Papers

In this paper we establish removable singularities results for Einstein metrics and for metrics with Ricci curvature bounded below. Let $n\geq 2$. On a closed $n$-manifold, we show that an $L^\infty$-Riemannian metric whose Ricci curvature is bounded below outside a singular set of codimension $> 3- \frac{1}{n-1}$ canonically extends to an $\mathrm{RCD}$ space. As a consequence, using a new removable singularity theorem for Einstein metrics, we prove that in dimension $4$ any Einstein metric with $L^\infty$ singularities of codimension $>3-\frac{1}{3}$ extends smoothly across the singular set, possibly after changing the smooth structure. In higher dimensions, we construct a $C^{1,\alpha}$-Riemannian manifold structure on the regular set of a non-collapsed $\mathrm{RCD}$ space that is a Riemannian manifold with bounded $|\mathrm{Ric}|$ outside a set of codimension $>2$. Our results can be used to give a proof of Schoen's conjecture on scalar curvature singularities for metrics that are either continuous, or $L^\infty$ and sufficiently close to a smooth background metric.

The classical Heisenberg Pauli Weyl (HPW) inequality exhibits a strong rigidity phenomenon on Riemannian manifolds i.e. on Cartan Hadamard manifolds and those with non-negative Ricci curvature, the validity of the Euclidean HPW inequality or the existence of extremizers strictly forces the manifold to be isometric to Euclidean space, $\mathbb{R}^n$. This geometric discrepancy motivates the study of curvature dependent corrections and their associated stability properties. In this article, we investigate the geometric stability of the HPW inequality. Specifically, given a sequence of pointed Riemannian manifolds and appropriately normalized functions with a vanishing HPW deficit, we address whether the sequence converges to the corresponding model space in the pointed Gromov Hausdorff topology. We prove that for pinched Cartan Hadamard manifolds with sectional curvature bounded above by $c < 0$, the manifolds converge to the model hyperbolic space $\mathbb{H}^n_c$. In the non-negative Ricci curvature setting, we establish convergence to $\mathbb{R}^n$, up to a metric rescaling factor governed by the sequence's moment term. Consequently, we deduce that the known HPW inequality formulated via the asymptotic volume ratio (AVR) is suboptimal for non negatively Ricci curved manifolds not isometric to Euclidean space. To resolve this, we introduce a curvature corrected HPW inequality for this setting, analogous to the Cartan Hadamard case. Finally, we establish quantitative rigidity estimates in both curvature regimes, demonstrating that the HPW deficit when evaluated at Gaussian profiles which explicitly controls an appropriately defined distance to the respective model space.

We introduce a new invariant associated to an action of a group $G$ on a (possibly non-commutative) group $K$. The invariant is based on the $G$-parametrized LS-category of the semi-direct product $K\rtimes G$ but it also depends on a choice of a derivation $d\colon G\to K$. The invariant is thus denoted $\operatorname{tc}(d)$ and it takes integer values between $\operatorname{cd}(K)$ and $\operatorname{cd}(K)+\operatorname{cd}(G)$ (where $\operatorname{cd}(G)$ stands for the cohomological dimension of a group). We develop a general theory and then compute its values for a number of important examples, including fundamental groups of mapping tori, of pure braid groups and of iterated semi-direct products of free groups. To achieve this, we relate $\operatorname{tc}(d)$ to the cohomological dimension of its factors, to the sectional category of the inclusion of $G$ in $K\rtimes G$, and to the properties of torsors that are associated to the derivation $d$.

We introduce a framework for establishing Blaschke-Santaló-type inequalities on $m$-tuples of Polish measure spaces coupled together by a continuous cost function. Central to our approach is a transference principle, which provides a mechanism to lift geometric weighted inequalities involving cost-polar sets into functional integral inequalities of Santaló-type. We call these equivalent inequalities cost-Santaló inequalities. This definition expands and includes previous notions in the literature. We apply this principle to deduce several new versions of functional Santaló inequalities, including on the space of rectangular matrices and a functional sine Santaló inequality. A surprising development is that probability spaces with log-concave isoperimetric functions fit into our framework, for example, Gauss space and spherical space, leading to new functional Santaló inequalities in these settings. In particular, we obtain results for $\operatorname{RCD}(K,\infty)$ spaces. As a discrete application, we obtain an inequality for the Hamming cube. Finally, we explore applications to optimal transport, utilizing our functional framework to establish generalized transport-entropy inequalities on arbitrary Polish spaces satisfying a cost-Santaló inequality, which we explicitly instantiate for matrix spaces.

