Papers

The repeated averages process is a stochastic averaging process on graphs whose mixing time is known for several structured families, but no general sharp expression is known for all connected graphs. It was conjectured that the $L^2\!\to\!L^1$ mixing time is of order $|E|\log n/\lambda_2$. We disprove this conjecture using the graph $G_n$ obtained by attaching one leaf to the complete graph $K_n$. We prove that $\lambda_2(G_n)=1$ and that \[ t_{\varepsilon,2\to1}(G_n)=\Theta_{\varepsilon}(\gamma(G_n))=\Theta_{\varepsilon}(n^2), \] with no additional $\log n$ factor. The example has a strongly localized Fiedler eigenvector: most of its squared $L^2$ mass lies on the leaf, while the balancing mass is spread across the clique.

We give explicit annular normal forms and classify Frobenius structures for $p$-adic Schwarzian equations defined by developing differentials. For a unit of the Robba ring, reduction of order expresses the associated module as a quadratic rank-one twist of a unipotent extension determined by the residue. Frobenius existence is characterized by a square-class condition. At odd primes, there are exactly four differential-module types, and a uniform dominant-monomial hypothesis gives a Frobenius formula in analytic families across the zero-residue locus. For rational nonintegral annular powers, we obtain a diagonal normal form and a congruence criterion at odd primes; both results hold at every prime when the unit factor is an explicit square. Applied to the reducible modular equations $\theta^2y-\ell^2E_4y/144=0$, with $\gcd(\ell,6)=1$, this gives exactly two annular types at every prime and determines the least $s\geq1$ for which Frobenius under $q\mapsto q^{p^s}$ exists. The subfamily $\ell=12n+1$ has a single annular type. Heine–Stieltjes residue cancellation and nondegenerate Jacobi configurations provide explicit examples. We also prove an obstruction to constant projective Frobenius equivariance of Robba-ring developing maps.

In this paper, we completely resolve the two total positivity problems raised by Li [J. High Energy Phys., 2023(7):Paper No. 167, 44 pp., 2023], thereby determining the precise scope of this positivity structure in the one-dimensional conformal bootstrap. First, the Gauss hypergeometric kernel $\mathcal{F}(\Delta,z) = {}_2F_1(\Delta,\Delta;2\Delta;z)$ is proved to be strictly totally positive of order infinity for $\Delta > 0$ and $z \in (0,1)$. Second, the sharp lower $\Delta$-parameter threshold for the associated one-dimensional $SL(2,\mathbb{R})$ conformal block $G_{\Delta}(z) = z^{\Delta}\mathcal{F}(\Delta,z)$ is shown to be $1/2$: the conformal-block kernel $G_{\Delta}(z)$ is strictly totally positive of order infinity for $\Delta \geq 1/2$. For every $\tau \in (0,1/2)$, there exists a strictly negative odd-order minor of $G_{\Delta}(z)$ such that all its $\Delta$-values are in $(\tau,1/2)$ and all its $z$-values can be chosen arbitrarily close to one. Consequently, no restriction $z > z_0$ with $z_0 < 1$ can restore total positivity of order infinity over all $\Delta > 0$.

Pulse interventions represent a highly effective measure for infection control, as they influence disease transmission through short-term actions. In addition, the habitat ranges of dengue vectors and hosts exhibit periodic variations driven by environmental and climatic factors. To investigate the effects of impulsive interventions and domain evolution on disease transmission, we propose a dengue fever reaction-diffusion model that incorporates impulsive perturbations in a periodically varying domain. By applying the Poincar$\acute{e}$ map and the Krein-Rutman theorem, we establish the existence of the principal eigenvalue for the periodic eigenvalue problem with impulses, thereby extending previous studies on reaction-diffusion equations in fixed domains without impulsive effects. Sufficient conditions governing the long-term dynamics of periodic solutions are derived using the comparison principle and monotone iteration theory. Numerical simulations corroborate the theoretical findings and elucidate the effects of impulsive-intervention intensity and periodic domain evolution on disease transmission patterns. Our results indicate that increasing the intensity of pulse interventions suppresses disease transmission, whereas a larger magnitude of domain variation impedes disease control.

