Tao (2018) showed that in order to prove the Lonely Runner Conjecture (LRC) up to $n+1$ runners it suffices to consider positive integer velocities in the order of $n^{O(n^2)}$. Using the zonotopal reinterpretation of the conjecture due to the first and third authors (2017) we here drastically improve this result, showing that velocities up to $\binom {n+1}{2}^{n-1} \le n^{2n}$ are enough.

We prove the same finite-checking result, with the same bound, for the more general shifted Lonely Runner Conjecture (sLRC), except in this case our result depends on the solution of a question, that we dub the Lonely Vector Problem (LVP), about sumsets of n rational vectors in dimension two. We also prove the same finite-checking bound for a further generalization of sLRC that concerns cosimple zonotopes with n generators, a class of lattice zonotopes that we introduce.

In the last sections we look at dimensions two and three. In dimension two we prove our generalized version of sLRC (hence we reprove the sLRC for four runners), and in dimension three we show that to prove sLRC for five runners it suffices to look at velocities adding up to $195$.

The smallest eigenvalue of a graph is the smallest eigenvalue of its adjacency matrix. We show that the family of graphs with smallest eigenvalue at least $-\lambda $ can be defined by a finite set of forbidden induced subgraphs if and only if $\lambda < \lambda ^*$, where $\lambda ^* = \rho ^{1/2} + \rho ^{-1/2} \approx 2.01980$, and $\rho $ is the unique real root of $x^3 = x + 1$. This resolves a question raised by Bussemaker and Neumaier. As a byproduct, we find all the limit points of smallest eigenvalues of graphs, supplementing Hoffman’s work on those limit points in $[-2, \infty )$.

We also prove that the same conclusion about forbidden subgraph characterization holds for signed graphs. Our impetus for the study of signed graphs is to determine the maximum cardinality of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Denote by $N_{\alpha , \beta }(d)$ the maximum number of unit vectors in $\mathbb {R}^d$ where all pairwise inner products lie in $\{\alpha , \beta \}$ with $-1 \le \beta < 0 \le \alpha < 1$. Very recently Jiang, Tidor, Yao, Zhang, and Zhao determined the limit of $N_{\alpha , \beta }(d)/d$ as $d\to \infty $ when $\alpha + 2\beta < 0$ or $(1-\alpha )/(\alpha -\beta ) \in \{1,\sqrt 2,\sqrt 3\}$, and they proposed a conjecture on the limit in terms of eigenvalue multiplicities of signed graphs. We establish their conjecture whenever $(1-\alpha )/(\alpha - \beta ) < \lambda ^*$.

We provide a second-order energy expansion for a gas of N bosonic particles with three-body interactions in the Gross-Pitaevskii regime. We especially confirm a conjecture by Nam, Ricaud, and Triay in [25], where they predict the subleading term in the asymptotic expansion of the ground state energy to be of the order $\sqrt {N}$. In addition, we show that low-energy states satisfy Bose-Einstein condensation with a rate of the order $N^{-\frac {3}{4}}$.

Schubert Vanishing is a problem of deciding whether Schubert coefficients are zero. Until this work it was open whether this problem is in the polynomial hierarchy ${{\mathsf {PH}}}$. We prove this problem is in ${{\mathsf {AM}}} \cap {{\mathsf {coAM}}}$ assuming the Generalized Riemann Hypothesis ($\mathrm{GRH}$), that is, relatively low in ${{\mathsf {PH}}}$. Our approach uses Purbhoo’s criterion [57] to construct explicit polynomial systems for the problem. The result follows from a reduction to Parametric Hilbert’s Nullstellensatz, recently analyzed in [2]. We extend our results to all classical types.

We gather evidence on a new local-global conjecture of Moretó and Rizo on values of irreducible characters of finite groups. For this we study subnormalisers and picky elements in finite groups of Lie type and determine them in many cases, for unipotent elements as well as for semisimple elements of prime power order. We also discuss subnormalisers of unipotent and semisimple elements in connected as well as in disconnected reductive linear algebraic groups.

Rigid meromorphic cocycles are defined in the setting of orthogonal groups of arbitrary real signature and constructed in some instances via a p-adic analogue of Borcherds’ singular theta lift. The values of rigid meromorphic cocycles at special points of an associated p-adic symmetric space are then conjectured to belong to class fields of suitable global reflex fields, suggesting an eventual framework for explicit class field theory beyond the setting of CM fields explored in the treatise of Shimura and Taniyama.

Motivated by classical Alexander invariants of affine hypersurface complements, we endow certain finite dimensional quotients of the homology of abelian covers of complex algebraic varieties with a canonical and functorial mixed Hodge structure (MHS). More precisely, we focus on covers which arise algebraically in the following way: if U is a smooth connected complex algebraic variety and G is a complex semiabelian variety, the pullback of the exponential map by an algebraic morphism $f:U\to G$ yields a covering space $\pi :U^f\to U$ whose group of deck transformations is $\pi _1(G)$. The new MHSs are compatible with Deligne’s MHS on the homology of U through the covering map $\pi $ and satisfy a direct sum decomposition as MHSs into generalized eigenspaces by the action of deck transformations. This provides a vast generalization of the previous results regarding univariable Alexander modules by Geske, Maxim, Wang and the authors in [16, 17]. Lastly, we reduce the problem of whether the first Betti number of the Milnor fiber of a central hyperplane arrangement complement is combinatorial to a question about the Hodge filtration of certain MHSs defined in this paper, providing evidence that the new structures contain interesting information.