In this paper, we construct confluence relations for $q$-analogues of multiple zeta values by adapting the classical construction to the $q$-setting. Our construction uses $q$-analogues of multiple polylogarithms depending on an auxiliary variable $z$, functional relations arising from their $q$-differential equations, and the (regularized) limit as $z$ tends to $1$. We prove that the resulting family of relations contains the stuffle product relations and, assuming a certain conjectural identity, the duality relations for both the Bradley–Zhao and Schlesinger–Zudilin models.

We consider three specific $q$-deformed matrix models (the Chern-Simons, $q$-Laguerre, and $q$-Gaussian matrix models) which can be solved explicitly using the property of superintegrability. We show that one- and two-point functions of inverse characteristic polynomials can be interpreted as quantizations of Lagrangian subvarieties in $(\mathbb{C}^*)^2$ and $(\mathbb{C}^*)^4$, respectively. While such a geometric picture is expected for the Chern-Simons matrix model, our results for the $q$-Laguerre and $q$-Gaussian models are new and exhibit additional features. In particular, specific anti-symplectic birational involutions play an essential role in the construction. This reformulation provides a geometric framework for analyzing the semiclassical, large-$N$ expansion of matrix-model correlators. This picture is suggestive of the geometry underlying open topological strings, although the two constructions are not identical and the precise relation between them remains to be understood.

Let $\mathbb{Z}_p$ be the ring of $p$-adic integers with respect to a prime $p$ and let $d$ be a positive integer. For each $\mathbf{z}=(z_1, z_2,..., z_d) \in \mathbb{Z}_p^d$, let $T_{\mathbf{z}}: \mathbb{Z}^d \times \mathbb{Z}_p \to \mathbb{Z}_p$ be an adding-type $\mathbb{Z}^d$-action on $\mathbb{Z}_p$ defined by $T^{\mathbf{n}}_{\mathbf{z}}(x):=x+ \sum_{i=1}^d n_i z_i$ for $\mathbf{n}=(n_1,n_2, \cdots, n_d) \in \mathbb{Z}^d$ and $x \in \mathbb{Z}_p$. Under some mild assumptions on $\mathbf{z}$, the action $T_{\mathbf{z}}$ is a free $\mathbb{Z}^d$-odometer (by odometer, we mean a minimal and equicontinuous action on a Cantor space). In this paper, we derive a necessary condition for continuous orbit equivalence between such $\mathbb{Z}^d$-odometers by constructing algebraic models for them. We then study the free minimal constant speedups of these $\mathbb{Z}^d$-odometers. It turns out that such a speedup of $T_{\mathbf{z}}$ is again an adding-type $p$-adic $\mathbb{Z}^d$-odometer $T_{\mathbf{w}}$ for some $\mathbf{w} \in \mathbb{Z}_p^d$. However, the necessary condition above may not hold for the speedup. This provides the first known examples of free minimal bounded speedups (of free $\mathbb{Z}^d$-odometers) which are not continuously orbit equivalent to the original ones and hence disproves a conjecture by Johnson and McClendon. Our result also indicates that continuous orbit equivalence is a rare phenomenon for free minimal constant speedups of $p$-adic $\mathbb{Z}^d$-odometers of adding type when $d \geqslant 2$.

We prove a uniform spectral-gap estimate at every even exponent for isometric representations on arbitrary closed subspaces of real $L_p$-spaces. As an application, this result, combined with the work of Bader, Furman, Gelander and Monod, gives an affirmative answer to their Question 4.1: Kazhdan's property $(T)$ implies property $(T_X)$ whenever $X$ is a closed subspace or a quotient of a real $L_p$-space, for every $1<p<\infty$.

Global representations are subdirect representations satisfying a sheaf-like local-to-global patching principle. We develop a unified and simplified theory of such representations. For quasivarieties with a near-unanimity term, this framework recovers the main classical representation theorems through a common argument and yields a general representation by factors of bounded subdirect width. When the relatively subdirectly irreducible members form a universal class, we obtain an optimal result: every relatively congruence-distributive algebra admits a global representation by relatively globally indecomposable factors, which we characterize explicitly. We also provide a converse: global representations by factors of bounded relative subdirect width force the existence of a near-unanimity term. Adding semisimplicity to the hypotheses of the main theorem yields optimal representation results for filtral quasivarieties and dual discriminator varieties, together with a simple description of the globally indecomposable algebras. The technical engine behind these results is a new infinitary extension of the congruence-system component of the Baker–Pixley theorem.

