Papers

We study the quantised version of the Kadomtsev-Petviashvili equation in the formulation proposed in [4]. We analyse two aspects. One is the conservation of the first higher charge $H_3$ (besides the total mass $H_0$, the momentum $H_1$ and the Hamiltonian $H_2$). We find the expression of $H_3$ and prove commutativity, provided we regularise a specific ultraviolet divergence. The result then does not depend on the choice of a regulator, under fairly general assumptions. We then study the Bethe ansatz in finite volume and obtain a general solution to the two-body problem. We have used AI supervised by us to aid the proofs.

Universality relies on combining operators with different algebraic properties, which depend on the physical structure of the Hilbert space encoding the information. Systems of identical symmetric particles—including bosons and symmetric collections of spin-$1/2$ particles—can be mapped to angular-momentum systems, where rotations supplemented by quadratic, spin-squeezing interactions provide universal control. We revisit universality in this framework and confirm that such linear and quadratic operations provide universal control on every fixed-particle-number subspace. At the abstract level, however, we show that the dynamical Lie algebra generated by all degree-at-most-two elements does not exhaust the full universal enveloping algebra of $\mathfrak{su}_{\mathbb{R}}(2)$, contrary to a stronger claim made previously: higher powers of the quadratic Casimir element cannot be generated by commutators and linear combinations. This obstruction does not conflict with fixed-sector universality because, under a fixed irreducible spin-$n/2$ representation, all powers of the Casimir element act as scalar multiples of the identity. We determine the resulting abstract dynamical Lie algebras exactly and show that the central-free and full degree-at-most-two algebras map, respectively, onto $\mathfrak{su}_{\mathbb{R}}(n+1)$ and $\mathfrak{u}_{\mathbb{R}}(n+1)$, establishing special-unitary and full-unitary universality. We further derive a practical degree-two universality criterion and discuss finite-dimensional SSRC universality.

Let $\Omega$ be a lattice in $\mathbb{C}$ with algebraic invariants and complex multiplication, let $\mathcal{E}$ be the elliptic curve associated with $\Omega$, and let $\wp$ be the Weierstrass function relative to $\Omega$. Set $ k:=\operatorname{End}(\mathcal{E}) \otimes_{\mathbb{Z}}\mathbb{Q}.$ We prove that if $t_1,\dots,t_s$ are $\mathbb{Q}$-linearly independent algebraic numbers and $p_1,\dots,p_n$ are $k$-linearly independent algebraic numbers, then the $s+n$ numbers \[ \mathrm{e}^{t_1},\dots,\mathrm{e}^{t_s}, \wp(p_1),\dots,\wp(p_n) \] are algebraically independent over $\overline{\mathbb{Q}}$. The proof uses the Tannakian description of the Lie algebra of the unipotent radical of the $1$-motive associated with these points.

For \(\omega\in\calD\), we give necessary and sufficient symbol conditions for the boundedness and compactness of the Riemann–Liouville family\(V^\varphi_{\alpha,\beta}:A^p_\omega\to H^q\) for every \(0<p,q<\infty\) and admissible \(\alpha,\beta>0\): without any additional weight assumption when \(\alpha\ge\beta\), and, when \(\alpha<\beta\), under the condition \(p(\beta-\alpha)<d_-(\omega)\), where \(d_-(\omega)\) is the critical reverse-doubling exponent. We prove that this inequality is exactly equivalent to the naturally shifted source weight retaining the two-sided doubling and tail geometry required by the reduction; an integrable logarithmic example shows that the endpoint fails. The extension from power weights is not formal, because the reduction replaces \(\omega\) by \(\omega(z)(1-|z|^2)^{p(\alpha-\beta)}\), and a negative shift may destroy even integrability. Combining moment-induced Littlewood–Paley theory for doubling Bergman spaces with an exact coefficient-multiplier comparison, we establish the required Riemann–Liouville transfer, including the finite-dimensional exceptional modes at positive integral shift orders. A finite-codimensional range decomposition and a fractional \(g\)-function reduction then reduce both questions to a single weighted area-map problem. Within the admissible parameter set, the effects of \(\alpha\) are absorbed by the shifted source weight and bounded model corrections, so the final symbol conditions depend on \(\beta\). The thresholds \(p=2\) and \(p=q\) yield pointwise, Carleson, tent-integral, and non-tangential maximal criteria. In the tent-integral range, boundedness already implies compactness.

