Papers

A graph is $H$-free if it has no induced subgraph isomorphic to $H$. In 2020, Chudnovsky, Goedgebeur, Schaudt, and Zhong characterized all graphs $H$ such that there are only finitely many minimal obstructions to $3$-colouring $H$-free graphs. In general, the minimal obstructions to $k$-colouring $H$-free graphs are the $(k+1)$-vertex-critical $H$-free graphs, those are, the $H$-free graphs $G$ with $\chi(G)=k+1$ but $\chi(G-v)=k$ for every vertex in $G$. In this paper we complete the characterization for all $k > 4$ by showing that there are onky finitely $k$-vertex-critical $H$-free graphs if and only if $H$ is an induced subgraph of $P_4+\ell P_1$ for some $\ell \geq 0$.

For the two-dimensional quantum Heisenberg ferromagnet with asymptotically radial power-law exchange, we prove the leading low-temperature free energy at every fixed spin, uniformly on the weak-field scale. We obtain fractional and logarithmically corrected laws associated with the dispersions predicted in [16, 17]. Quantum Monte Carlo simulations support the associated anomalous magnon dispersions for spin one-half [20]. For kernels with finite second moment, we also extend the nearest-neighbour theorem [14], generalizing Takahashi's quadratic law [21] and its weak-field dependence [11].

We show that every orientation-preserving action of a finite group on a closed nonpositively curved 3-manifold preserves a nonpositively curved metric.

We give a criterion for the box product of a Fourier–Mukai kernel to induce a fully faithful functor to the derived category of a symmetric quotient stack. This leads to many new examples of embeddings of derived categories of moduli spaces. We study in more detail the case of the moduli space of stable bundles on a curve. Concretely, we identify pieces of the left-orthogonal complement of our embedding, and extend the known semi-orthogonal decomposition of the moduli space of bundles with rank 2 and fixed determinant to the moduli space of bundles with varying determinant.

Superconductivity and intrinsic topological order can intertwine in ways that go beyond simple coexistence. Here we develop a classification framework for (2+1)D gapped superconductors with condensate charge $2ne$. By focusing on the topological properties of superconducting vortices, we precisely characterize this interplay. We obtain a classification with increasing levels of intertwinement, from the well known invertible superconductors without topological order, to untwisted and twisted SC$^\ast$, and finally to SC$^{\sharp}$. Our main finding is the family SC$^{\sharp}$, in which the elementary vortex carrying $h/(2ne)$ flux cannot be factorized into an invertible superconductor vortex dressed by bulk anyons, and every elementary vortex is necessarily non-Abelian. For such phases, we also obtain no-go results: the elementary vortex sector has quantum dimension at least $\sqrt{6}$ and cannot satisfy Fibonacci fusion rules. We develop a systematic algebraic framework to obtain all these superconducting phases through anyon condensation, allowing us to construct several infinite families of examples and make connections to anyon superconductivity. Finally, we introduce primitive SC$^\sharp$ phases for which factorization of every nontrivial vortex sector is obstructed, and argue that they cannot arise by disordering the relative phases of lower-charge superconducting condensates. Our classification scheme based on vortex properties allows us to uncover new distinctions between topologically ordered superconductors and constrain how they can be realized.

We argue that the scattering amplitude of a single minus helicity gluon with arbitrarily many positive helicity gluons is topological. The amplitude is constant in chambers of kinematic space, its resonances are located at walls and its factorization is determined by wall-crossing. Yangian invariance dictates the amplitude to be determined by a 2d null Wilson loop that can be lifted into a positive and transverse 3d link. By reinterpreting wall-crossing and the Weinberg soft theorem as skein relations and Reidemeister moves of the subknots, we construct a state-counting description consisting of adding the `corners' of their HOMFLY-PT polynomials. Beyond the classical limit, we argue that this provides a conjectured but precise connection to topological field and string theory. The new formula is rigorously derived via a Lean-formalized dependency graph (DAG) as detailed in a companion paper, laying out the precise assumptions including Reidemeister III invariance. It is further tested exhaustively, including novel `star' kinematics where the amplitude and state count are shown to become Catalan numbers, in agreement with knot theory literature.

