Papers

We prove that a K3 surface over an algebraically closed field admits a nonzero global symmetric differential of positive degree if and only if the characteristic is $p=2$, and it is supersingular of Artin invariant $\sigma_0=1$. Vanishing was previously only known in characteristic zero by a result of Kobayashi. For the exceptional case we show that there is a unique (up to scaling) nontrivial global symmetric differential in every positive even degree. Along the way, we extend a theorem of Jang and show that a supersingular K3 surface in characteristic $p>0$ is isomorphic to a smooth quartic surface if and only if $p\geq3$ or $p=2$ and it is of Artin invariant $\sigma_0\geq3$.

Christ [Chr01] looked at a certain boundedness problem for trilinear operators. In this paper we set up a framework of mixed-norm Brascamp-Lieb inequalities and identify a new interesting regime not covered by the study of classical Brascamp-Lieb. Positive results in [Chr01] can be viewed as the first nontrivial progress in this new regime. We will also prove another mixed-norm Brascamp-Lieb inequality, showcasing how one can use a tensor product trick to slightly sharpen Christ's argument and also obtain the endpoint case. We then discuss the connections between mixed-norm Brascamp-Lieb and the Kakeya Conjectures, as well as unique new difficulties for these mixed-norm Brascamp-Lieb inequalities compared to the classical setting.

We study the problem of guessing a realization of an i.i.d. random sequence given element-wise correlated side-information. We use type-counting to provide estimates of the tail probabilities of the number of guesses for the case without side-information, which was shown earlier through large-deviation techniques. We then extend the same counting argument to the conditional setting, obtaining new explicit expressions for the corresponding guesswork exponents as divergences involving conditional tilted distributions. Finally, we provide an application of these exponents to brute-force password guessing with side-information.

Let k and l be two non-negative integers with k > l. A graph G is (k,l)-stable if alpha(G - S) >= alpha(G) - l for every subset S of V(G) with |S| = k, where alpha(G) denotes the independence number of G. Dong and Wu established that alpha(G) <= floor((n - k + 1)/2) + l for a (k, l)-stable graph G, where n is the order of G. A (k, l)-stable graph G is tight if alpha(G) = floor((n - k + 1)/2) + l. In this paper, we provide a complete characterization of tight (k, 0)-stable graphs for k >= 4. In particular, we prove that tight (k, 0)-stable graphs are K_{k+1} and K_{k+2} for k >= 5, which not only extends the result of Liu, Song and Wang [J. Graph Theory 110(2) (2025), 193-199] from k >= 24 to k >= 5, but also proves the conjecture of Dong and Luo [Electron. J. Comb. 32(4) (2025), 4-45] once more.

Given a graph $G$, we show that the mixed metric dimension of $G$ is exactly $\ell(G)+2c(G)$ if and only if $G$ is either a cactus graph in which every cycle has precisely one vertex of degree at least $3$, or a balanced $\Theta$-graph, where $\ell(G)$ and $c(G)$ denote the number of leaves and the cyclomatic number of $G$, respectively. This provides an affirmative answer to a conjecture proposed by Sedlar and Škrekovski (2021).

We prove that the monodromy group of every singular hyperbolic metric on a non-hyperbolic Riemann surface, in the sense of potential theory, is Zariski dense in ${\rm PSL}(2,\mathbb{R})$, confirming a conjecture of the authors. The main new step is to show that a singular hyperbolic metric on an arbitrary parabolic Riemann surface cannot have monodromy contained in a conjugate of the real affine subgroup of ${\rm PSL}(2,\mathbb{R})$. The same argument also gives a direct proof in the compact case. Combined with the nonexistence results for the remaining proper subgroup types, this proves the conjecture.

