Papers

We denote by $\mathcal{R}_{g;m}$ the moduli space of $m$–pointed Prym curves of genus $g$, that is, tuples $[\widetilde C / C; x_1, \dots, x_m]$ where $[C, x_1, \dots, x_m]$ is an $m$–pointed curve of genus $g$ and $\widetilde C/ C$ is an étale double cover of $C$. In this paper, we address the problem of the non–tautology of the Chow ring of $\mathcal{R}_{g;m}$. The locus which allows us to achieve earlier bounds for the non–tautology of $\mathrm{CH}^\bullet(\mathcal{R}_{g})$ compared to $\mathcal{M}_g$ is the component $\mathcal{R}\mathcal{B}_g^0$ of the locus of bi–elliptic Prym curves. This parametrises covers $[\widetilde C/ C]$ such that, if $C \rightarrow E$ is the bi–elliptic structure, the composition $\widetilde C \rightarrow E$ factors through an elliptic cover of $E$. Our main contribution is thus the non–tautology of the class $[\mathcal{R}\mathcal{B}_8^0] \in \mathrm{CH}^*(\mathcal{R}_8)$. In the course of establishing this theorem, a similar result for the compact moduli spaces $\overline{\mathcal{R}}_{g; 2m}$ for $g + m \geq 8$ is proven.

We introduce generalized Fofana spaces and we give some of their basic properties. These spaces are a kind of generalization of generalized Morrey spaces. As application, we establish the boundedness of the Hardy-Littlewood maximal operator on these new spaces.

We study several distinct but related Fourier analytic variants of the well-known Kakeya and Furstenberg set problems in the plane. For example, given $0<s,t<1$, we call a set $K \subseteq \mathbb{R}^2$ an $(s,t)$-Kakeya set if there exists a set of directions $E \subseteq S^1$ with Hausdorff dimension at least $t$ such that, for each $e \in E$, the set $K$ contains a subset of a unit line segment in direction $e$ whose Fourier dimension, viewed as a subset of $\mathbb{R}$, is at least $s$. For $\Delta(s,t)$ defined to be the infimum of the Fourier dimension among all $(s,t)$-Kakeya sets in $\mathbb{R}^2$, we prove that \[ \Delta(s,t) = \min\{s,2t\}. \] We also obtain similar results in the Furstenberg set version of the problem and in the case where the Hausdorff dimension of the collection of lines is replaced by the Fourier dimension.

We develop a relational duality for semilattices with adjunctions (SLatas) based on binary meet-relations. First, we introduce the category of MoS-spaces and establish a dual equivalence with modal semilattices. Then, by means of A-relations, we define the category RelSP and prove a dual equivalence between SLata and RelSP. To compare this framework with the multirelational semantics previously developed for SLatas, we introduce the notion of normal mS-space and show that, under this condition, the multirelational structure can be canonically recovered from a meet-relation, and conversely. As a consequence, we prove that the categories RelSP and SLataSp are isomorphic.

The study of arboreal Galois representations (that is, Galois groups arising from iteration of polynomial and rational functions) originated with work of Odoni in the 1980s. Beginning in the early 2000s it underwent a period of renewed interest, which continues to this day. Written in 2007, this survey article gives a sense of the subject from the early days of this renewal. It is presented here as a document of historical interest – precisely as originally written – and because some recent work has referenced specific pieces of it. It was written as an informal document, and not intended to be published. Much, though not all, of the content overlaps with the 2013 survey article “Galois representations from pre-image trees: an arboreal survey" of the author.

In this paper, we study the long-time behavior of solutions to the two-dimensional Navier-Stokes equations in the presence of Couette flow on the half plane with Navier-slip boundary conditions. We prove that the total vorticity will approach \begin{align*} -1+\frac{M_2(\omega_{0})}{\nu^{3/2}(1+t)^{5/2}} \bar{\Omega}\left( \frac{x}{\sqrt{\nu(1+t)^3}}, \frac{y}{\sqrt{\nu(1+t)}} \right), \end{align*} where $-1$ is the vorticity of the Couette flow and $\bar{\Omega}$ is the kernel of a Fokker-Planck type operator $\mathcal{L}=\partial_Y^2+\frac32 X\partial_X+\frac12 Y\partial_Y+\frac52-Y\partial_X$. In the proof, we introduce a new idea of studying the spectrum of such type operators with boundary.