Motivated by mirror symmetry, we study exponential sums over the $\mathbb{F}_q$-points of the $A_1$ Coulomb branch. We show that their fiberwise structure is governed by polynomial Gauss sums, providing a geometric realization of Selberg character sums studied by Evans. Using Evans's evaluation, we obtain explicit formulas for the Coulomb-branch exponential sums as products of classical Gauss sums.

We prove that any finite time collapsing Kähler Ricci flow on ruled surfaces develops a Type I singularity, such singularity is modeled on the standard product shrinker $\mathbb{P}^1\times \mathbb{C}$. As an application, we obtain the optimal collapse rate of fibers on ruled surfaces.

It is well known that the Young lattice is the Bratelli diagram of the symmetric groups, expressing how irreducible representations restrict from $\mathfrak{S}_{N}$ to $\mathfrak{S}_{N-1}$. In 1975, Stanley discovered a similar lattice called the Young-Fibonacci lattice which was identified as the Bratelli diagram of a family of algebras $\{\mathbf{O}_N(X,Y)\}_{N \geq 0}$ by Okada in 1994. In this paper, we first realize the Okada algebra $\mathbf{O}_N(X,Y)$ and the associated monoid $\mathbf{O}_N$ using a labelled version of non-crossing arc-diagrams appearing in the description of the Temperley-Lieb algebra and Jones monoid. We establish, for general parameters $(X,Y)$, that the dimension of the Okada algebra $\mathbf{O}_N(X,Y)$ is $N!$, noting that Okada proved this result only in the semisimple case. We interpret a natural bijection between permutations and labelled arc-diagrams as an incarnation of Fomin's version of the Robinson-Schensted correspondence associated to the Young-Fibonacci lattice. The arc-diagram formalism allow us to probe the structure of the Okada monoid and algebra. In particular we prove that the Okada monoid is a regular, aperiodic $*$-monoid and we describe its Green relations and order. These results allow us to construct a cellular basis of the Okada algebra and to show that $\{\mathbf{O}_N(X,Y)\}_{N \geq 0}$ forms a coherent tower of cellular algebras in the sense of Goodman and Graber. We present some conjectures expressing the Gram determinant of the invariant bilinear form attached to each cell module in terms of Okada's clone Schur functions. We conclude the paper by presenting two follow-up, ongoing projects along with a series of questions pushing further the analogy between the symmetric groups and the Okada algebras.

For every $n$, we construct a 2-dimensional $n$-element poset whose Hasse diagram has independence number $n\exp(-\Omega(\sqrt{\log n}))$, and consequently, chromatic number $\exp(\Omega(\sqrt{\log n}))$. This also yields an $n$-point planar set whose box-Delaunay graph satisfies the same bounds.

We prove that semi-strong convergence to a Bose–Einstein equilibrium below the critical temperature implies finite-time Bose–Einstein condensation (BEC) for conservative isotropic measure solutions of the spatially homogeneous quantum Boltzmann equation with low-momentum scattering degeneracy exponent $0\leq\eta<1$. For every admissible initial measure with $\overline T/\overline T_c<1$, there exists a conservative solution that has a positive zero-energy atom after a finite time without any assumption of initial concentration near zero energy. We also show that it persists and converges to the equilibrium condensate mass, while the full solution converges strongly. The proof introduces a method to obtain condensation from relaxation estimate to Bose–Einstein equilibrium. The relaxation estimate first gives a fixed amount of mass below a small energy level. We then use collision estimates at smaller and smaller energy scales and add their effects in a weighted sum functional with a fixed upper bound. If no atom forms at zero energy, enough mass remains in the positive-energy shells to force this sum to exceed its upper bound in finite time, which leads to condensation in finite time.

