

Let $(X,\Delta )$ be a normal pair with a projective morphism $X \to Z$ and let A be a relatively ample $\mathbb {R}$-divisor on X. We prove the termination of some minimal model program on $(X,\Delta +A)/Z$ and the abundance conjecture for its minimal model under assumptions that the non-nef locus of $K_{X}+\Delta +A$ over Z does not intersect the non-lc locus of $(X,\Delta )$ and that the restriction of $K_{X}+\Delta +A$ to the non-lc locus of $(X,\Delta )$ is semi-ample over Z.

Given a family of pairs over a smooth curve whose general fiber is a log Calabi–Yau pair in a fixed bounded family, suppose there exists a divisor on the family whose restriction on a general fiber is ample with bounded volume. We show that if the total space of the family has relatively trivial log canonical divisor and the special fiber has slc singularities, then every irreducible component of the special fiber is birationally bounded.

In this paper we solve a long-standing problem which goes back to Laurent Schwartz’s work on mean periodic functions. Namely, we completely characterize those locally compact Abelian groups having spectral synthesis. So far a characterization theorem was available for discrete Abelian groups only. Here we use a kind of localization concept for the ideals of the Fourier algebra of the underlying group. We show that localizability of ideals is equivalent to synthesizability. Based on this equivalence we show that if spectral synthesis holds on a locally compact Abelian group, then it holds on each extensions of it by a locally compact Abelian group consisting of compact elements, and also on any extension to a direct sum with a copy of the integers. Then, using Schwartz’s result and Gurevich’s counterexamples, we apply the structure theory of locally compact Abelian groups to obtain our characterization theorem.

We prove a new tableaux formula for the symmetric Macdonald polynomials $P_{\lambda }(X;q,t)$ that has considerably fewer terms and simpler weights than previously existing formulas. Our formula is a sum over certain sorted non-attacking tableaux, weighted by the queue inversion statistic $\operatorname {\mathrm {\texttt {quinv}}}$. The $\operatorname {\mathrm {\texttt {quinv}}}$ statistic originates from a formula for the modified Macdonald polynomials $\widetilde {H}_{\lambda }(X;q,t)$ due to Ayyer, Martin, and the author (2022), and is naturally related to the dynamics of the asymmetric simple exclusion process (ASEP) on a circle.

We prove our results by introducing probabilistic operators that act on non-attacking tableaux to generate a set of tableaux whose weighted sum equals $P_{\lambda }(X;q,t)$. These operators are a modification of the inversion flip operators of Loehr and Niese (2012), which yield an involution on tableaux that preserves the major index statistic but fails to preserve the non-attacking condition. Our tableaux are in bijection with the multiline queues introduced by Martin (2020), allowing us to derive an alternative multiline queue formula for $P_{\lambda }(X;q,t)$. Finally, our formula recovers an alternative formula for the Jack polynomials $J_{\lambda }(X;\alpha )$ due to Knop and Sahi (1996) using the same queue inversion statistic.

In this paper we prove a general uniqueness result in the inverse boundary value problem for the weighted p-Laplace equation in the plane, with smooth weights. We also prove a uniqueness result in dimension 3 and higher, for real analytic weights that are subject to a smallness condition on one of their directional derivatives. Both results are obtained by linearizing the equation at a solution without critical points. This unknown solution is then recovered, together with the unknown weight.

We prove motivic versions of mass formulas by Krasner, Serre, and Bhargava concerning (weighted) counts of extensions of local fields.

Normal matrices, or matrices which commute with their adjoints, are of fundamental importance in pure and applied mathematics. In this paper, we study a natural functional on the space of square complex matrices whose global minimizers are normal matrices. We show that this functional, which we refer to as the non-normal energy, has incredibly well-behaved gradient descent dynamics: despite it being nonconvex, we show that the only critical points of the non-normal energy are the normal matrices, and that its gradient descent trajectories fix matrix spectra and preserve the subset of real matrices. We also show that, even when restricted to the subset of unit Frobenius norm matrices, the gradient flow of the non-normal energy retains many of these useful properties. This is applied to prove that low-dimensional homotopy groups of spaces of unit norm normal matrices vanish; for example, we show that the space of $d \times d$ complex unit norm normal matrices is simply connected for all $d \geq 2$. Finally, we consider the related problem of balancing a weighted directed graph – that is, readjusting its edge weights so that the weighted in-degree and out-degree are the same at each node. We adapt the non-normal energy to define another natural functional whose global minima are balanced graphs and show that gradient descent of this functional always converges to a balanced graph, while preserving graph spectra and realness of the weights. Our results were inspired by concepts from symplectic geometry and Geometric Invariant Theory, but we mostly avoid invoking this machinery and our proofs are generally self-contained.

We present bounds on the maximal gain of adaptive and randomized algorithms over nonadaptive, deterministic ones for approximating linear operators on convex sets. If the sets are additionally symmetric, then our results are optimal. For nonsymmetric sets, we unify some notions of n-widths and s-numbers, and show their connection to minimal errors. We also discuss extensions to nonlinear widths and approximation based on function values, and conclude with a list of open problems.

We compare the marked length spectra of some pairs of proper and cocompact cubical actions of a nonvirtually cyclic group on $\mathrm {CAT}(0)$ cube complexes. The cubulations are required to be virtually co-special, have the same sets of convex-cocompact subgroups, and admit a contracting element. There are many groups for which these conditions are always fulfilled for any pair of cubulations, including nonelementary cubulable hyperbolic groups, many cubulable relatively hyperbolic groups, and many right-angled Artin and Coxeter groups.

For these pairs of cubulations, we study the Manhattan curve associated to their combinatorial metrics. We prove that this curve is analytic and convex, and a straight line if and only if the marked length spectra are homothetic. The same result holds if we consider invariant combinatorial metrics in which the lengths of the edges are not necessarily one. In addition, for their standard combinatorial metrics, we prove a large deviations theorem with shrinking intervals for their marked length spectra. We deduce the same result for pairs of word metrics on hyperbolic groups.

The main tool is the construction of a finite-state automaton that simultaneously encodes the marked length spectra of both cubulations in a coherent way, in analogy with results about (bi)combable functions on hyperbolic groups by Calegari-Fujiwara [14]. The existence of this automaton allows us to apply the machinery of thermodynamic formalism for suspension flows over subshifts of finite type, from which we deduce our results.

In a previous paper, the authors extended Mirzakhani’s (almost-everywhere defined) measurable conjugacy between the earthquake and horocycle flows to a measurable bijection. In this one, we analyze the continuity properties of this map and its inverse, proving that both are continuous at many points and in many directions. This lets us transfer measure convergence between the two systems, allowing us to pull back results from Teichmüller dynamics to deduce analogous statements for the earthquake flow.

We sharpen and generalize the dimension growth bounds for the number of points of bounded height lying on an irreducible algebraic variety of degree d, over any global field. In particular, we focus on the affine hypersurface situation by relaxing the condition on the top degree homogeneous part of the polynomial describing the affine hypersurface, while sharpening the dependence on the degree in the bounds compared to previous results. We formulate a conjecture about plane curves which provides a conjectural approach to the uniform degree $3$ case (the only remaining open case). For induction on dimension, we develop a higher-dimensional effective version of Hilbert’s irreducibility theorem, which is of independent interest.