For an AR($1$) sequence with coefficient $0 < a < 1$, the probability of staying above zero decays exponentially in the number of steps; at the unit root $a=1$ it decays polynomially, with order $n^{-1/2}$ for symmetric increments. We study this transition for AR($1$) sequences driven by symmetric and asymmetric $\alpha$-stable innovations. We write $\Lambda(a,\alpha)$ for the symmetric exponential persistence rate and $\Lambda(a,\alpha,\rho)$ for the rate of a strictly stable law with positivity parameter $\rho$. The entire chain admits an exact representation through a single stable Lévy process observed on a geometrically expanding time grid. In the symmetric case, comparison with continuous half-line survival gives $\Lambda(a,\alpha) \leq \frac{\alpha}{2}\log{(1/a)}$. For $0 < \alpha < 2$, this bound disproves the stable specialization of a conjecture of Hinrichs, Kolb and Wachtel concerning regularly varying innovation tails. Combining stable closure under subsampling with a monotonicity coupling yields a lower bound of the same near-unit order. This proves $\Lambda(a,\alpha) \asymp \log{(1/a)}$ as $a \uparrow 1$ and shows that $\Lambda(a,\alpha)/\log{(1/a)}$ converges to a limit in $(0,\alpha/2]$ equal to the supremum of that ratio over $0 < a < 1$. A Lamperti transformation reduces identification of this constant to a dense-sampling problem for a stationary stable Ornstein-Uhlenbeck process. Existing Gaussian theory determines the sharp value at $\alpha=2$; for $0 < \alpha < 2$, rescued crossings between observations remain the obstacle. For asymmetric strictly stable innovations with $\alpha \neq 1$ and admissible $\rho\in(0,1)$, the corresponding bounds are $\Lambda(a,\alpha,\rho) \leq \alpha(1-\rho)\log{(1/a)}$ and $\Lambda(a,\alpha,\rho) \asymp \log{(1/a)}$. At $\alpha=2$, only the symmetric case $\rho=1/2$ occurs.

The complex-step method is traditionally derived from the Taylor expansion of an analytic function and is widely used as a numerical technique for derivative approximation. We present an alternative formulation based on the Cauchy–Riemann equations and show that the classical complex-step relation arises naturally from the harmonic structure of holomorphic functions. In particular, the complex-step method admits two complementary harmonic interpretations: as a Cauchy problem, in which the derivative is identified with the normal datum of the imaginary component on the real axis, and as a reconstruction problem in a strip, in which the finite imaginary perturbation provides the upper-boundary data. The latter formulation leads explicitly to the strip Poisson and conjugate Poisson kernels and their derivatives. A related harmonic reconstruction framework in the upper half-plane leads to the Poisson, conjugate Poisson, and Cauchy kernels as elementary reconstruction operators for harmonic and holomorphic functions. Extending this reconstruction from ordinary boundary functions to finite measures yields the classical Stieltjes transform and its inversion formula. The same measure-theoretic structure appears in spectral theory, where scalar matrix elements of the resolvent are Stieltjes transforms of the associated spectral measures. These results establish a common complex-analytic structure connecting complex-step differentiation, harmonic reconstruction, Stieltjes inversion, and spectral reconstruction, while distinguishing the boundary-value problems through which the corresponding information is recovered.

An edge-weighted graph $H$, possibly with loops, is antiferromagnetic if its adjacency matrix is entrywise nonnegative and has at most one positive eigenvalue, counted with multiplicity. We show that, for any graph $G$ with $d_v:=\operatorname{deg}_G(v)$, $$\operatorname{hom}(G,H) \ge \prod_{v\in V(G)} \operatorname{hom}(K_{d_v+1},H)^{\frac{1}{d_v+1}},$$ whenever $H$ is antiferromagnetic. In fact, we prove a vertex-inhomogeneous strengthening of this inequality, allowing a different fugacity vector at each vertex of $G$. This gives a common generalization of the lower-bound inequalities of Sah, Sawhney, Stoner, and Zhao for independent sets, of Csikvári for $q$-colorings, and of the authors for semiproper colorings with at most two proper colors. Furthermore, it confirms recent conjectures of the authors and of Davies and LeBlanc. A key ingredient, of independent interest, is a strengthening of the delete-one form of Shearer's inequality for Lorentzian measures, which provides a new approach to graph homomorphism inequalities.