Reliable epidemic monitoring often requires inferring regional infection burden and transmission heterogeneity from noisy, spatially aggregated, and potentially sparse surveillance data. Agent-based models (ABMs) are attractive for this task because they represent individual behavior, contact heterogeneity, and localized interventions, but these same features make them difficult to calibrate online. We develop a generative AI-based data-assimilation (GenDA) framework for partially observed epidemic ABMs that estimates both the epidemic state and a heterogeneous parameter field while respecting the gap between observable macrostates and latent agent-level microstates. GenDA combines a training-free, score-based generative update for macrostate correction with a direct parameter update based on macrostate discrepancies, followed by a macro-micro reassignment step that restores consistency with the ABM. In controlled and geographically explicit synthetic experiments, the framework recovers regional epidemic burden, dominant hotspot structures, and effective transmission heterogeneity from aggregated observations, while improving post-assimilation forecasts relative to state-only assimilation.

In this paper, we establish a homological existence criterion for the discrete $L_p$ Minkowski problem for all $p<0$, where $p=-n$ corresponds to the celebrated centro-affine case. For discrete measures with antipodal support, we obtain a complete existence characterization for arbitrary positive masses. In the even case, a solution can be chosen origin-symmetric.

We consider convex optimization with nonlinear inequality constraints and develop a primal-dual multiplier framework that is consistent in continuous and discrete time. We first propose continuous-time dynamics with Nesterov-type vanishing damping $\alpha/t$, together with compatible extrapolations of the dual variable and the nonlinear constraint mapping. Under convexity assumptions and $\alpha\geq3$, we establish $\mathcal O(t^{-2})$ convergence rates for both nonlinear feasibility and the objective residual. In the noncritical regime $\alpha>3$, with an admissible choice of the extrapolation parameter, we further prove that the entire primal-dual trajectory converges to a KKT pair and sharpen both continuous-time estimates to $o(t^{-2})$. We then derive an inexact accelerated primal-dual algorithm through a compatible discretization of a perturbed version of the dynamic. For composite convex objectives, a weighted summability condition on the primal inexactness yields $\mathcal O(k^{-2})$ rates for feasibility and the objective residual. In the corresponding noncritical regime, the discrete primal-dual sequence converges to a KKT pair and both residual estimates improve to $o(k^{-2})$. Thus the continuous and discrete results exhibit matching accelerated rates and matching asymptotic improvements. To the best of our knowledge, this is the first Nesterov-type primal-dual multiplier framework for convex optimization with nonlinear inequality constraints.

We study the rational canonical form of sparse random matrices over a finite field. Suppose $A_n\in\mathrm{Mat}_n(\mathbb{F}_p)$ has independent and $\alpha_n$-balanced entries. We prove that if $n\alpha_n-\log n\to+\infty$, then, for every fixed collection of distinct monic irreducible polynomials over $\mathbb{F}_p$, the corresponding primary partitions of $A_n$ converge jointly to the same asymptotically independent Cohen-Lenstra distributions as in the uniform random matrix model. This strengthens the finite-field cokernel universality theorem of Lee (arXiv:2511.13070), which assumes $\alpha_n=c\log n/n$ for a fixed $c>1$, by resolving the additive endpoint case explicitly left open in his work. Moreover, our result extends universality from ordinary cokernels to the full rational canonical form. The condition is optimal for universality uniformly over fixed irreducible polynomials. Our proof uses the surjection moment method developed by Sawin and Wood (arXiv:2210.06279) over $\mathbb{F}_p[t]$, together with a refinement of Lee's Fourier estimates that retains the full leading-order decay. We also construct degree-$d$ critical sparse obstructions, suggesting a polynomial-dependent threshold of $1/d$ for irreducible polynomials of minimal degree $d$.

The set of the triangulations with vertex set $X$ of a simple polygon $\mathrm{P}$ can be structured into a flip-graph $\mathcal{F}(\mathrm{P},X)$ whose edges connect two triangulations that differ by a single arc. The geometry of flip-graphs has been thoroughly studied and it is known that the subgraph $\mathcal{F}_\varepsilon(\mathrm{P},X)$ induced by the triangulations that contain a given arc $\varepsilon$ is strongly convex in $\mathcal{F}(\mathrm{P},X)$ when $\mathrm{P}$ is convex and $X$ contains no puncture (points in the interior of $\mathrm{P}$) and at most one flat vertex (points in the interior of an edge). When $X$ contains at least two punctures or flat vertices, it is also known that this strong convexity property fails. Here, we close the last open case by showing that, for any convex polygon with sufficiently many vertices, one can always place a single puncture in $X$ in such a way that $\mathcal{F}_\varepsilon(\mathrm{P},X)$ is not strongly convex in $\mathcal{F}(\mathrm{P},X)$. We prove a similar result for simple polygons with a single reflex vertex. The main ingredients in our proofs are a decomposition lemma for a class of $3$-dimensional triangulations and a hyperbolic volume argument regarding their embedding into $\mathbb{H}^3$.