Agustín-Aquino solved, in terms of the table of marks of $\Aff(\mathbb{Z}/2k\mathbb{Z})$, the problem of enumerating the classes of bicolour self-complementary and rigid patterns in $\mathbb{Z}/2k\mathbb{Z}$ (also known as strong dichotomy classes). In particular, the rigid pattern-inventory polynomial appeared, for odd $k$, to yield the number of strong classes with negative sign when evaluated in $-1$, and it was conjectured that this is true for $k$ a power of an odd prime. Here we prove the conjecture is true for $k$ odd in general.

Let $A$ be a simple separable exact $C^*$-algebra that has traces. We show the following existed regularity properties are equivalent: \quad(1) $l^\infty(A)/J_A$ has real rank zero, where $J_A$ is the trace kernel ideal. \quad(2) $A$ is tracially almost divisible. \quad(3) $A$ is tracially $m$-almost divisible for some $m\in\N\cup\{0\}.$ \quad(4) $A$ has tracial approximate oscillation zero. \quad(5) $A$ has Property (TM). We also show that for an algebraically simple separable stable rank one \CA\ $B$ with non-empty compact ${\rm T}(B)$ and locally finite nuclear dimension, its uniform tracial completion $(\ol B^{\rT(B)}, \rT(B))$ is hyperfinite, type ${\rm II_1},$ and isomorphic to $({\cal R}_{\rT(B)},\rT(B))$. Furthermore, $\ol{B}^{{\rm T}(B)}$ is pure, has real rank zero and stable rank one, and satisfies $\rT (\ol B^{\rT(B)} )= \rT(B).$ Consequently, every simple separable unital diagonal AH-algebra $V$ (e.g. Villadsen algebras of the first type) has the following tracial strict comparison: For every $a,b\in V_+,$ if $d_\tau(a)<d_\tau(b)$ holds for all traces $\tau\in\rT(V),$ then there is a sequence $\{r_n\}\subset V$ such that $\lim_n\|a-r_n^*br_n\|_{2,\rT(V)}=0.$

Recovering probability measures from moments is a central theme in statistics and optimization. In particular, we focus on the recovery of measures from moments and pseudo-moments, which may come from solving the moment-SOS hierarchy in one dimension. A typical strategy when recovering a measure from moments is to verify the flat-extension property, which certifies that the underlying measure is finitely atomic and ultimately leads to recovery. For many classes of measures, however, the flat extension never occurs and thus if one aims to recover the measure corresponding to the moments, assumptions need to be made. We formulate a new kind of recovery problem, where one assumes that the measure has compact support and a fulfills a mild separation criterion. The key feature of this recovery problem formulation is that it covers not only finitely atomic measures, but also measures with continuous components. We study this new problem and describe three situations in which different guarantees can be proven. These guarantees are developed by studying the spectral representation of the Gelfand-Naimark-Segal construction and its connection to orthogonal polynomials, which ultimately allows us to provide several additional insights, which apply to algorithms widely used for the recovery of atomic measures from moments. Furthermore, the statements proven lead to novel algorithms, which we benchmark, further confirming the theoretical findings.

Understanding ideal points in the character varieties of knot complements has led to a number of important invariants for 3-manifolds. Ohtsuki (1994) counted the ideal points for character varieties of 2-bridge knot complements, and he made his techniques more concrete in an ensuing paper (1996). Drawing on these ideas, for all 2-bridge knots $K$, we explicitly determine the structure of a Serre tree for each essential surface in the knot complement directly from the knot diagram. Using these trees, we derive a formula for the number of ideal points associated to each incompressible surface.