Artificial intelligence threatens to outrun human understanding and control. New mathematics is needed to design AI that is legible, steerable, and cooperative with humanity. I organize this invitation by mathematical field, so you can turn straight to your own: logic and game theory for cooperation; probability for agency and world-models; algebra and representation theory for learned features; analysis and geometry for generalization and training dynamics. Each section ends with an open problem that is accessible to a working mathematician with no prior experience in AI safety.

We establish sharp finite-ball shooting classifications for radial nodal Dirichlet solutions of the logarithmic equation and, within the maximal simple-zero shooting class, of the sublinear scalar-field equation. Recent whole-space uniqueness and phase-transition theorems are used as external inputs, while the finite-ball zero-curve ranges, endpoint asymptotics, and spectral consequences are proved here. The two models require different singular analyses: in the logarithmic problem the nonlinearity is not locally Lipschitz at a nodal zero and the linearized potential diverges there, whereas in the sublinear problem the limiting whole-space profile reaches a double zero at a finite support radius, beyond which continuation is nonunique. For every $R>0$ and $k\ge0$, the logarithmic problem has, up to sign, a unique radial Dirichlet solution with exactly $k$ interior zeros. Its shooting height $\beta_k^{\log}(R)$ is a strictly decreasing $C^1$ bijection from $(0,\infty)$ onto $(\alpha_k^{\log},\infty)$ and satisfies \[ \beta_k^{\log}(R)\longrightarrow\alpha_k^{\log} \quad(R\to\infty),\qquad \log\bigl(\beta_k^{\log}(R)^2\bigr) =\frac{\rho_{k+1}^2}{R^2}+\kappa_{k+1,n}+o(1) \quad(R\downarrow0), \] where $\kappa_{k+1,n}>0$ is given by an explicit Bessel integral. For the sublinear problem, the maximal simple-zero branch exists precisely for $R\in(\rho_{k+1},S_k)$, where $\rho_{k+1}$ is the $(k+1)$-st positive zero of the regular Bessel profile $\Phi_n$, and $S_k$ is the support radius of the unique compactly supported $k$-node whole-space bound state.

We introduce a fully–discrete variational approximation of Mather measures and sets for Tonelli Lagrangians on the flat torus, together with a numerical procedure for approximating the entire Mather set. The scheme is based on a fully–discrete Lax–Oleinik operator with integer winding labels. We prove an $O(\tau+h/\tau)$ error estimate for the critical value, convergence of critical solutions, and a finite-dimensional characterization of fully–discrete Mather measures. Accumulation points of the reconstructed minimizing measures are continuous Mather measures, while the supports satisfy complementary upper and lower convergence results involving the Mañé and Mather sets. To avoid the selection of only some minimizing components by exact discrete minimizers, we introduce a mass-threshold approximation based on almost-minimizing holonomic measures. This yields a finite-dimensional constrained optimization procedure designed to recover the whole Mather set.

We investigate the $k$-variation bounding property of the discrete-time Hankel operator, i.e., its invariance under the set of signals with a variation (number of sign changes) of at most $k$. Building on existing sign-consistency criteria, it is shown that this property is equivalent to the external positivity of $k+1$ explicitly realized linear discrete-time systems. Thus, making the property tractable via numerical and analytic certificates. We also derive dominant-pole restrictions for the case of $k=1$. A three-node thermal example illustrates the results and distinguishes variation bounding from variation diminishing.

We prove a structural theorem for point sets in $\mathbb R^2$ which determine few pinned angles. More precisely, we prove the existence of an absolute constant $c>0$ such that if $n$ is sufficiently large and $P$ is a set of $n$ points then there exists a point $q \in P$ which determines at least $n^{1+c}$ distinct angles to other pairs of points of $P$, provided that $P$ is not of one of the following exceptional forms: all but at most one of the points of $P$ lie on a line; all but two points of $P$ lie on a line, and the two exceptional points are symmetric with respect to the line; all the points of $P$ lie on a circle; all but one of the points of $P$ lie on a circle, and the exceptional point is the centre of the circle. As a consequence, we answer a question of Corrádi, Erdős and Hajnal by showing that if $n$ is sufficiently large and $P \subseteq \mathbb R^2$ has cardinality $n$ and is not contained on a single line, then $P$ determines at least $n-2$ angles. Moreover, we prove that the unique point set attaining this minimum is the regular $n$-gon.