We determine the generalized and local derivations of the mirror Heisenberg-Virasoro algebra. As a by-product, we give a complete characterization of $\delta$-(bi)derivations.

Let $S(Q,M,N,(a_n))=\sum_{q\le Q}\sum_{a\bmod q^2,(a,q)=1}|\sum_{M<n\le M+N}a_ne(an/q^2)|^2$ be Zhao's large sieve sum with square moduli. Zhao conjectured that $S\ll(QN)^\epsilon(Q^3+N)\sum|a_n|^2$. The bound of Baier and Zhao (2008) exceeds this conjecture by a factor which reaches $Q^{1/2}$ at the critical point $N=Q^3$, and there the exponent $1/2$ has not been lowered since. We prove that, under Hooley's Hypothesis $R^*$ for short Salié sums, $S\ll(QN)^\epsilon(Q^3+N+Q^{4/3}N^{2/3})\sum|a_n|^2$. This replaces the excess over Zhao's conjecture by its $2/3$-th power throughout the range $Q^{5/2}<N<Q^4$; at $N=Q^3$ it gives $S\ll Q^{1/2-1/6+\epsilon}N\sum|a_n|^2$. The proof is short. Following Zhao's original treatment, we approximate the point under consideration by fractions $b/r$ with denominators up to $Q^{2/3}N^{1/3}$, which for $N<Q^4$ is beyond the customary $\sqrt{N}$. If the approximation is poor, an elementary count suffices. If it is good, the analytic part of the problem disappears: after Poisson summation in both variables the count is expressed through quadratic Gauss sums, and their exact evaluation at every modulus turns the sum over the dual variable into short Kloosterman and Salié sums, to which Hypothesis $R^*$ applies and gives the square root of the elementary count. At odd moduli the hypothesis is needed only in the form of Conjecture (C*) of Friedlander and Iwaniec. The Gauss-sum evaluations, including even moduli and coefficients sharing a factor with the modulus, are proved in full. This version supersedes earlier versions, which obtained the exponent $1/2-1/28$ at $N=Q^3$ by a much longer route through sums of modular square roots. The paper was prepared in collaboration with Claude (Anthropic).

If $U$ is a minimal, diskbusting, finite list of words in a free group $F_n$ of rank $n$ such that the sum of the lengths of words in $U$ is at most $2n+4$, we prove that the natural presentation complex of the Baumslag double of $F_n$ along $U$ virtually contains a $\pi_1$-injective embedded closed hyperbolic surface. This verifies the Tiling Conjecture of Kim and Wilton for this type of lists of words, and in particular, implies that the corresponding Baumslag double contains a hyperbolic surface subgroup.

We study the asymptotic density of $\mathbb{F}_q$-point-free linear sections of geometrically irreducible projective varieties over finite fields. We then apply these results to rank-metric codes via determinantal varieties. Our approach recovers the known cases in which the density tends to $0$ or $1$ and determines the limit in the cases where it was previously unknown. To compute these previously unknown limits, we extend the notion of quasireflexivity to higher-dimensional varieties and show that determinantal varieties satisfy this property. This allows us to invoke the Chebotarev density theorem for varieties over finite fields to obtain the desired estimate.