We prove exponential decay of pair correlations for 1D stationary point processes when spacings satisfy a Markov condition, geometric ergodicity, and a condition on exponential moments. The conditions are phrased for stationary sequences of spacings (intervals between consecutive points) whose law comes from the Palm distribution of the point process. The key technical ingredient is a Markov renewal theorem with exponential convergence rate. The proofs combine classical regeneration techniques with the notion of geometric ergodicity for Markov chains with general state space. We apply the result to two models from statistical mechanics: (1) Gibbs point processes with a hard-core, finite-range pair potentials and (2) a harmonic chain of atoms, related to an autoregressive Gaussian process.

A key property of the Delaunay filtration is that it is topologically (i.e., weakly) equivalent to the offset (union-of-balls) filtration. Recently, this filtration has been extended to point clouds equipped with an $\mathbb{R}$-valued function, yielding a computable 2-parameter filtration that satisfies an analogous weak equivalence. Motivated in part by the study of time-varying data, we introduce a 3-parameter extension of the Delaunay filtration for point clouds equipped with an $\mathbb{R}^2$-valued function, also satisfying an analogous weak equivalence. For a point cloud $X \subset \mathbb{R}^d$, our trifiltration has size $O\bigl(|X|^{\lceil(d+1)/2\rceil+1}\bigr)$. We present an algorithm that computes this trifiltration in time $O\bigl(|X|^{\lceil d/2\rceil+2}\bigr)$, together with an implementation. Our experiments demonstrate that implementation can handle thousands of points in $\mathbb{R}^3$, with memory growth that is nearly linear.

Dissipative quantum chaos is an emerging theory that is expected to extend the ideas, concepts, and methodology of conventional Hamiltonian quantum chaos from coherent evolution to open quantum dynamics. The new theory should provide a set of tools to distinguish chaotic open quantum systems from integrable ones, as well as quantitative measures of their chaoticity (or, conversely, integrability). The foundations of this theory were laid in the late 1980s, and from the very start it was clear that, like its Hamiltonian predecessor, it had to be based on the spectral properties of the operators governing open quantum evolution. After these first steps, the field remained relatively quiet for many years and it is only over the last decade that the development of dissipative quantum chaos has received a strong boost, as confirmed by a large number of publications on this topic and, very recently, the first experiments performed to test its theoretical predictions. In this chapter, we review these recent developments and outline the basic foundations of dissipative quantum chaos.

We prove a variant of Rauch's hot spots conjecture for hyperbolic planar domains with small Neumann or mixed Dirichlet-Neumann eigenvalues. We conclude, for instance, that on bounded convex domains in the hyperbolic plane with sufficiently large area, second Neumann Laplace eigenfunctions have no interior critical points.

Optimization is widely used to determine the physical and financial exchange of wholesale electricity in organized markets. Guarantees of solution optimality and feasibility rest largely on convexity, which is not in general a characteristic of the governing equations for power grid and gas pipeline networks. Policy decisions that base the scheduling and locational pricing of electricity transactions on optimization rely on the guarantee of revenue adequacy, which ensures that the market administrator will collect enough payments in congestion rents to settle financial transmission rights. Developing a similar mechanism for locational trade valuation of natural gas also requires assurance that pricing outcomes are revenue adequate, and also cover the costs of gas compressor operation. However, it has been shown that the AC power flow equations are in general non-convex and hence conditions for guaranteeing revenue adequacy in optimal power flow solutions are challenging to generalize. In this study, we develop a general formal mathematical setting for nonlinear physical network flows and examine the conditions for revenue adequacy. The result is verified for DC and AC power flow as well as steady-state gas flow in a pipeline network.

In 1982, S.-T. Yau conjectured that there exist four distinct embedded minimal two-spheres in any manifold diffeomorphic to $S^3$. Wang-Zhou confirmed this conjecture for Riemannian three-spheres when the metric is bumpy or has positive Ricci curvature. We prove the following quantitative version of their theorem. Suppose that $(S^3,g)$ has positive Ricci curvature and scalar curvature $R_g\ge \Lambda_0>0$. Then there exist four distinct embedded minimal two-spheres $\Sigma_1,\ldots,\Sigma_4\subset (S^3,g)$ such that $\operatorname{area}_{g}(\Sigma_i)\le 12\pi(i+1)/\Lambda_0$ for every $i=1,\ldots,4$. We apply this result to a problem posed by S.-T. Yau in 1987 on whether the planar two-spheres are the only minimal spheres in ellipsoids centered at the origin in $\mathbb R^4$. Haslhofer-Ketover proved that ellipsoids with one sufficiently large semi-axis contain at least one non-planar embedded minimal two-sphere. We prove that such ellipsoids contain at least three non-planar embedded minimal two-spheres.