We study the invasion dynamics of populations exhibiting positive density-dependent effects. We start with a single individual and consider a single-type birth and death process. The initial individual growth rate vanishes but it increases with the population density, proportionally to the number of individuals divided by a scaling parameter $K$. Before reaching the macroscopic scale~$K$, the population process is almost critical. We prove that the probability for the population to reach the macroscopic level $K$ decreases as $1/\sqrt{K}$ as $K$ goes to infinity. We also describe the associated trajectories and show that invasion can be split into three time periods. First, the process needs to escape from zero, and conditioning on survival, it grows linearly until the order $\sqrt{K}$. The scaled process is approximated by a diffusion, as for critical branching process, with an additional drift term coming from cooperation, which breaks the branching property. Second, in intermediate scale $\sqrt{K}$, we observe another diffusion, surviving with positive probability, without conditioning. Finally, beyond $\sqrt{K}$ scale, the process can be approximated by a classical macroscopic ODE limit. The proof of the first phase involves change of probability and characterization of uniform integrability of martingales, while the two other phases rely on uniform approximations on polynomial time scales.

We prove that the bound \[ \max \{ |16A|,|16f(A)| \} \gg_m |A|^{\frac{3}{2}+\frac{1}{162}} \] holds for any polynomial $f$ with degree $m \geq 2$ and any finite $A \subset \mathbb R$. This shows that the classical Jarník obstruction to growth beyond exponent $3/2$, which occurs for general strictly convex functions, cannot occur for polynomial functions.

The qualitative analysis of nonlinear dynamical systems relies on identifying geometric and topological structures that govern phase-space organization. We develop a computational framework based on novel topological and geometric descriptors of vector fields that enables robust comparison of dynamical regimes from both analytical models and sampled data. The framework comprises the Begin-End Point Embedding (BEPE), Density of Directions (DOD) and Euler characteristic-based summaries (Euler Characteristic Curve (ECC) and Euler Characteristic Profile (ECP)). BEPE captures local variations in vector field, while DoD characterizes the global organization of vector directions. ECCs and ECPs provide filtration-based topological summaries of vector-field structure, with ECPs extending these summaries to multiparameter settings. Together, these descriptors encode complementary geometric and topological information about flow organization. We formulate both continuous and sampled versions of the descriptors and establish their robustness with respect to perturbations, finite sampling and selected transformations. The effectiveness of the proposed approach is demonstrated on a collection of benchmark systems (the Hopf bifurcation, the FitzHugh-Nagumo model, the Lorenz system). Experiments on the discrete Hénon map demonstrate the applicability of the framework to displacement fields generated by discrete dynamical systems, while experiments on high-dimensional turbulent flows assess its scalability beyond low-dimensional phase spaces. Comparisons with classical approaches, including Conley-index-based methods and $L_p$-type metrics, show that the proposed descriptors provide complementary structural information while offering favorable robustness, interpretability and computational efficiency, establishing a practical connection between dynamical systems theory and topological data analysis.

We analyze a stochastic algorithm with Halpern-type anchoring for constrained convex-concave problems and monotone variational inequalities. This single-loop and single-call algorithm uses one unbiased sample of the gradient operator at every iteration, to be applicable to monotone games with noisy feedback. With $t$ denoting the iteration counter, we prove an anytime last-iterate convergence rate of $O(t^{-1/4})$ for both the gradient-mapping norm and restricted gap, bypassing the $O(t^{-1/5})$ constrained-anytime bottleneck in the literature. Specializing then to multi-point oracles, we use variance reduction to achieve the $O(t^{-1/2})$ rate with an anytime single-loop algorithm using $2$ samples per iteration. Our results allow constrained problems with a potentially unbounded feasible set; as well as a structured class of stochastic oracles whose variance need not be uniformly bounded.

Advanced artificial intelligence models think in tokens; contemporary communication systems carry bits. The direct way to bridge this gap is to transmit tokens, but that makes a model-specific representation part of the air interface, coupling the endpoints through a shared tokenizer, codebook, and often a neural transceiver. We take a different route: keep bits in the payload and let tokens control how those bits are generated and protected. The resulting token-bit interface transition aligns task-side tokens with source- and channel-coding units, translates token relevance into codec controls, and preserves the induced priority order across the coding chain. We instantiate it for image classification, where a vision transformer scores the task relevance of each image region from its attention maps: those scores steer block-wise JPEG rate allocation, then group the compressed bits for protection at different polar-code rates. The payload remains an explicit, reconstructable bitstream recovered by a correspondingly configured decoder. Over-the-air experiments on a software-defined radio testbed show improved accuracy–latency tradeoffs over separate source-channel coding, performance competitive with far more memory-intensive neural joint source-channel coding, and graceful degradation under channel mismatch. Token communication, then, need not transmit tokens explicitly; what it needs is an interface through which tokens determine how bits are communicated.