In this paper, through the introduction of Lorentz–Muckenhoupt classes, we systematically investigate the boundedness of maximal operators on multiplier weighted Lorentz spaces. As applications, we give the characterization of the commutators of fractional integrals, which yields a partial answer to an open question proposed by D. Cruz-Uribe. Second, Hardy inequalities in Lorentz spaces are established, with the critical case $p=d$, in which the classical Hardy inequality fails. Finally, we apply the Lorentz estimates to fractional Schrödinger equations with singular potentials.

Chernoff information is a fundamental divergence measure characterizing the optimal error exponent in Bayesian binary hypothesis testing, with applications in information fusion, time-series analysis, and statistical learning theory. However, closed-form expressions exist only for simple parametric families, and nonparametric estimation remains difficult because the quantity is defined as an optimization of the unnormalized Rényi divergence over its order. We reformulate this optimization via a derivative condition, whose zero locates the optimal mixture parameter, and estimate the derivative directly using a $k$-nearest-neighbor method. We prove the $L_2$-consistency of the derivative estimator under mild regularity conditions on the densities and their domain. Coupled with a bisection procedure that locates the optimal parameter up to arbitrary precision, this yields an estimator for Chernoff information.

We study the critical exponent of the Falconer functional for projective Anosov representations with Lipschitz limit sets and its relationship to the Hausdorff dimension of the limit set. Extending a result of Pozzetti, Sambarino, and Wienhard, we show, under density and irreducibility assumptions on the representation, that this exponent equals the Hausdorff dimension of the limit set. We also show that these assumptions are necessary for the strategy used there and in the present paper. To this end, we study the action of $\mathrm{SO}(p,p)$ on the space of maximal isotropic subspaces of $\mathbb R^{p,p}$ and provide examples of $\mathbf H^{p,q}$-convex-cocompact representations of lattices with virtual cohomological dimension $p$. Besides yielding counterexamples to seemingly innocent claims about irreducible representations in $\mathrm{SO}(p,q+1)$, these examples are of independent interest: they admit equivariant spacelike embeddings of Riemannian symmetric spaces into $\mathbf H^{p,q}$ on which uniform lattices act cocompactly.

This paper introduces a robust W-infinity optimal control framework for linear Itô diffusions using a weighted Sobolev-space performance measure. Because the sample paths of Itô diffusions are nondifferentiable, the formulation leverages the weak derivative of the expected state. An LMI-based semidefinite program is developed for dynamic output-feedback synthesis, and a rigorous stability analysis guarantees mean-square ultimate boundedness with minimized ultimate bound. A numerical example demonstrates that the proposed approach provides effective disturbance attenuation with fast transient performance.

The numerical simulation of time-harmonic wave propagation in elastic media plays an important role in applications such as geophysics and non-destructive testing. Accurate discretization of the elastic Helmholtz equation at high frequencies is challenging due to numerical dispersion and pollution effects. In this work, we develop an asymptotic dispersion correction for a Marker-And-Cell (MAC) discretization of the isotropic elastic Helmholtz equation. We characterize the discrete dispersion relation of the scheme and determine the leading-order term in the dispersion error in both two and three spatial dimensions. Based on this analysis, we derive a correction that is asymptotically optimal in the limit of vanishing mesh size. The proposed approach improves the agreement between the discrete and continuous wave propagation properties while preserving the structure of the underlying discretization. We also establish a connection between the factorization of the dispersion relation and the structure of the grad-div operator symbol, providing additional insight into the algebraic structure of the elastic problem. Numerical experiments finally demonstrate a substantial reduction of relative errors and confirm the effectiveness of the proposed correction. We further provide numerical evidence that the corrected discretization improves the convergence behavior of multigrid solvers.

This short work describes an extension of the permanental process model which includes fixed effects. By starting with a prior on the fixed effects coefficients we show that, in the diffuse prior limit, the intensity function of the permanental process can be found using the representer theorem and naturally decomposed into a fixed effects term and a function which is an element of a Reproducing Kernel Hilbert Space (RKHS). We show that the limiting equivalent kernel defines an RKHS whose squared norm is exactly the limiting penalty. This allows for straightforward scientific interpretation of permanental process models and the easy incorporation of domain knowledge into the estimation process.