We study magnets composed of molecules with an arbitrary number $m$ of magnetically active atoms in a chain configuration interacting with nearest-neighbor couplings. We derive the thermodynamics of a chain of spin-$j$ $SU(2)$ magnetic atoms using a mean-field approach and determine the equilibrium states and stability conditions. The system exhibits a rich phase structure with a cascade of ferromagnetic phase transitions, their number ranging from $m$, for maximal ferromagnets, to zero, for paramagnets, depending on the relative strengths of the couplings. In the presence of impurities (modified couplings) at the ends of the chain, boundary states emerge that coexist with the paramagnetic phase or the fully ferromagnetic phase. For long chains, a continuum approximation can be taken that clarifies the structure of the phases. In the continuum description, magnetized states at temperatures lower than the critical temperature at which they appear correspond to kink solitons.

Understanding why deep neural networks make a given prediction is of great importance for their safe deployment. In computer vision, saliency maps, which highlight the image region most influential for a prediction, remain a widely-used form of explanation. An empirical observation is the apparent sparsity of gradient saliency maps of adversarially-trained neural networks. In this paper, we propose a theoretical explanation of this phenomenon for two-layer ReLU networks. We build on the established equivalence of adversarial training to the minimization of the empirical risk with weight-decay penalization and an added adversarial total variation term – valid for certain loss functions. As the number of data points and neurons grows and the regularization parameters are sent to zero at appropriate rates, we prove that minimizers converge to a Bayes classifier with minimal gradient and Barron norm. Sparsity appears since for adversarial training with $\ell_\infty$-attacks the gradient norm is anisotropic and favors axis-aligned / sparse gradients. We illustrate our theoretical findings experimentally by evaluating the gradient $\ell_1$-norm and thresholded sparsity of naturally versus adversarially trained models.

In this paper we propose and study what we call the Erdős-Falconer distance problem in the integer lattice. One motivation is Magyar's discrete analog of the result of Furstenberg, Katznelson, and Weiss, who showed that all arbitrarily large distances had to appear for a set of positive upper Banach density in $\mathbb{R}^d$, $d\geq 2$. Another motivation is the Falconer distance problem, which can be viewed as a refinement of a result of Steinhaus, dropping assumptions from positive Lebesgue measure to merely asking about the dimension of sets. In this context we thus want to lower the assumption of positive upper Banach density and yet obtain statements that guarantee that we have lots of distinct distances. All of these questions are also related to the Erdős distinct distance problem. Our naming is as for the finite field variant of the problem, acknowledging that parts of the problem share similarities with the Erdős distinct distance problem while others share similarities with the Falconer distance problem. The main tool we use are $\ell^p$-improving inequalities for discrete spherical averages, which to the best of our knowledge is the first application of such bounds. One can use results on the Erdős distinct distance problem to make progress on our conjecture but surprisingly, even if using the strongest available bounds in high dimensions using the mechanism of Solymosi and Vu, then in certain regimes our methods do better. While a resolution of the Erdős distinct distance problem would imply our unpinned conjecture then no results on either the Erdős or Falconer distance problem seem to imply any results for our pinned variant. We view this problem as a model problem that might ultimately contribute back to the original pinned distance problems.

Let $S$ be a finite sum of independent Bernoulli random variables. We ask how large the mean must be before the atom at $k+r$ can strictly exceed the atom at $k$. For the threshold $T_{k,r}:=\inf\{\mathbb{E}S:\mathbb{P}(S=k+r)>\mathbb{P}(S=k)\}$, the adjacent case recovers the classical Darroch boundary $k+1/(k+2)$, while every gap $r\ge3$ has threshold $k+1$. Distance two is the unique nonadjacent exception: $T_{k,2}=k+\Delta_k$, where $\Delta_k$ and its staircase optimizer are explicit and $\Delta_k=1/2+1/\sqrt{2k}-1/(4k)+O(k^{-3/2})$. The proof combines the classical fixed-mean reduction to shifted binomial laws with an exact discrete optimization. The result also yields sharp restrictions on nonadjacent coefficient reversals in real-rooted polynomials with nonnegative coefficients.