The manuscript provides a description of a theoretical analysis of a non-Newtonian Prandtl fluid subject to peristaltic flow through an inclined asymmetric channel. We explore the effect of an inclined magnetic field on the peristaltic flow. This is relevant for applications involving fluid flow in narrow, inclined (tilted) tubes similar to blood vessels or the digestive system. The model also includes thermodynamic aspects such as heat diffusion (the Soret effect) and viscous dissipation resulting from wall-fluid slip conditions, which may help optimize medical devices such as lab-on-a-chip systems and dialysis machines. In this study, the concentration of a generic chemical, temperature, and fluid velocity are taken into account through mass, heat, and momentum balances, respectively. The solution is approximated using numerical techniques suitable for long wavelengths (low frequency) and low Reynolds numbers. The study also discusses trapping phenomena, which are crucial from a clinical point of view. The developed insights can improve the understanding of physiological flows in the gastrointestinal tract and blood vessels. By understanding how the fluid moves and how particles are trapped, these insights may contribute to the design of improved medical pumps and artificial organs. Graphical visualizations are provided for the fluid velocity profile, temperature distribution, and concentration of a generic chemical. Furthermore, the numerical results are validated through comparison with a closed-form solution from a benchmark problem.

Let $R$ be a commutative ring with 1, $P$ a locally finite partially ordered set, and $G$ a group. We derive necessary and sufficient conditions for an $R$-algebra isomorphism between the incidence algebra $I(P,R)$ and the group algebra $RG$. Then, for an indecomposable ring $R$, a finite poset $P$ and an arbitrary group $G$, we classify the $G$-gradings of $I(P,R)$ up to graded isomorphism. The classification rests on a complete set of primitive orthogonal homogeneous idempotents. The corner algebras are split group algebras of finite abelian subgroups of $G$, and the off-diagonal Peirce blocks are multiplicity-free sums of bimodules induced from characters of double coset stabilizers. Graded isomorphisms are shown to have a rigid form, and a grading is determined up to graded isomorphism by the poset of idempotents, the corner groups, the types of the atomic bimodules and the structure constants of their multiplication. The data which occur are characterized by polynomial conditions, and over an algebraically closed field of characteristic zero only finitely many graded isomorphism classes share given partial invariants. An example shows that the structure constants cannot be omitted. Some previous results are extended and enhanced, while providing alternative proofs for some known facts.

In this work, we develop a second-order, linear, decoupled, and structure-preserving numerical scheme for the modified Landau–de Gennes model of smectic-A (SmA) liquid crystals. The main contributions are threefold. First, to the best of our knowledge, we propose the first integration of the generalized scalar auxiliary variable (GSAV) approach with a second-order exponential time-differencing Runge–Kutta (ETDRK2) discretization, leading to a second-order GSAV–ETD2 scheme. Second, we prove that the proposed scheme satisfies an unconditional energy-dissipation law, thereby closing the theoretical gap in the energy-stability analysis of second-order GSAV exponential integrators of this class. Third, by deriving a coercive discrete reformulation, we establish a fully discrete error estimate without imposing any coupling condition between $\tau$ and $h$, achieving the optimal convergence rate $\mathcal{O}(\tau^2+h^2)$. Numerical experiments are presented to verify our theoretical results and to simulate the self-assembly dynamics of the SmA phase.

This work presents theoretical advances in the study of cyclic and quasi-cyclic lattices. First, we provide elementary facts regarding cyclic and quasi-cyclic lattices, and discuss about the cyclicity of some notable lattices. The main contributions of the paper is in the algebraic setting: we investigate cyclic lattices arising from $\mathbb{Z}$-modules in Galois number fields via the Minkowski embedding. We establish necessary and sufficient conditions for an algebraic lattice to be cyclic over both cyclic and general Galois number fields, expressed in terms of naturally associated groups. Moreover, we derive a necessary and sufficient condition for ideal lattices to be cyclic, depending on the factorization of the ideal in the underlying number field.