Chaotic itinerancy (CI), brought to attention, among others, by K. Ikeda, I. Tsuda and K. Kaneko in the early 1990s, is a phenomenon in which trajectories in a dynamical system experience periods of ordered motion near quasi-attractors interspersed with chaotic transitions between them. Possible maps in which CI was found include coupled map lattices (CML) and globally coupled one-dimensional chaotic maps (GCM). We study such maps using numerical methods and graph algorithms. Specifically, we partition the state space into a finite grid of compact subsets, and we represent the map using a multivalued mapping of grid elements. This mapping can be perceived as a directed graph, with grid elements as vertices and individual mappings between them as weighted edges. This setup provides a coarse view of global dynamics and opens the opportunity for using Markov chains and efficient graph algorithms to study dynamical features. In particular, invariant sets can be found by computing strongly connected components in the graph. Analysis of the transition matrix of the graph makes it possible to find its stationary distribution and to compute local entropy as a measure of expansion or instability in the system. Using these tools, we propose an algorithm for assessing whether a certain map possesses the CI property and show its application to dynamical systems: a globally coupled system of logistic maps and a variant of a CML system for which we conduct computations for a large range of parameters.

We present a reformulation of four-dimensional chiral gravity in terms of a chiral gauge connection and a Clifford-valued coframe. The exterior algebra generated by $e^a \gamma_a$, together with the gauge curvature $F$, provides a natural and systematic construction of gauge-covariant differential forms. We show that suitable combinations of the chiral actions reproduce Einstein–Cartan–Holst gravity with a cosmological term. The original superconnection invariants yield only algebraic fermionic equations, so the fermionic one-forms do not propagate but instead generate a mass-like mixing between the two chiralities. Spinor propagation is supplied by a spinorial BF-type term, which under the matter ansatz reduces to the standard first-order Weyl kinetic action, including the torsion coupling. We also review the symmetry and variational properties of the matter ansatz, including its Lorentz equivariance, its relation to residual supersymmetry and twistor spinors, and the connection between the reduced and unrestricted field equations.

We establish an asymptotic formula for squarefree polynomials over a finite field whose coefficients are in $\{0,1\}$. This resolves a function-field analogue of a conjecture of Erdős, Mauduit and Sárközy. More generally, we estimate the probability of a polynomial $\sum \varepsilon_it^i$ being squarefree, where $\varepsilon_i$ are random variables sampled according to measures on $\mathbb{F}_q$.

We prove that every finite simple group of Lie type over a field of order at most 8 contains a conjugacy class whose square is the whole group. Together with the previously established cases, our results complete the proof of Thompson's conjecture. We explicitly construct the required class, address all low-rank identifications, and provide the precise computational certificates used in the exceptional character calculations.

We study Suslin's cancellation conjecture on smooth real affine varieties whose real locus is empty or, more generally, of small cohomological dimension.

In this work, we investigate the inverse problem of determining a semilinear term in a nonlinear parabolic equation from one single boundary flux measurement taken on an arbitrary subset of the boundary. More precisely, we address both uniqueness and stability issues of the inverse problem and establish new Hölder-type stability estimates. The Hölder exponent depends explicitly on the measurement configuration as well as on regularity properties of the semilinear term. The analysis relies on a novel approach based on the derivation of a suitable integral identity involving solutions of the associated adjoint equation. This allows reformulating the inverse problem as an inverse source problem with a sign-changing source term. The main results are obtained by combining fundamental properties of parabolic equations, including maximum principle and appropriate energy estimates. Finally, we complement the theoretical analysis with an iterative reconstruction algorithm inspired by inverse source problems, and illustrate its accuracy on several numerical experiments.

We construct irreducible smooth SU(2)-monopoles on asymptotically cylindrical 3-manifolds with vanishing first and second Betti numbers by a gluing construction. We also construct reducible singular U(1)-Dirac monopoles on arbitrary asymptotically cylindrical 3-manifolds.