We study algebraic locality principles on a 2+1D closed lattice in the presence of a Gauss law for a non-invertible symmetry. Prior work in arXiv:2509.03589 showed that when enforcing the Gauss law of an invertible symmetry, the principle of "Haag duality" is preserved exactly, and "disjoint additivity" is preserved after appropriate treatment of discreteness artifacts. Here we show that for a large class of non-invertible on-site symmetries, Haag duality is preserved exactly only for sufficiently nice, "cuspless" regions. For cusped regions, we instead have a weak form of Haag duality that requires adding a collar. Our results apply to double models with a purely magnetic constraint, and to the more general framework of constraints induced by the on-site action of a Hopf algebra. In particular, we treat a class of extended string-net models explicitly. We also demonstrate disjoint additivity for double models based on a group, and a weakened form of disjoint additivity in the setting of a general Hopf algebra.

Let $\theta$ be a non-constant inner function and let $\phi=\overline{u}v$, where $u$ and $v$ are inner functions such that $v$ divides $\theta$. In this paper we characterize the partially isometric truncated Toeplitz operators $A_{\phi}$ and dual truncated Toeplitz operators $D_{\phi}$ with symbols of the form $\phi=\overline{u}v$. Along with that, we obtain a few more characterization results, including the space of extremal vectors for non-zero partially isometric truncated and dual truncated Toeplitz operators.

Tokens are becoming the basic units through which foundation models represent and process information for understanding and inference. However, traditional wireless communication, centered on bit-level fidelity, faces a mismatch between what is transmitted reliably and what downstream models actually consume. This mismatch calls for a communication design that directly accounts for token-level task relevance and downstream model requirements, rather than treating all transmitted bits as equally important. In this paper, we propose TONIC, a token-centric semantic communication framework for task-oriented wireless systems. The transmitter converts each source sample into a sequence of tokens, estimates token-level task relevance, and allocates protection through utility-aware unequal error protection under a fixed channel-use budget. At the receiver, token-level confidence is used to gate unreliable decisions, turning harmful substitutions into recoverable erasures before a Transformer-based completion model restores the masked tokens for final task inference. Our framework combines transmitter-side semantic-aware protection with receiver-side confidence-aware gating in a modular and interpretable architecture, rather than relying solely on fully black-box end-to-end learning. We further establish a utility-aware Bayes-risk interpretation for the receiver-side gating rule and study its interaction with unequal protection and completion. Experimental results on image classification show that TONIC consistently outperforms separation-based schemes, the pixel-domain DeepJSCC baseline, and token-domain baselines under matched communication budgets over AWGN, Rayleigh, and Rician channels.

Inspired by the well-known result stating that if any iterate of a mapping is a Banach contraction on a complete metric space, then the mapping itself possesses a unique fixed point, we investigate that claim for a Chatterjea contraction but by retaining the left-hand side of the inequality as per the mapping itself. With an additional assumption of k- continuity, the existence and uniqueness of a fixed point is obtained for a new class of contractions, m-Chatterjea contraction, on a complete metric space. Several examples are given in order to substantiate many theoretical claims such as discontinuity at the unique limit point of the iterative sequence, as well as examples demonstrating that this new class strictly contains the class of Chatterjea mappings.