We prove the existence of traveling wave solutions for a diffusive Nicholson model with a delayed linear harvesting term and two delayed-response channels. Our approach combines monotone heteroclinic connections of the associated scalar delay equation with the large-speed traveling-wave theorem of Faria, Huang and Wu. For \(0<\sigma<r\), we use a previous upper- and lower-solution construction under an explicit parameter condition. For \(0<r\leq\sigma\), we establish a new monotone heteroclinic connection. To fit the abstract framework, we embed the scalar equation into an auxiliary two-dimensional reaction–diffusion system and prove that every relevant traveling wave lies on the invariant diagonal. We then determine an explicit parameter region in which the required spectral hypotheses hold. Consequently, the original diffusive Nicholson equation admits traveling waves connecting the trivial and positive equilibria for all sufficiently large wave speeds.

We present a simplified proof of the Erdős-Sós conjecture on trees in graphs, which is based on its recent resolution by GPT-6 Astra. At the same time, we also give another proof of the corresponding conjecture for antidirected trees in digraphs, posed by Addario-Berry, Havet, Linhares Sales, Reed, and Thomassé, and we simplify recent proofs of Kalai's conjecture for tight trees in hypergraphs. Our proof uses the language of random cyclic orderings in order to take advantage of symmetries that were obscured in the original argument, and we hope that the reader will find it intuitive. We also characterize the extremal examples for Kalai's conjecture, and we generalize it to the setting of so-called fully directed $r$-uniform hypergraphs, whose edges are ordered $r$-tuples.

Zavarzina asked whether every bijective non-contractive self-map of a precompact quasi-metric space is an isometry. We answer the question in the negative, in a stronger form: on a hereditarily precompact quasi-metric space (a tolled extension of the irrational rotation over Z), the shift is a bijective non-contractive map that is not an isometry. The construction generalizes: for every metric space Y, every bijective isometry $\sigma$ of Y, and every point whose $\sigma$-orbit is aperiodic and totally bounded, the tolled Z-extension is such a counterexample; instances include the irrational rotation and the p-adic odometer. The conjugate space is not precompact, so the two-sided hypothesis of Zavarzina's positive result is sharp, and the space is not expand-contract plastic. The core results, including the general scheme and the circle instance, are machine-checked in Lean 4 over mathlib, sorry-free (no unproved steps), under a pinned toolchain. A structural section proves a monochromy theorem: for every strict pair on a hereditarily precompact space, one endpoint's backward self-returns have self-defect at least a quarter of the pair's defect, uniformly on a chain-closed displacement set. Boundary results delimit what remains open: one-sided compactness together with separation.

We consider the identification of memory and stochastic forcing in a one-dimensional generalized Langevin equation with state-dependent Brownian noise and optional additive symmetric $\alpha$-stable jumps. A finite-dimensional Markov embedding provides Kramers–Moyal conditional moment relations that separate memory drift, Brownian diffusion, and jump contributions. From increment ensembles conditioned on the observed velocity history, truncated statistics estimate the stable parameters, corrected second moments determine the Brownian amplitude, and first moments lead to a Volterra equation for the memory kernel, solved by RKHS regularization. We establish conditional finite-sample bounds for the moment estimators and a conditional $L^2$ error bound for kernel recovery with a data-selected regularization parameter. Numerical experiments examine Brownian and Brownian–Lévy models under different forces, memory kernels, observation lengths, measurement noise, and approximate history conditioning.

Let $q=3q_0^2$, where $q_0=3^s$ and $s\ge3$. We prove that a smooth, projective, geometrically irreducible curve over $\F_q$ with $q^3+1$ rational points and genus \[ \frac32q_0(q-1)(q+q_0+1) \] is $\F_q$-isomorphic to the Ree curve.