Erdős problem 647 asks whether any $n > 24$ satisfies $\max_{m<n}(m + \tau(m)) \le n + 2$, where $\tau$ is the divisor-count function. Computational searches have excluded solutions up to $10^{12}$ by direct sieve and up to roughly $9.17 \times 10^{18}$ within a modular reduction whose Lean component relies on native_decide; those computations sit outside any proof kernel. We give the first exclusion checked end to end by one: no solution exists with $24 < n \le 10^9$, proved in Lean 4 with axiom closure exactly {propext, Classical.choice, Quot.sound} – no sorry, no native_decide, no problem-specific axiom. The proof replays a chain of 6,685,922 factorization witnesses whose excluded intervals concatenate across $(24, 10^9]$; it needs no primality facts beyond primes below 1024, and it is the finite, fully proved form of a domination-interval argument whose asymptotic step was the identified gap in a withdrawn January 2026 claim on this problem. The generation pipeline is cross-checked by two further independent implementations, the compiled development replays through the standalone lean4checker, and two from-source verification legs – Lean toolchains compiled from source by gcc and by clang, mathlib rebuilt with no cache – reproduce the committed certificates byte for byte, with olean digests identical across three builds on two architectures. Our range is three to ten orders of magnitude below the computational frontiers we cite; the contribution is the trust base, not the range.

We study Cohen–Macaulayness, in the sense of Hamilton–Marley, of non-Noetherian rings arising as big Cohen–Macaulay algebras. Motivated by Bhatt's notion of cohomological Cohen–Macaulayness, we call a locally finite-dimensional ring CCM if its structure sheaf satisfies this condition. Our main comparison theorem shows that every locally finite-dimensional CCM ring is locally HMCM. Using this theorem, we prove that if $A$ is Noetherian and $R$ is an integral $A$-algebra that is locally balanced big Cohen–Macaulay over $A$, then $R$ is CCM and hence locally HMCM. In particular, if $A$ is an excellent Noetherian domain, $p$ is a prime, $n\geq1$, and $A/pA\neq0$, then $A^+/p^nA^+$ is CCM and locally HMCM. We also show, using finite-dimensional valuation domains, that CCM is strictly stronger than locally HMCM. Finally, for a Noetherian ring $A$ of characteristic $p>0$ and its perfection $A_{\mathrm{perf}}$, we prove that the following conditions are equivalent: $A$ is locally weakly $F$-nilpotent; $A_{\mathrm{perf}}$ is a locally balanced big Cohen–Macaulay $A$-algebra; and $A_{\mathrm{perf}}$ is CCM. Under these equivalent conditions, $A_{\mathrm{perf}}$ is locally HMCM.

A finite group is totally $3$-closed if every faithful permutation representation of it is $3$-closed. We study this property for the finite simple projective special linear groups. We prove that $\PSL_2(q)$ is totally $3$-closed if and only if $q\geq 7$ is prime, and that $\PSL_3(q)$ is totally $3$-closed if and only if either $q=3$, or $q$ is prime and $q\equiv 2\pmod 3$. We further prove that $\PSL_4(q)$ is never totally $3$-closed and that $\PSL_n(q)$ is not totally $3$-closed whenever $n\geq 5$ and $q>2$. Within the family $\PSL_n(q)$, only the groups $\PSL_n(2)$ with $n\geq 5$ remain unresolved. In particular, this answers Problem~20.2 of the Kourovka Notebook affirmatively.

Bilevel optimization is a powerful tool for modeling hierarchical decision-making processes, which arise in various real-world applications. Due to their nested structure, however, bilevel problems are intrinsically hard to solve, even if all variables are continuous and all parameters of the problem are exactly known. Further challenges arise if mixed-integer aspects and problems under uncertainty are considered. In this article, we summarize selected results from the author's dissertation. We study mixed-integer linear min-max problems with a $\Gamma$-robust treatment of uncertain data, for which we present exact and heuristic solution approaches. The performance of the methods is assessed in a computational study on 560 instances of the knapsack interdiction problem. Our results show that the heuristic closes the optimality gap for a significant portion of the considered instances and often practically outperforms both heuristic and exact benchmark approaches.