The Kučera–Gács theorem is a fundamental result in algorithmic randomness. It states that every infinite sequence $X$ is Turing reducible to a Martin-Löf random $R$. This paper studies resource-bounded analogues of the Kučera-Gács Theorem, at the resource bounds of polynomial-time and finite-state computation. We prove a {quasi-polynomial-time}{ Kučera-Gács Theorem}, showing that every infinite sequence $X$ is quasi-polynomial-time reducible to a polynomial-time random sequence $R$. We also show that for any $X$, the oracle use of $R$ is $n+o(n)$ bits for obtaining the first $n$ bits of $X$. We then study the relationship between compressibility and Turing reductions, in the polynomial-time setting. We establish that $\rho^-_{\mathsf{poly}}(X) = K_{poly}(X)$, demonstrating that the lower polynomial-time Turing decompression ratio is precisely characterized by the polynomial-time Kolmogorov complexity rate. We note that this characterization fails for the polynomial-time dimension if one-way functions exist, resolving an open problem from Doty's work. We use these results to strengthen the {quasi-polynomial-time}{ Kučera-Gács Theorem}. We show that every infinite sequence $X$ is quasi-polynomial-time reducible to a {polynomial-time random} sequence $R$, where the lower oracle use rate of the reduction is less than ${K}_{poly}(X)$. We also show that any sequence extracted from the (even larger) set of normal sequences by a finite-state reduction must have a convergent asymptotic frequency for its symbols. Since sequences lacking this invariant property exist, they cannot be finite-state reduced from any normal sequence. Hence we show that the Kučera-Gács theorem fails for finite-state reductions.

A metric probability space $M$ admits thresholds if the random geometric graph on $M$ has a threshold for every monotone graph property. We connect the existence of thresholds to the uniform expansion of $M$ and prove that all standard tori, spheres, and cubes admit thresholds.

We establish omega results for the divisor and circle problems that are conjecturally sharp, while also determining the sign of the large values obtained. This improves on the work of Soundararajan and on the subsequent independent refinements of Sourmelidis and Mahatab, and gives the first improvement on Hafner's 1981 $\Omega_+$ result for the divisor problem and his $\Omega_-$ result for the circle problem. The main new ingredient is a resonance method which works directly with the phase appearing in the Voronoï summation formula. This is achieved by replacing the usual positive kernels by a one-sided sectorial kernel, namely the density of a Gamma distribution, whose Fourier transform lies in a suitable sector of the complex plane.

Recently, the authors introduced the Gamified Katětov order on filters over $\omega$. This was shown to be strictly coarser than the classical Katětov order, and in fact collapses all MAD families to a single equivalence class. In the opposite direction, the present paper shows that the Gamified Katětov order also embeds $\mathcal{P}(\omega)/\mathrm{Fin}$, and thus contains an antichain of size continuum. The analysis brings into focus some interesting connections with Ramsey theory. As part of a broader programme investigating the interplay between combinatorial and computable complexity, we then apply our construction to produce a large new family of non-modest degrees in the extended Weihrauch hierarchy, which arise from associated effective subtoposes.

Let $mH$ be the graph formed by $m$ vertex-disjoint copies of a graph $H$. Let $G \to (H)_r$ denote that, in any $r$-colouring of the edges of $G$, there exists a monochromatic copy of $H$. In 1975, Burr, Erdős, and Spencer showed that if $H$ is a graph on $k$ vertices whose independence number is $\alpha$, then $K_n \to (mH)_2$, where $m\sim n/(2k-\alpha)$, and that the $1/(2k-\alpha)$ factor is best possible. In the 1990s, Rödl and Ruciński proved that, for all but a few graphs~$H$, the threshold for the property $\mathbb{G}(n,p) \to (H)_r$ is $n^{-1/m_2(H)}$. In this paper, generalizing the result of Burr, Erdős, and Spencer, we prove that $n^{-1/\max\{m_2(H),1\}}$ is the threshold for the property $\mathbb{G}(n,p) \to (mH)_2$, where $m\sim n/(2k-\alpha)$. This threshold matches the one found by Rödl and Ruciński for most graphs $H$, extending their result in the case $r=2$.