We show that for a fixed $\mathrm{SL}_3(\mathbb Z)$ Hecke–Maass cusp form $F$, there are infinitely many primitive Dirichlet characters $\chi$ such that $L(1/2, F\times \chi)$ and $L(1/2, \chi)$ do not vanish simultaneously. More precisely, we obtain an asympotic formula with a power-saving error term for the product of these central values averaged over $\{\chi\bmod q: q\in \mathcal Q\}$ where $\mathcal Q$ is a set of squarefree integers with a small divisor of suitable size.

We study three quadratic Collatz-type maps on the non-negative integers. Each of them halves an even number, and they send an odd $n$ to $n(n-1)/2$, to $n(n+1)/2$ and to $(n^2 - 1)/4$, respectively. Along an orbit, the odd terms grow until they reach a number of the form $2^m + 1$ or $2^m - 1$, depending on the map, and an orbit with an odd term greater than $1$ is bounded exactly when this happens. The step that reaches such a number gives a solution of an exponential Diophantine equation, which becomes the equation $2^a \pm 2^b + 1 = z^2$ after completing the square. With Szalay's classification of the solutions of this equation, we classify all orbits of the maps. Every orbit either is eventually periodic or tends to infinity, and we determine exactly the initial values with bounded orbits. For the map with odd branch $n(n-1)/2$, Sedaghat stated the classification, and we give a new and complete proof. For this map, we also give an elementary criterion for unbounded orbits, based on divisibility alone. To our knowledge, the classifications for the other two maps are new.

Threshold scattering below the ground-state energy and gradient is established for the three-dimensional focusing energy-critical nonlinear Schrödinger equation. A single solution-dependent Gaussian interaction scale, mass-free angular coercivity, and a frequency-gap source estimate control the projected dynamics. These estimates yield quartic spacetime integrability and a zero-mass contradiction, without radial symmetry or an a priori $L^2$ bound.

We characterize Fourier quasicrystal measures with positive integer masses as real sections of effective Lee-Yang cycles, with weighted local intersection degrees. The section matrix has positive maximal minors, and the mass at each root is its weighted local intersection degree. For Delone support the reduced support of the cycle can be chosen strict. The support of every integer-mass Fourier quasicrystal measure is the common complex zero set of finitely many real-valued trigonometric polynomials. For unit masses the representing cycle can be chosen reduced, with local degree one at every physical root. These results answer Questions (1), (2), and (4) of Alon, Kummer, Kurasov, and Vinzant. The proof recovers a finite-dimensional torus from the spectrum, proves that the resulting analytic set is algebraic, and reconstructs its complex branches and their integer coefficients from their moments. A support-height theorem for holonomic sequences gives the required linear spectral bound. Singular Lee-Yang sections are Fourier quasicrystal measures with their natural local intersection degrees; discarding those degrees can destroy the Fourier quasicrystal property.

The Szemerédi–Trotter theorem, a fundamental theorem in incidence geometry, states that $n$ points and $m$ lines in $\mathbb{R}^2$ form at most $O(m+n+m^{2/3}n^{2/3})$ incidences. Recently, Lewko and independently Miao–Xie extended the Szemerédi–Trotter theorem to arbitrary fields with a correction term $O(\frac{mn}{\mathfrak{p}})$ if the field has positive characteristic $\mathfrak{p}$. In this paper, we provide some geometric intuition for the two proofs and compare them. Moreover, we show how both proofs can be improved using Hasse derivatives. As a consequence, we obtain a tight upper bound over any algebraically closed field. To be specific, when the field has positive characteristic $\mathfrak{p}$, $n\leq m$ and $\mathfrak{p}^{2(k-1)}\leq m<\mathfrak{p}^{2k}$ for some positive integer $k$, the number of incidences can be bounded by \[O\left(\mathfrak{p}^{k-1}n+m+m^{2/3}n^{2/3}+\frac{mn}{\mathfrak{p}^{k}}\right).\]