For a klt singularity, C. Xu and Z. Zhuang [33] proved the associated graded algebra of a minimizing valuation of the normalized volume function is finitely generated, finishing the proof of the stable degeneration conjecture proposed by C. Li and C. Xu. We prove a family version of the stable degeneration: for a locally stable family of klt singularities with constant local volume, the ideal sequences of the minimizing valuations for the normalized volume function form families of ideals with flat cosupport, which induce a degeneration to a locally stable family of K-semistable log Fano cone singularities. In the proof, we give a method to construct families of Kollár models, which are a crucial tool introduced by Xu–Zhuang to prove finite generation for valuations of higher rational rank.

The aim of this paper is to illustrate both analytically and numerically the interplay of two fundamentally distinct non-Hermitian mechanisms in the deep subwavelength regime. Considering a parity-time symmetric system of one-dimensional subwavelength resonators equipped with two kinds of non-Hermiticity – an imaginary gauge potential and on-site gain and loss – we prove that all but two eigenmodes of the system pass through exceptional points and decouple. By tuning the gain-to-loss ratio, the system changes from a phase with unbroken parity-time symmetry to a phase with broken parity-time symmetry. At the macroscopic level, this is observed as a transition from symmetrical eigenmodes to condensated eigenmodes at one edge of the structure. Mathematically, it arises from a topological state change. The results of this paper open the door to the justification of a variety of phenomena arising from the interplay between non-Hermitian reciprocal and nonreciprocal mechanisms not only in subwavelength wave physics but also in quantum mechanics, where the tight-binding model coupled with the nearest neighbour approximation can be analysed with the same tools as those developed here.

We establish several new results on the existence of probability distributions on the independent sets in triangle-free graphs where each vertex is present with a given probability. This setting was introduced and studied under the name of “fractional coloring with local demands” by Kelly and Postle and is closely related to the well-studied fractional chromatic number of graphs.

Our first main result strengthens Shearer’s classic bound on independence number, proving that for every triangle-free graph G there exists a distribution over independent sets where each vertex v appears with probability $(1-o(1))\frac {\ln d_G(v)}{d_G(v)}$, resolving a conjecture by Kelly and Postle. This in turn implies new upper bounds on the fractional chromatic number of triangle-free graphs with a prescribed number of vertices or edges, which resolves a conjecture by Cames van Batenburg et al. and addresses yet another one by the same authors.

Our second main result resolves Harris’ conjecture on triangle-free d-degenerate graphs, showing that such graphs have fractional chromatic number at most $(4+o(1))\frac {d}{\ln d}$. As previously observed by various authors, this in turn has several interesting consequences. A notable example is that every triangle-free graph with minimum degree d contains a bipartite induced subgraph of minimum degree $\Omega (\log d)$. This settles a conjecture by Esperet, Kang, and Thomassé.

The main technique employed to obtain our results is the analysis of carefully designed random processes on vertex-weighted triangle-free graphs that preserve weights in expectation. The analysis of these processes yields weighted generalizations of the aforementioned results that may be of independent interest.

We consider a new family $(\mathcal {T}_n)_{n\geq 1}$ of aperiodic sets of Wang tiles and we describe the dynamical properties of the set $\Omega _n$ of valid configurations $\mathbb {Z}^2\to \mathcal {T}_n$. The tiles can be defined as the different instances of a square-shaped computer chip whose inputs and outputs are 3-dimensional integer vectors. The family include the Ammann aperiodic set of 16 Wang tiles and gathers the hallmarks of other small aperiodic sets of Wang tiles. Notably, the tiles satisfy additive versions of equations verified by the Kari–Culik aperiodic sets of 14 and 13 Wang tiles. Also configurations in $\Omega _n$ are the codings of a $\mathbb {Z}^2$-action on a 2-dimensional torus like the Jeandel–Rao aperiodic set of 11 Wang tiles. The family broadens the relation between quadratic integers and aperiodic tilings beyond the omnipresent golden ratio as the dynamics of $\Omega _n$ involves the positive root $\beta $ of the polynomial $x^2-nx-1$, also known as the n-th metallic mean. We show the existence of an almost one-to-one factor map $\Omega _n\to \mathbb {T}^2$ which commutes the shift action on $\Omega _n$ with horizontal and vertical translations by $\beta $ on $\mathbb {T}^2$. The factor map can be explicitly defined by the average of the top labels from the same row of tiles as in Kari and Culik examples. The proofs are based on the minimality of $\Omega _n$ (proved in a previous article) and a polygonal partition of $\mathbb {T}^2$ which we show is a Markov partition for the toral $\mathbb {Z}^2$-action. The partition and the sets of Wang tiles are symmetric which makes them, like Penrose tilings, worthy of investigation.

For a family $\mathcal {F}$ of graphs, let ${\mathrm {ex}}(n,\mathcal {F})$ denote the maximum number of edges in an n-vertex graph which contains none of the members of $\mathcal {F}$ as a subgraph. A longstanding problem in extremal graph theory asks to determine the function ${\mathrm {ex}}(n,\{C_3,C_4\})$. Here we give a new construction for dense graphs of girth at least five with arbitrary number of vertices, providing the first improvement on the lower bound of ${\mathrm {ex}}(n,\{C_3,C_4\})$ since 1976. As a corollary, this yields a negative answer to a problem in Chung-Graham [3].

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.

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 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.

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

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 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.