Entanglement cost is the asymptotic rate of Bell pairs required to prepare a quantum state. Its regularized definition has made exact evaluation difficult, even for isotropic states (i.e., depolarized maximally entangled states). In this work, we determine the entanglement cost of every qubit isotropic state and, more generally, every two-qubit Bell-diagonal state. Our proof uses a family of suitably tuned qubit semigroups to transform a general log-Sobolev entropy bound into supporting lines of Wootters' function. A recent exact tensorization theorem of Dong et al. [arXiv:2606.17729] extends these bounds to arbitrarily many copies, yielding a general lower bound on entanglement cost. Matching this bound with the entanglement of formation also determines the exact cost for a broader class of two-qubit states. For qudit isotropic states, we derive another general lower bound from the full depolarizing $2\to 3$ norm. This bound substantially improves on the PPT-relative entropy of entanglement and nearly matches the entanglement of formation, leaving a gap of at most $2\%$ of $\log d$ that vanishes as $d$ grows. Finally, the exact qubit results and qudit bounds extend to the entanglement cost of preparing quantum depolarizing channels under both parallel and sequential strategies.

Let A be a nonempty compact subset of the plane and let A (r) be its parallel set at distance r. We prove that the number of holes of A (r) the number of bounded components of its complement, which is its rst Betti number satises $\infty$ r 0 $\beta$1(A (r) ) dr $\le$ 4050 (diam A) 4 r -3 0 for every r0 > 0, the integrand vanishing for r $\ge$ diam A/ $\sqrt$ 3. The proof rests on Fu's theorem that the critical values of the distance function of a planar compact set form a set of vanishing half-dimensional Hausdor measure, on two lemmas of Rataj, Spodarev and Meschenmoser, and on a square-root summability estimate for the gaps of the critical-value set, of which we give a complete proof. We show by an explicit family of curves two combs facing each other that no analogous bound can hold at a xed radius: a connected curve of bounded length, diameter, oscillation count, parallel-set area and parallel-set perimeter can have arbitrarily many holes at one radius, so the integrated estimate cannot be replaced by a xed-radius bound in terms of these coarse geometric quantities. We also bound the hole count uniformly in the radius by the number of components of local maxima of the distance function, and record a bound on the boundary length of a parallel set by its area. The results supply the deterministic input for limit theorems on the persistent homology of the Wiener sausage.

We demonstrate that forking in monadically NIP theories satisfies a very strong form of triviality over arbitrary sets, generalizing Baldwin-Shelah in monadically stable theories and analogous to Shelah's result for finite satisfiability over models in monadically NIP theories. This implies in particular that strict non-forking independence is particularly well behaved, and that the forking relation induces a canonical semilinear partial order on singletons. We also give an example of an $\omega$-categorical monadically NIP theory so that no finite set is an extension base.

Kaplansky's zero-divisor conjecture asserts that the group ring of a torsion-free group over a field has no zero divisors. It holds for every group with the unique-product property, so a counterexample can only come from a torsion-free group without unique products. In lectures in 2021, Gardam announced that the torsion-free $\tilde{A}_2$ lattice $\Gamma = \langle a, b \mid a b a^2 b^{-1} a^2 b^{-2}, a b^3 a b^4 a^{-1} b \rangle$ does not have unique products and presented it as a new candidate group: it has property (T), and the known methods for proving the conjecture do not apply to it. To our knowledge, no proof of the announcement has been published. We give a proof checked by the Lean 4 kernel and stated against Mathlib's UniqueProds class. The witness is an explicit pair of finite subsets with $|A| = 32$ and $|B| = 28$ in which each of the 896 products coincides with another product. For 658 products the certificate is an identity in the free group; the other 238 certificates are explicit products of conjugated relators, 970 conjugates in all, checked by free reduction. A homomorphism onto $\mathbb{Z}/42$ shows that each pair $(u,v)$ differs from its partner $(u',v')$ as a pair of group elements, which is all the theorem requires. Together with a homomorphism onto the alternating group $A_4$ it also shows that the listed words are pairwise distinct, so the sets have exactly 32 and 28 elements. The witness and certificates come from an untrusted search program and are re-checked by Lean. The development uses only the axioms propext, Classical.choice and Quot.sound, with no sorry and no native_decide. The mathematical statement is Gardam's. To our knowledge this is the first verification in a proof assistant of a unique-product failure in a torsion-free group; torsion-freeness of $\Gamma$ is taken from Gardam and is not formalized here.