Conformal counterfactual prediction constructs prediction sets with finite-sample coverage guarantees for counterfactual outcomes and individual treatment effects under the no-interference assumption. In this work, we relax this assumption by allowing each unit's potential outcomes to depend on other units' treatments and covariates. In this setting, propensity-score reweighting does not restore weighted exchangeability, and existing methods may fail to achieve valid coverage. To address this issue, we develop interference-adjusted weighted conformal prediction that accounts for interference by constructing an observable upper bound on the ideal and unobserved conformal $p$-value under the target intervention. The resulting prediction sets provide finite-sample marginal coverage guarantees for counterfactual outcomes and individual treatment effects in both transductive and inductive settings. We also derive a sharper construction when intervention-induced changes in nonconformity scores are bounded. Numerical experiments show that our methods preserve nominal coverage, whereas existing methods may not.

At every prime $p$, we disprove the algebraic generating hypothesis for compact objects in Hovey's stable category of $BP_*BP$-comodules and its local analog at every positive height, answering questions of Barthel and Heard. In both settings, for every $r,n\geq1$, we construct a ghost whose first $n$ composition powers are all nonzero and have exact additive order $p^r$. The same construction applies to the stable category of $E_*E$-comodules for even $p$-local Landweber exact theories $E$ that are not rational. Through the motivic deformation of Gheorghe–Wang–Xu, these examples give $C\tau$-linear counterexamples to the cellular $\mathbb{C}$-motivic generating hypothesis, with the same order and composition properties. We also construct nonzero non-$C\tau$-linear ghosts between $C\tau$-modules and a nonzero ghost on a motivic spectrum with noncontractible Betti realization.

We construct, for every odd prime $p$, a bijection between the $p'$-spin characters $\widetilde S_n$ and those of a Sylow normalizer which, in particular, is equivariant under the Galois action detecting $p$-rationality. An analogous result holds for $\widetilde A_n$. We also obtain a blockwise refinement for height-zero spin characters and their Brauer correspondents.

We propose a dissipative, irreversible damage model for anisotropic media in large deformations to address the damage process of the myocardium following a cardiac infarction. We model damage to the collagen in the cleavage planes resulting from an increased load to the passive tissue. Starting from variational principles, we derive a quasi-static differential model governing the irreversible evolution of the damage. We apply our model to two test cases. First, a rectangular passive slab geometry to which an indenter-like load is applied. Then, we couple our model with a model for the numerical simulation of left ventricular electromechanics. Our results show that a redistribution of the load within a left ventricle occurring following a cardiac infarction can produce the damage to the collagen scaffold in the infarcted area consistent with experimental results. This is indicative of the fact that passive load-dependent dissipative phenomena are implicated in post-infarction remodelling.

We study nonlinear Landau damping for the Vlasov–Yukawa equation in the large-box regime, as a periodic box of length $2\pi L$ expands to the whole space. First, for $d\ge 3$, we prove global stability and scattering in an $L$-dependent Gevrey class, with estimates uniform for $L\ge L_0$. The density obeys a two-regime upper bound: the whole-space decay rate $\langle t\rangle^{-d}$ before the critical time scale $T_{\rm disp}(L)\sim L$, and a periodic phase-mixing bound in the rescaled time $t/L$ thereafter. Second, to quantify the nonlinear resonance effect, we identify an echo-Volterra operator governing the nonlinear density memory, and provide sharp estimates for its non-collinear and collinear parts. The non-collinear time-frequency interactions remain uniformly bounded, whereas the collinear interactions do not become comparable to the background until the much later time scale $T_{\rm col}(L)\sim L^{d-1}$. Third, we investigate the stability and convergence of Sobolev initial data. For data of size $\varepsilon$, polynomial Sobolev smallness persists up to the time scale $\varepsilon^{-1} T_{\rm col}(L)$. Within the shorter time scale $T_{\rm disp}(L)\sim L$, compatible periodic solutions converge locally, as $L\to\infty$, to a global solution of the whole-space Vlasov–Yukawa equation.