In this paper, we study the fundamental solution of the mixed local-nonlocal operator $ -\Delta+(-\Delta)^s,$ $ 0<s<1. $ We derive precise asymptotic expansions of the fundamental solution and its gradient, up to the first nontrivial correction term, both near the origin and at infinity. The expansions describe the transition between the local and nonlocal diffusion scales and include the critical regimes in which logarithmic terms occur. As applications of these asymptotics, we establish a distributional Bôcher-type theorem for nonnegative supersolutions with an isolated singularity. We further obtain quantitative positive and antisymmetric maximum principles on punctured balls.

Let $X$ be a compact connected Riemann surface and $S\,\subset\, X$ a finite subset. We consider parabolic principal $G$–bundles $\mathcal{E}_{G}$ on $X$ with parabolic structure on $S$, where $G$ is a connected complex reductive affine algebraic group. Let $P\, \subset\, G$ be a parabolic subgroup and $\mathcal{E}_{P}\, \subset\, \mathcal{E}_{G}$ a reduction of structure group of $\mathcal{E}_{G}$ to $P$. We give a criterion for the existence of a parabolic Lie algebroid connection on $\mathcal{E}_{P}$ for any given parabolic Lie algebroid on $(X,\,S)$ whose anchor map is not surjective. More precisely, $\mathcal{E}_{P}$ admits a parabolic Lie algebroid connection if the reduction $\mathcal{E}_{P}\, \subset\, \mathcal{E}_{G}$ is parabolically infinitesimally rigid. In particular, the Harder–Narasimhan reduction of $\mathcal{E}_{G}$ admits a parabolic Lie algebroid connection.

We develop a theory of first order differential calculi in braided monoidal categories and classify braided covariant and bicovariant calculi on braided Hopf algebras. We show that, under certain conditions, bicovariant calculi can be transmuted to braided bicovariant calculi. For Radford–Majid biproducts, we combine bicovariant calculi on a Hopf algebra and braided bicovariant calculi on the corresponding braided Hopf algebra to covariant smash product calculi. The associated Maurer–Cartan form is shown to decompose into a direct sum of the Maurer–Cartan forms of the structure Hopf algebra of the quantum principal bundle and the braided Hopf algebra on the base. Geometrically, this construction realises the quantum affine extension of a given Hopf algebra, and we prove that the resulting quantum principal bundle is equipped with a frame resolution induced by the quantum Maurer–Cartan form. Building on this correspondence, we introduce and develop the notion of quantum $\textrm{G}$-structure, proving that quantum $\textrm{G}$-structures are quantum frame resolutions on the reduction. The theory is illustrated by examples based on transmutations of higher analogues of Sweedler's Hopf algebra and on the braided quantum plane, seen as a Yetter–Drinfeld module of $O_q(\mathrm{GL}_2)$.

We prove the Quadratic Easy Coefficients Conjecture stated as Conjecture 1 in T. W. Cusick, Recursions for quadratic rotation symmetric functions weights, Discrete Applied Mathematics 378 (2026), 93–101. For an arbitrary finite sum of quadratic monomial rotation symmetric Boolean functions, we identify the recurrent part of the rules matrix with a signed binary de Bruijn transfer matrix $B$. We then give a one-step presentation of the finite-type shift associated with the Boolean function in the symbolic-dynamics construction of Chirvasitu and Cusick. Fourier transformation in an auxiliary $\F_2$ coordinate decomposes the adjacency matrix of this shift into an unsigned de Bruijn block and the signed block $B$. Consequently the dynamical zeta function is \[ \zeta_{X_f}(z)=\frac{1}{\det(I-z\cR(f))}, \] where $\cR(f)$ is the rules matrix. This equality identifies, with their algebraic multiplicities, the characteristic values, supplied by symbolic dynamics, with the roots of the characteristic polynomial of the rules matrix. The desired easy coefficients formula follows from the trace of $B^n$. We also prove nonsingularity and justify the unique backward extension of the weight recurrence.