We prove that the Conway knot has infinite order in the concordance group, and in particular that it is not slice. Ours is an alternative, independent proof with respect to those of Piccirillo and of Donatone, Kegel, Lewark, and Truöl.

For a monic irreducible $f \in \mathbb{Z}[x]$ of degree $n$ and an integer $c$, we study the palindromic transform $F(x) = x^n f(x + x^{-1} + c)$, which produces a polynomial of degree $2n$. We give a discriminant formula $\operatorname{disc}(F) = f(c+2)f(c-2)\operatorname{disc}(f)^2$, sufficient conditions for the irreducibility of $F$, and a criterion showing that $F$ is monogenic when $f$ is monogenic and $f(c+2)f(c-2)$ is squarefree, together with a matching non-monogenicity criterion. Iterating the transform yields monogenic number fields of degree $2^k n$ from a fixed base polynomial. As an explicit example, we construct a monogenic field of degree 512 from $x^4 + x + 1$.

A Stieltjes clock allows effective time to advance continuously, remain unchanged over intervals, or jump. We study whether a compact-valued set-valued map evolving relative to a Stieltjes clock admits a single-valued selection that passes through a prescribed graph point while retaining the regularity and variation of the multifunction. For g-Hölder exponents $\alpha \geq 1$, we prove that regularity can be preserved without increasing the g-Hölder seminorm, with no monotonicity assumption on g. If g is additionally nondecreasing, one prescribed-point selection preserves both this regularity and the Hausdorff variation on every subinterval. The same selection consequently preserves all finite Riesz p-variations associated with nondecreasing external clocks. For $\alpha>1$, the structure becomes jump-driven. For left-continuous nondecreasing Stieltjes clocks, continuous clock evolution cannot generate variation: all variation is carried by jumps. We obtain exact jump decompositions for both the set-valued map and its selection, together with explicit atomic formulas for Riesz p-variation. Examples show that the principal regularity, variation, and jump bounds are attained. For compact-convex Euclidean-valued maps, we also examine the case $0<\alpha<1$. In this setting, the Hölder exponent can still be preserved, but preservation of the same constant through a prescribed point holds in one dimension and can fail in higher dimensions. Finally, without a quantitative Hölder bound, we characterize exactly when every compact-valued Hausdorff g-continuous map admits a prescribed-point g-continuous selection: precisely when the clock image is zero-dimensional.

We construct commutative rings whose derived categories have non-spatial frames of smashing ideals, thereby answering in the negative questions of Balchin–Stevenson and Aoki.

For a $k$-uniform hypergraph $\mathcal H$, the shadow of $\mathcal H$ is the graph whose edges are the pairs covered by a hyperedge. In this paper, for all sufficiently large $n$, we determine the $n$-vertex $k$-uniform hypergraphs of maximum adjacency-tensor spectral radius whose shadow has no $K_t$ minor, for every $t\ge k+1$, and those whose shadow has no $K_{s,t}$ minor, for every $2\le s\le t$ with $s+t\ge k+1$ and every residue of $n-s+1$ modulo $t$; outside these ranges the problems are trivial. This extends to uniform hypergraphs the theorem of Tait on graphs with no $K_r$ or $K_{s,t}$ minor, whose remaining residues were settled by Zhai and Lin. In each case the extremal hypergraph is unique, and it is the $k$-clique hypergraph of the join of a clique with a graph that we call the light part. For $K_{s,t}$ the answer depends on $j=k-s+1$. When $j\le1$, the maximum has order $n^{(k-1)/k}$, and the light part is the one found by Zhai and Lin for the adjacency matrix, including its exceptional components. When $j\ge2$, a regime that does not occur for graphs, the maximum has order $n^{(s-1)/k}$ and $t$ enters its leading constant. The light part then consists of copies of $K_t$ and one smaller clique, with a single exception: for $(k,s,t)=(9,8,8)$ and $n-s+1\equiv2\pmod 8$, the complement of the Petersen graph appears. When the smaller clique has between $1$ and $j-1$ vertices, the extremal graph is not unique. In particular, for $t=8$, $4\le s\le7$, $k=s+1$ and $n-s+1\equiv2\pmod 8$, the clique hypergraph of the extremal graph of Zhai and Lin is not extremal. For $j\ge2$ the light part is determined by a weighted clique inequality, which for $j\ge3$ follows from a weighted form of the closed-neighborhood counting of Chao and Dong.