Quasiparabolic sets extend parabolic coset spaces while retaining a length filtration and natural Hecke algebra deformations. Gelfand models built from induced linear characters ask when such spaces can account for every irreducible representation exactly once. We classify, up to equality of the individual induced characters, all quasiparabolic Gelfand models for finite irreducible Coxeter groups with inducing characters restricted from their ambient parabolic subgroups, including the additional models of type \(D_{4r+2}\) and \(B_3\). We derive the associated finite Gelfand-pair and commutant consequences, construct canonical Hecke structures for the additional classical models.

Let $B(\mathcal H)$ be the $*$-algebra of all bounded linear operators on a Hilbert space $\mathcal H$. A classical result due to Stamplfi shows that, for any $a\in B(\mathcal H)$, the norm of the inner derivation $\delta_a=[a,\cdot]$ on $B(\mathcal H)$ is given by $\|\delta_a\|_{B(\mathcal H)\to B(\mathcal H)}=2\inf\{ \|a-c{\bf 1}\|_{B(\mathcal H)}:c\in \mathbb{C}\}$. In the present paper, we provide a formula for the norm of a generalized derivation implemented by self-adjoint operators from an ideal in $B(\mathcal H)$, which partially answers a question posed by Fialkow and Loebl in the 1980s. We also consider this question in a more general setting of a symmetrically normed (not necessarily complete or separable) space affiliated with a properly infinite semifinite von Neumann algebra.

Inspired by Grothendieck's foundational work, we establish that the sequence of functors $(Ext^n_{pBan}(E,-))_{n \ge 0}$ on the category of $p$-Banach spaces ($0 < p < 1$) forms a universal $\delta$-functor. To this end, for any pair of $p$-Banach spaces $E$ and $X$, we construct a $p$-Banach space $\mathcal N_E^X$ that renders each functor $Ext^n_{pBan}(E,-)$ effaceable. Furthermore, we show that this space serves as a relative injective object for $E$ and $X$, enabling the construction of relative injective presentations and proving that the functors $Ext^n_{pBan}(E,-)$ are the right derived functors of $\mathcal{L}(E,-)$. We will also examine the situation in the category of quasi-Banach spaces.

In this paper we propose augmented Lagrangian finite element methods for cavitation in the Reynolds and Stokes models of lubrication, for which the discrete complementarity conditions hold exactly and the augmentation parameter drops out of the method. For the Reynolds equation this follows from nodal quadrature in a mixed piecewise linear method, for which we prove the classical first-order error estimate. We also determine the computable stability threshold of a multiplier-free stabilised alternative. For Stokes flow we show that the constrained scalar is the mechanical pressure when the deviatoric stress, with zero bulk viscosity, is used, but not with the customary incompressible stress. We discretise with a jump-stabilised Crouzeix-Raviart element and piecewise constant pressure, and prove well-posedness, stability in two and three dimensions, and a first-order error estimate. We present numerical examples which verify these properties and show that close pressure profiles in the two models do not imply close cavity predictions, the latter also being sensitive to the end conditions of the computational domain.

Let $H$ be a fixed $d$-regular graph with $d\ge2$, and let $t(H,\cdot)$ denote its homomorphism density. We study the upper-tail event $t(H,G(n,p))\ge r^{|E(H)|}$ for fixed $0<p<r<1$ in a dense Erdős–Rényi random graph $G(n,p)$. Near the Lubetzky–Zhao replica-symmetric phase boundary and away from the exceptional target density $(d-1)/d$, we prove that the optimizer of the Chatterjee–Varadhan variational problem on the symmetry-breaking side is bipodal (two-block) and unique up to relabeling. Its block parameters depend analytically on $(p,r)$. To treat the exceptional boundary point, where the nonexceptional theory degenerates, we construct an analytic curve approaching that point from the symmetry-breaking side along which the unique optimizers are nonconstant rank-one bipodal graphons. In both settings, we derive asymptotic expansions of the edge-density deficit and the rate function that governs the exponential decay of the upper-tail probability. Moreover, the conditioned random graph converges in cut distance to the corresponding bipodal optimizer as $n\to\infty$. As $(p,r)$ approaches the phase boundary from the symmetry-breaking side, the optimizers converge to their constant limits through two distinct mechanisms. For each fixed nonexceptional target density, one block shrinks to zero measure, giving convergence in $L^1$ but not in $L^\infty$. Along the exceptional curve, both blocks remain macroscopic: their sizes tend to $1/2$ and all three block densities tend to $(d-1)/d$, yielding convergence in $L^\infty$.