We establish a discrete-to-continuum variational theory for area-preserving parameterization of surfaces based on the stretch energy. For orientation-preserving diffeomorphisms between compact, connected, oriented Riemannian surfaces of equal total area, we prove that area-preserving maps are precisely the critical points of the stretch energy under boundary-fixing variations. We then establish discrete-to-continuum convergence of the optimal energy values: within a uniformly geometrically controlled admissible class, the discrete infimum converges to the smooth minimum with second-order accuracy, while discrete almost minimizers have first-order decay of their $L^2$ area distortion. Thus, the discrete problem not only approximates the smooth stretch energy but also recovers area preservation in the refinement limit. We further derive the $L^2$-gradient flow of the stretch energy and its simplicial counterpart, whose projected quasi-implicit iteration is shown to be globally convergent. The resulting method applies to open and closed surfaces of several topological types, and numerical experiments on benchmark surfaces illustrate its improvement over existing methods.

Among finite simple groups, character values of alternating and sporadic groups have relatively low irrationality at any prime $p$, whereas those of simple groups of Lie type can have arbitrarily high $p$-irrationality. We provide concrete evidence supporting this phenomenon. In particular, we show that if $\chi:=R_{\mathbf{T}}^{\mathbf{G}}(\theta)$ is a Deligne–Lusztig character of a finite reductive group $\mathbf{G}^F$, with $\theta$ an irreducible character of a maximal torus $\mathbf{T}^F$, and if $\chi$ has degree prime to $p$, then the so-called $p$-rationality level of $\chi$ coincides precisely with that of $\theta$. We present further evidence suggesting that Lusztig induction preserves $p$-rationality for characters of $p'$-degree.

Davidson and Kennedy introduced the dilation order in connection with commutative C*-algebras and classical Choquet theory. Its noncommutative counterpart continues to detect the unique extension property of GNS representations. Motivated by the failure of Arveson's hyperrigidity conjecture and the amended theorem of Clouâtre and Thompson, we develop a state-space approach to this rigidity phenomenon. First, we introduce the strong dilation relation on the state space of a C*-algebra and characterize the unique tight extension property through maximality of states. Second, we define the integral subdivision relation and prove that maximality of all pure states in the dilation order implies maximality of every state in the integral subdivision relation. This provides a state space-level amendment of Arveson's hyperrigidity conjecture and yields an alternative proof of the Clouâtre-Thompson theorem. Finally, we show that the strong dilation and integral subdivision relations coincide in the commutative setting and they both agree with the abstract Choquet order associated with a function system.

Grounded arithmetic is a family of formal systems for reasoning about computation in which a statement may be asserted only when a terminating computation backs it; the logics are paracomplete - for a sentence whose backing computation never settles, neither the sentence nor its negation is derivable, so paradoxes like the Liar are harmless rather than explosive. The reflective member of the family, RGA, can quantify over its own computations, but cannot certify that its own unbounded searches have definite yes-or-no answers. This paper studies what happens when that openness is closed by exactly one rule - ATI, the $\omega$-grounded universal: if every numeric instance of a universal sentence is certified decided, the universal is certified decided. The resulting system, OGA, shares RGA's syntax and rules symbol-for-symbol otherwise, and every consequence is developed as a machine-checked theorem. Decidedness certificates become abundant - every totality question about a computable function is certified to have an answer, whether or not anyone can produce it - and this is exactly the provable separation between the two systems. OGA is complete for its own semantics; certified-but-unresolved sentences receive values built from the system's own open questions. Provability remains recursively enumerable, with a primitive-recursive certificate checker, while $\omega$-truth deliberately is not. Within that asymmetry, incompleteness takes a new form. The Gödel sentence is classified, unconditionally, as a genuine fiction: neither provable nor refutable, yet valued, and carrying a computable pedigree recording exactly what adopting it as an axiom commits one to. The adoption is itself a theorem suite: extending OGA by any finite stock of true fictions is consistent, and independently certified adoptions can never collide.