Paulsen and West associate perfect cuboids with triples of rational points on a congruent-number elliptic curve and assert that the corresponding curve has rank at least two. The proof of their Theorem 4.2 contains a gap: translation of a generator by a point of order two interchanges only the odd-indexed terms of two coordinate sequences, rather than the entire sequences. We exhibit two explicit counterexamples to that intermediate assertion and give a complete proof of the rank obstruction. The division-polynomial argument reduces a hypothetical rank-one configuration to one remaining parity pattern. Ingram's primitive-divisor theorem then forces an additive relation among its indices, which is incompatible with a positivity lemma on the elliptic curve. More precisely, the three points of every positive cuboid configuration generate a subgroup of rank at least two, even when the ambient curve has larger rank. The proof is unconditional.

Diffusion samplers can reduce computation by generating selected spectral coordinates and filling the remaining directions with noise. How many directions must they retain? We study this question for data with power-law covariance spectra. For Gaussian data compared to a smoothed target, we prove matching bounds on the required number of retained directions, provided that the ambient dimension is sufficiently large. The truncation error depends on the combined Wiener gains of the omitted directions, regardless of the accuracy of the sampler on the retained coordinates. Keeping only directions whose signal exceeds the output noise level can therefore leave a non-vanishing error: many individually weak directions remain significant in aggregate. Combining this characterization with a diffusion convergence bound yields sufficient sampling-step complexity under exact scores. The upper bounds also extend to estimated principal components and, componentwise, to Gaussian mixtures. The practical prescription is to select the retained subspace using an aggregate spectral-tail error budget, then to choose the diffusion noise level accordingly.

A $\lambda$-self-expander $F: M^n\to \mathbb{R}^{n+1}$ is the solution of the isoperimetric problem of weight $e^{\frac{|x|^2}{4}}$. In this paper, for sufficiently large $\lambda$, we construct an immersed non-embedded spherical $\lambda$-self-expander in $\mathbb{R}^{n+1}$ with the mean curvature $H>0$. Moreover, we obtain the explicit solution for $n=2$.

The balanced Hellinger conjecture compares the overlap of two output laws induced by a Boolean decision under binary symmetric noise. We prove that a single input coordinate minimizes this overlap among antipodal decisions, whose values reverse when all input signs are reversed, throughout the noise interval. The proof uses a normalized nuclear norm of the cross-class block of the Gram matrix of square-root channel densities. Multiplication by an antipodal decision exchanges the even and odd Fourier spaces preserved by the Gram matrix. A coordinate exchange attains the minimum cost of matching their spectra. A matrix contraction extends the sharp bound to arbitrary independent binary-input channels. Fixed coordinate flips and randomized decisions follow from the same argument and concavity. These extensions give bounds for information and testing, including linear Gaussian observations, and a weighted edge-boundary inequality.

For a positive integer $n$, we study the irreducibility of the generalised Laguerre polynomials \[ L_n^{(a)}(x) = \frac{1}{n!}\sum_{j=0}^{n} \binom{n}{j} \frac{(n+a)!}{(j+a)!}(-1)^j x^j, \] where $a \in \mathbb{Z}_{\ge 0}$. Earlier works established that these polynomials are irreducible over $\mathbb{Q}$ for $0 \le a \le 50$. In this paper, we substantially strengthen these results by establishing that $L_n^{(a)}(x)$ are irreducible for all $a \le 1000$. We derive sharper constraints on potential integer roots and make systematic use of the geometry of the associated Newton polygons. This approach allows us to extend the irreducibility range significantly while keeping explicit computations to a minimum.

The discovery that the generalized Riemann functional equation from the literature can be extended to real $b \ge 0$, by exploiting symmetries of a new formula for the Hurwitz zeta function with respect to the shift parameter $(b)$, led to the search for a similar relation for the Lerch $\Phi$ function, $\Phi(e^z,k,b)$, where a new formula is used as a starting point and the symmetries with respect to $b$ are exploited again. This search resulted in a generalization of the Riemann functional equation with three variables, whose domain is then extended to $b>1$, with the aid of a new self-contained Lerch $\Phi$ formula, making the relation hold for all real $b$. As a secondary objective, we address the problem of obtaining the analytic continuation of a finite summation that appears in a formula for the Lerch $\Phi$ function. Above all, this paper demonstrates, with clear elementary mathematics, how a functional equation can be derived.