The logic FOT is a team-based logic whose expressive power coincides with first-order logic at the level of both sentences and open formulas. In contrast to dependence and independence logics, which can define stronger second-order team properties, FOT is designed to capture exactly elementary team properties, modulo the empty team. In this paper we consider two modifications of FOT. First, we investigate essentially the inclusion atom free fragment of FOT. Our main result establishes quantifier elimination for the fragment in the empty signature. Second, we study an extension of FOT by the intuitionistic implication. The main conclusion is that adding this single connective increases the expressive strength so that every second-order sentence can be encoded by an open formula evaluated on the full team. Consequently, validity of formulas is equivalent to validity of full second-order logic.

For the Jordan quiver, we construct a radial quantum trace morphism from the quantum preprojective algebra to the algebra of invariant differential operators on the Cartan subalgebra. Its classical limit recovers the classical trace morphism associated with the commuting quotient, and we prove that both the quantum and classical trace morphisms are surjective. We further relate noncommutative quantizations of the Jordan preprojective algebra to the spherical rational Cherednik algebra of type A. Finally, we derive explicit formulas for the Harish–Chandra homomorphism and its radial part in power-sum coordinates.

Fraction-based information measures have received considerable attention for describing complex systems, as they enable the investigation of signals with high sensitivity \citep{machado2014fractional}. In this paper, we introduce quantile versions of fractional cumulative past inaccuracy (FCPI) and dynamic fractional cumulative past inaccuracy (DFCPI) measures based on the inverse Mittag-Leffler function (MLF) or fractional logarithm function, which are the extensions of the fractional cumulative past and dynamic fractional cumulative past entropies, respectively. The various properties of quantile-based FCPI and its dynamic version are provided. We also propose a nonparametric estimator for the proposed measure, and simulation studies are carried out for validation. Finally, we bring out real data application of the newly introduced quantile-based FCPI.

The estimates of intermediate derivative play important role in the theory of partial differential equations. The modern approach to this problem is via pointwise estimates, since this allows to obtain results for various function norms. The paper aims to extend the pointwise estimate for intermediate derivative in terms of the averaging operator for higher orders derivatives.

Beck and Pixton expressed the Ehrhart polynomial of the Birkhoff polytope as a weighted sum of constant terms of several multivariate rational functions. Xin and Zhang studied a class of constant terms $h_n(t)$, which can be regarded as a variation of the Morris constant term. They proved that $h_n(t)$ is a polynomial of degree $(n-1)^2$ and obtained many nice properties involving the Morris constant term identity. Let $h_n^*(y)=(1-y)^{(n-1)^2+1}\sum_{t\geq0}h_n(t)y^t$. For fixed $n\geq 3$, we obtain the following results for $h_n^*(y)$: (i): $h_n^*(y)$ is a polynomial with positive integer coefficients. (ii): $h_n^*(y)$ is real-rooted. In particular, all its roots are non-positive real numbers. (iii): $h_n^*(y)$ is Gamma-positive. Furthermore, $h_n^*(y)$ is palindromic, unimodal, and ultra log-concave. This confirms Xin and Zhang's conjecture regarding $h_n^*(y)$. In order to resolve this conjecture, we also developed an operator on the space of symmetric polynomials that preserves real stability. As a byproduct, we prove that every root of a Gamma polynomial associated with $h_n^*(y)$ is a negative real number.

Exceptional points provide a mechanism for producing strong spectral sensitivity in non-Hermitian wave systems. For subwavelength resonators with highly dispersive material parameters, however, the resonance problem is not a linear eigenvalue problem in the frequency. In this paper, we use the Lippmann-Schwinger formulation and the associated volume-potential reduction to study exceptional points in dispersive resonator arrays. The reduction leads to a finite-dimensional nonlinear matrix pencil whose entries encode both the local material response and the volume-mediated interaction between the resonators. We characterize exceptional points as defective characteristic values of this pencil and derive explicit conditions for dimers and higher-order systems. We show that passive damping and detuning can generate exceptional points without imposed gain, and we formulate an inverse-design condition for placing such degeneracies at prescribed frequencies. We also prove the corresponding Puiseux splitting laws and identify how material dispersion enters the leading sensitivity prefactor. Numerical experiments, including halide-perovskite-inspired dimers and trimers, illustrate the theory and show that the exceptional-point mechanism persists for explicit highly dispersive passive material laws.