We study a new class of preferential attachment trees with self-reinforcement. At each time, each vertex is assigned a weight equal to the cumulative sum over past times of an affine function of its degree. A new vertex attaches itself via a single edge to an already present vertex with a probability proportional to the current weight of that vertex. This “integrated popularity” rule builds long memory directly into the attachment mechanism, thereby destroying the Markov and partial-exchangeability features that underlie the classical analysis of affine preferential attachment models. More broadly, the model connects to applied-probability work on long-memory self-interacting processes (such as the elephant random walk), emphasizing how non-Markovian reinforcement reshapes asymptotic behaviour. Despite this loss of structure, we identify an explicit exponent $\phi=\phi(\delta)$ governing both local and global growth: typical degrees at time $n$ scale as $n^{1/\phi}$, and the empirical degree distribution converges to a power-law with a tail exponent $\phi+1$. We further prove Benjamini–Schramm local convergence to an infinite random rooted tree characterized via an embedded continuous-time branching process. The limiting tree is a sin-tree, and is not the Pólya-type limiting tree arising in the non-reinforced setting. Our results provide a tractable probabilistic description of a natural “memoryful” network-growth mechanism, and quantify precisely how reinforcement renormalizes the classical preferential-attachment exponents.

Holland-Simpson thin-wire finite-difference time-domain (FDTD) simulations of obliquely oriented closed-loop antennas exhibit persistent low-frequency parasitic currents because the current-deposition operator fails to conserve charge. This deposition operator, together with an interpolation operator that samples the tangential electric field along the wire, can be realized as regularizations of distributions: the wire current is deposited as a source term by integrating it against a regularized delta function along the wire, and the electric field is sampled back to the wire by integrating it against the same regularized delta function. We show that charge conservation requires the deposited current to be discretely divergence-free when the wire carries a constant current, and we introduce a family of composite B-spline regularizations that satisfy this condition to machine precision. Exact evaluation of the coupling line integrals is possible because the B-spline kernels are piecewise polynomial with a priori-known breakpoints, allowing composite Gauss-Legendre quadrature with subinterval breakpoints at every grid-plane crossing. Taking the interpolation operator as the discrete adjoint of the deposition operator preserves skew-symmetry and ensures that a discretely irrotational electric field drives no net electromotive force around a closed loop. Numerical experiments on a center-fed dipole and on circular and square loop antennas show that the proposed regularizations yield orientation-independent impedance values consistent with known characteristics, whereas a simple trilinear regularization produces unphysical parasitic low-frequency currents in closed loops.

The paper is concerned with the error analysis of a numerical scheme for the approximation of parametric mean curvature flow. The scheme we study is based on a reparametrization using the DeTurck trick and was proposed by Elliott and Fritz in [15]. In the semidiscrete case, for a spatial discretization by finite elements of order $k \geq 2$ we prove an optimal $H^1$-error estimate for the position vector. We present numerical experiments that confirm this error bound and demonstrate that the scheme has good properties with respect to the distribution of mesh points as already observed in [15].

We present a "black box" proof of mean-field near-critical behaviour for a family of functions on $\mathbb Z^d$ (${d>2}$) satisfying a short list of assumptions. The functions represent two-point functions of a lattice statistical mechanical model in the subcritical or critical regimes, and are proved to have decay of the form $|x|^{-d+2+\varepsilon}\exp[-c|x|/\xi]$, for any $\varepsilon>0$. The black box applies to several models for which commonplace methods can be used to verify the assumptions. Applications include models of self-avoiding walk, percolation, spins (Ising, XY, $|\varphi|^4$), and lattice trees, all above their upper critical dimensions. The proof is based on random walk techniques, and provides a new, unified, probabilistic, and relatively simple proof of mean-field near-critical behaviour.

Graph Neural Networks (GNNs) have become the de facto standard for learning on relational data. While traditional GNNs' message passing is well suited for vector-valued node features, there are cases in which node features are better represented by probability distributions than real vectors. Concretely, when node features are Gaussians, characterized by a mean and a covariance matrix, naively concatenating their parameters into a single vector and applying standard message passing discards the geometric and algebraic structure that governs means and covariances. We propose Gaussian Sheaf Neural Networks (GSNNs), a principled framework that incorporates these inductive biases into graph-based learning. Building on the theory of cellular sheaves, we derive a new Laplacian operator that generalizes the sheaf Laplacian to this setting and preserves its key properties. We complement our theoretical contributions with experiments on synthetic and real-world data that illustrate the practical relevance of GSNNs.