We study the $(n-2)$-plane Kakeya problem in $\mathbb{R}^n$ ($n\geq4$) subject to convex Wolff axioms. In other words, we study finite families of $\delta$-neighborhoods of $(n-2)$-plane segments ($\delta$-flakes) in $\mathbb{R}^n$, with the property that not too many of them can be contained inside a common convex set. Our main result is that the union of $\delta$-flakes from such a set must have almost maximal volume. This can be viewed as a higher-dimensional generalization of Wang and Zahl's work for $\delta$-tubes in $\mathbb{R}^3$. The latter implies the Kakeya set conjecture in $\mathbb{R}^3$. As a consequence, we resolve Rogers' $2$-plane Kakeya problem in $\mathbb{R}^4$ subject to certain curvature conditions on two-dimensional families of direction subspaces in the Grassmannian $\mathrm{Gr}(4,2)$.

The ribosome flow model (RFM) is a nonlinear compartmental model for unidirectional transport along an mRNA chain. We study a generalized RFM in which the classical linear free-space factor is replaced by a monotone nonlinear steric-hindrance response. After recalling the global equilibrium and periodic entrainment properties needed for the analysis, we derive an exact weighted edge-dissipation identity for the gain of entrainment (GOE). This yields a global no-GOE criterion formulated through the curvature of the nonlinear hindrance law. In particular, the quadratic hindrance model admits no positive GOE in any dimension and for arbitrary periodic forcing amplitudes, with equality only when all rates share a common multiplicative modulation. In one dimension, we obtain a sharp convexity-based classification and construct positive-GOE examples when the relevant convexity fails. In higher dimensions, the second variation is governed by a symmetric Jacobi matrix whose diagonal terms encode local curvature and whose off-diagonal entries represent nearest-neighbor transport coupling. This leads to an exact Riccati criterion, a dimension-uniform condition for heterogeneous chains, and a closed-form critical dimension in the homogeneous case. We also exhibit a nonlinear hindrance for which low-dimensional chains have no positive second-order GOE, while a higher-dimensional chain admits a realizable positive GOE. These results reveal a dimension-induced gain mechanism that is absent from the classical RFM.

For every positive integer $n$, we prove that there is a constant $C_n$, depending only on $n$, such that $Vol(-K_X) \leq C_n \epsilon^{-(2^n-n-1)}$ whenever $X$ admits an $\epsilon$-lc log Fano boundary. The exponent is optimal in every dimension at least two, even for toric Fano varieties of Picard number one. In particular, the optimal exponent for fourfolds is eleven. We also prove an anticanonical interpolation theorem with optimal exponent $2^n-1$. The proof combines signed discrepancy estimates under projection, finite morphisms to projective space, and the canonical bundle formula. A reduction to a base bounded independently of $\epsilon$, followed by a volume estimate along a flag, yields the sharper volume exponent.

Heavy-tailed noise has been widely observed in modern machine learning, motivating the use of methods like gradient clipping and normalization. While these methods are well understood in centralized settings, much less is known in decentralized ones, where applying a nonlinearity to local gradients affects both optimization and consensus. Recent works on decentralized non-convex optimization have studied both clipping and normalization under heavy-tailed noise, with clipping yielding suboptimal rates and normalization needing local momentum or mini-batches to converge. This raises the question: can a baseline decentralized method using a nonlinearity achieve optimal convergence rates under heavy-tailed noise? We answer affirmatively with clipped decentralized SGD ($\mathtt{DSGD}$). For smooth non-convex costs under bounded $p$-th moment noise, $p \in (1,2]$, we show that clipped $\mathtt{DSGD}$ achieves order-optimal rates both with high probability and in expectation. Moreover, we establish a linear speed-up in the number of agents, which, to our knowledge, has not been shown for decentralized methods with clipping. The key technical ingredient is a sharp analysis of the consensus gap that exploits the structure of clipping, relegating network effects to higher-order terms. Our results highlight an important distinction between clipping and normalization in decentralized settings: while normalized $\mathtt{DSGD}$ can fail to converge, clipping retains magnitude information, enabling $\mathtt{DSGD}$ to be convergent and order-optimal. Numerical experiments validate our theory.