We consider damping optimization for vibrating systems described by a second-order differential equation. The goal is to determine the damping positions and viscosity values of the dampers in the system so that they produce the lowest average total energy. We propose a new framework for determining the number and positions of the dampers, using re-weighted $l_1$ minimization and pruning techniques. The efficiency and performance of our new approach are verified and illustrated in several numerical examples.

We show that any birationally rigid well-formed and quasismooth Fano 3-fold weighted complete intersection is K-stable.

Persistence diagrams summarize the multiscale topological structure of data, and in applications they often arrive paired with covariates. We develop nonparametric methodology and theory for estimating the expected weighted persistence diagram conditional on a Euclidean covariate. Representing each weighted diagram as a finite random measure on a compact window, we take the density of its conditional expectation as the regression target, the conditional weighted persistence intensity. For a conditional double-kernel estimator we establish finite-sample sup-norm rates with a matching minimax lower bound, uniform rates in partial optimal transport, and an unbiased-risk cross-validation criterion for bandwidth selection. Simulations with analytically known intensities corroborate the theory and show that cross-validation selects the oracle candidate bandwidth in the exact-intensity design. The method is illustrated by studying how radial geometry in cerebral artery trees varies with age.

On the once-punctured torus, we give an exact formula for the number of curves in any given mapping class group orbit of given word length. This settles a conjecture of Chas in [Cha16].

We investigate the effect of the clasp move of Legendrian submanifolds of dimension at least two on the moduli spaces of pseudo-holomorphic curves that contribute to the differential of the Chekanov-Eliashberg algebra. As an application, we compute the differential of Chekanov-Eliashberg algebra of twist spheres $\Lambda_k$ of dimension at least two. Moreover, we construct an infinite family of Legendrians $\mathcal{T}(\Lambda,k)$ from any horizontally displaceable Legendrian $\Lambda$ in $P\times\mathbb{R}$, such that the Chekanov-Eliashberg algebra of $\mathcal{T}(\Lambda,k)$ admits an augmentation, while $\mathcal{T}(\Lambda,k)$ has no exact embedded Lagrangian filling of vanishing Maslov class in the symplectization of $P\times \mathbb{R}$ for sufficiently large $k$.

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d.$ We introduce and study the notion of the generalized type of $A$ with respect to an $\mathfrak{m}$-primary ideal $I$ denoted by $\operatorname{type}_I(A).$ Let $e_i(I)$ denote $i$th Hilbert coefficients of $A$ w.r.t. $I$. Assuming $I$ is integrally closed and $e_{2}(I)=e_{1}(I)-e_{0}(I)+\lambda(A/I) \neq 0$, we establish a sharp lower bound for $\operatorname{type}_I(A)$ in terms of the multiplicity and certain lengths associated to $I.$ We further show that when this lower bound is attained, the associated graded ring $G(I)$, is Cohen Macaulay. In the case of Buchsbaum local rings of dimension $d$ and depth at least $d-1$, we obtain an optimal lower bound for $e_{2}(\mathfrak{m})$ using the technique of $S_{2}$-fication. Additionally, for an integrally closed $\mathfrak{m}$-primary ideal $I,$ we also study the second extremal case $e_{2}(I)=e_{1}(I)-e_{0}(I)+\lambda(A/I)+1$ and its consequences on $G(I).$ We also investigate bounds on $e_3(I)$ and for $d=3,$ we study the consequences when these bounds are attained for.

We investigate the cohomology of line bundles on the incidence correspondence, the partial flag variety parametrizing pairs consisting of a point in projective space and a hyperplane containing it. In characteristic zero, this cohomology is governed by the Borel-Weil-Bott theorem. In characteristic p>0, however, it becomes considerably subtler, and admits an equivalent reformulation in terms of cohomology tables for divided powers of the cotangent bundle on projective space. Our approach to the problem involves passing to infinitesimal thickenings of the incidence correspondence inside the ambient product of projective spaces. This leads to recursive formulas for the cohomology, generalizing earlier work of Donkin, of Liu, and of Gao-Raicu. We obtain generating functions for cohomology characters, expressed using truncated Schur polynomials and symmetric polynomials encoding the higher structure constants of the Verlinde algebras of SU(2) at levels (p-2) and (2p-2). Along the way, we exploit two important connections with multiplication in the graded Green-Han-Monsky representation ring: one relates this ring to cohomology, and another connects to the Verlinde algebras through the work of Coulembier-Etingof-Ostrik.