Understanding eco-evolutionary dynamics presents significant mathematical challenges, particularly regarding predictability, equilibrium, and the mathematical representation of fitness. Using Open System Dynamics and Stochastic Analysis, we develop a theoretical framework to characterize fitness change and its relationship with predictability. We explore the limiting behavior of biomass abundances modeled by stochastic processes, deriving continuous-time diffusion counterparts from discrete-time models. Focusing on neutrality and the search for equilibrium, we analyze diffusions in two distinct state spaces. For relative abundances on a d-dimensional simplex, we demonstrate the central role of orthogonal polynomials in model construction. For non-proportional abundances in the positive orthant of d-dimensional space, we detail two specific families of models. Finally, we address statistical inference by presenting numerical examples for relative abundances, illustrating how orthogonal polynomials facilitate maximum likelihood parameter estimation.

I believe that the development of AI tools will lead to a transformation of mathematics, both professionally and culturally. However, it will be a tragic missed opportunity if the massive shifts in culture and practice that are underway are motivated and influenced solely by AI. We must aspire to more than this. We can and should collaboratively strive for a future where our cultures, practices, and values are grounded first and foremost in empathy, compassion, justice, and humanity. In this article, I call to action every member of the mathematical community, whether you view yourself as a researcher, teacher, scholar, student, professional, hobbyist, or a combination of these and more.

In this paper, we study the $p$-adic zeros of the Tribonacci sequence $T_n$, for a prime $p$. In particular, we complete the analysis that was done in a 2024 paper of Yuri Bilu by investigating the case of $p=11$. In addition, we investigate the primes $p$ for which at the first two values $T_n$ and $T_{n-1}$ which are both divisible by $p$, both of these are actually divisible by $p^2$, proving under the assumption of a generalisation of the $abc$ conjecture that there are infinitely many $p$ which are not such.

Every smooth closed Riemannian surface of genus zero is isometric to the unit sphere carrying a conformal metric, the round metric multiplied by a positive factor. This paper solves the Poisson equation and the eigenvalue problem of the Laplace–Beltrami operator of such a metric with smooth spherical splines, taking the conformal factor as the only description of the geometry. Because the Dirichlet energy is conformally invariant in two dimensions, the stiffness matrix is the round-sphere matrix for every conformal metric, and the factor enters only the load vector, the mean-value constraint and the weighted mass matrix. A surface embedded in space with a conformal parametrization, a surface of revolution, and a surface given only as a triangle mesh are instances of the same problem that differ in how the factor is obtained; the first two are treated here, and the factor is checked against an area identity before any equation is discretized. The unknown is a spline on a fixed spherical triangulation in broken Bernstein–Bézier form, with $C^r$ smoothness imposed algebraically through edge functionals and the conforming space realized by a null-space matrix computed once and reused for every metric. Each step of the computation is stated with the formulas it needs. Experiments on the round metric, a prolate spheroid, a dumbbell of revolution, an Evans–Fung red-blood-cell profile and a prescribed factor with no embedding show the rates $d+1$ in $L^2$ and $d$ in energy for spline degree $d$, the rate $2d$ for eigenvalues with reproduction of the even-order eigenvalues of the round metric up to degree $d$ to roundoff, and, against parametric surface finite elements on the same triangulations, smaller errors at comparable numbers of unknowns for degrees four and six.