We provide an event-level Hawkes microfoundation for a multitype inverse-Gaussian stochastic clock. We show that the event counts and integrated intensities of nearly critical multivariate linear Hawkes processes converge jointly to a multivariate pure-jump subordinator when reproduction delays have finite mean and immigration is balanced so that rare large families remain visible. The dependence among the limiting coordinates is inherited from microscopic cross-excitation, and the limit admits a Brownian additive-field first-passage representation. We call this process the multitype inverse-Gaussian subordinator. Its diagonal case consists of independent classical inverse-Gaussian subordinators, with the univariate model as a further special case. The square-root specialization recovers the inverse-Gaussian clock obtained at the boundary of hyper-rough square-root models by Abi Jaber–Attal–Rosenbaum (Ann. Appl. Probab. 36(4): 3635–3660, 2026). Our cluster proof exposes the underlying mechanism: finite-variance near-critical branching creates rare but macroscopic families, while their internal timing disappears on the observation scale, so each family becomes a jump. We also clarify the relation with the multivariate Hawkes scaling theory of Xu (arXiv:2412.14459): its diagonal-atom condition yields the diagonal specialization, whereas the genuinely coupled limit requires a separate uniqueness argument. In addition, under a common tilted-stability condition, we replace the high-intensity assumption in that theory by a weaker accumulated-activity condition and provide the Riccati identification needed when the potential has an atom. We strengthen convergence of the count and compensator and extend the scalar conclusion to finite-variance age-dependent branching clusters.

Machine unlearning asks for a deletion algorithm whose output is close to retraining from scratch without the selected forget examples. In this work, we study forget-only unlearning, where the deletion algorithm receives only the trained model and the examples to forget, with no retained data or extra training information. We ask whether forget-only unlearning is always possible. We first show that this depends on the learning method: different datasets can produce the same trained model but require very different outputs after the same examples are removed. Using this observation, we derive lower bounds on how accurately unlearning can match retraining and instantiate them for several standard learning algorithms. We then ask what must be true when forget-only unlearning succeeds. To this end, we derive lower bounds on what an algorithm must memorize about the training data to handle arbitrary deletion requests. For simple threshold learners, the required information can be as large as the entire dataset, even though ordinary training keeps only one boundary point. Overall, our results show that information discarded during ordinary learning may be needed later for deletion, so models designed for forget-only unlearning may need to retain more information than standard training does.

A generalization of the exact overlaps conjecture predicts a formula for the dimension of self-similar measures in terms of the random walk entropy and the Lyapunov exponent. We prove the exact overlaps conjecture and its generalized version in dimension one. The proof relies on a new entropy inequality quantifying the information lost under addition of independent random variables in terms of a quantity averaging variance across scales.

We prove that a very general hypersurface of degree at least five and any positive dimension does not have a decomposition of the diagonal, over an uncountable algebraically closed field of characteristic not $2$. In particular, such hypersurfaces are not stably or retract rational. For the proof, we construct (in each dimension $\geq 8$) a certain quintic hypersurface with a nonzero unramified cohomology class, and which is rationally fibered by quadrics over a lower-dimensional projective space. From our construction, the main result follows using methods developed by Schreieder.

We prove that in the disk, sphere, whole-plane topology, spanning tree weighted planar maps converge to $\sqrt{2}$-Liouville quantum gravity disk, sphere or cone under circle packing and Riemann uniformization embedding as the number of faces of the map goes to infinity. As a byproduct, we also prove that the natural path on faces of the embedded tree-weighted planar maps converge to SLE$_8$. The proof is based on our earlier work on circle packing and Riemann uniformization embedding for random planar maps in ergodic scale-free environments, comparisons of circle packings in different domains in a companion paper, and the convergence of tree-weighted planar maps to $\sqrt{2}$-LQG by the first author.