Papers

In this short note we prove that, given a $\mathbb{C}$-elliptic operator $\mathcal{A}$ and a map $u\in \mathrm{BV}^{\mathcal{A}}(\Omega;V)$ satisfying \( \nabla_{\mathrm{ap}}u\in L^1(\Omega;V\otimes\mathbb{R}^d), \) then $u\in \mathrm{BV}(\Omega;V)$. The result is quantitative and follows from the Korn-type estimate \[ |Du|(\Omega) \leq C_{d,\mathcal{A}}\left( \|\nabla_{\mathrm{ap}}u\|_{\mathrm{L}^1(\Omega)} + |\mathcal{A}u|(\Omega) \right), \] valid for every such $u\in \mathrm{BV}^{\mathcal{A}}(\Omega;V)$. As a consequence, we obtain the characterization \[ \mathrm{BV}^{\mathcal{A}}(\Omega;V)\setminus \mathrm{BV}(\Omega;V) = \left\{ u\in \mathrm{BV}^{\mathcal{A}}(\Omega;V): \nabla_{\mathrm{ap}}u\notin L^1(\Omega;V\otimes\mathbb{R}^d) \right\}. \] Generative AI has been exploited. The usage is detailed in a specific Section.

For the canonical stack $\mathcal{X}$ of a log del Pezzo surface with cyclic quotient singularities, we compute its categorical genus as \[ g_{\mathrm{cat}}(\mathcal{X})= 1+\frac{1}{2}\sum_j w_j(\ell_j-1), \] where $w_j=\gcd(n_j,q_j+1)$ and $\ell_j=n_j/w_j$ are the local widths and Gorenstein indices of the singularities $\frac{1}{n_j}(1,q_j)$. Serre duality and inertial Riemann–Roch separate the smooth contribution from the local canonical characters, and identify categorical genus with $1+\dim H_{\mathcal{X}}^{\mathrm{age}<1}$. Passing to a sufficiently general $\mathbb{Q}$-Gorenstein deformation defines the residual invariant $g_{\mathrm{cat}}^{\mathrm{qG}}(X)$; its difference from $g_{\mathrm{cat}}(\mathcal{X})$ is exactly the weighted T-content contribution. For any surface admitting a toric $\mathbb{Q}$-Gorenstein degeneration, this identifies the residual categorical invariant with Tveiten's mutable genus, the genus of a general maximally mutable Laurent-polynomial fiber for the chosen degeneration. The locally $\mathbb{Q}$-Gorenstein rigid comparison is the special case in which the drop vanishes. For toric surfaces, the categorical genus of the given canonical stack counts all interior lattice points and equals the genus of a general Laurent-polynomial fiber. Explicit Oneto–Petracci mirrors give direct checks.

We prove that the knot Floer complex, along with some restrictions on the flip map, determines the dual knot to $\pm 1$ surgery on the Borromean knot. In particular, this implies that Heegaard Floer homology detects boundary Dehn twists.

The Chern-Simons-Schrödinger equation (in the temporal gauge) arises as the Hamiltonian flow of the abelian Higgs energy on $\mathbb R^2$. Manton (arXiv:hep-th/9701027) introduced the equation as a model for the dynamics of the critical points of the energy, known as vortices. He conjectured that, for a certain range of coupling constants, the vortex motion under the Chern-Simons-Schrödinger flow can be effectively captured by a first-order ODE on the moduli space of self-dual vortices constructed by Jaffe-Taubes (1980) and Samols (1992). As a first step towards a rigorous proof of Manton's conjecture, we formulate the Cauchy problem in DeTurck gauge within the natural energy space and prove global well-posedness. We also obtain, as a corollary of the well-posedness theory and the stability results in our previous work (arXiv:2603.24900), orbital stability of the self-dual vortices under the Chern-Simons-Schrödinger flow near self-dual coupling. The heart of our analysis lies in developing a geometric Littlewood-Paley theory based on the covariant heat equation in caloric gauge, which we use to perform a paradifferential-style decomposition of the Chern-Simons-Schrödinger equation.

We obtain an asymptotic formula for a cubic moment of self-dual $GL_2$ $L$-functions studied by Petrow and Young, with a small extra averaging over characters. Our asymptotic consists of the main term conjectured by Conrey–Farmer–Keating–Rubinstein–Snaith and a power-saving error term whose strength depends on the amount of extra averaging.

Gabai proved that any Murasugi sum of $\pi_1$-essential Seifert surfaces is also $\pi_1$-essential, and Ozawa extended this result to unoriented spanning surfaces. We show, however, that the analogous statement about geometrically essential surfaces is untrue. (A spanning surface is geometrically essential if it cannot be compressed or boundary compressed to another spanning surface.) We also ask when plumbing an unknotted annulus (with any number of twists) onto a compressible spanning surface yields a compressible surface. We provide positive and negative examples, and we establish a simple sufficient condition. As an application, we obtain a new constructive proof of a result of Hatcher and Thurston about essential spanning surfaces for 2-bridge knots and links.

Let $\Sm_q$ and $\Rm_q$ denote the Suzuki and Ree curves. Motivated by the Giulietti–Korchmáros curve, Skabelund constructed cyclic covers $\tSm_q$ and $\tRm_q$ of these curves and proved that they are maximal over $\F_{q^4}$ and $\F_{q^6}$, respectively. In the same paper he associated to the Suzuki and Ree curves certain ray class field covers $\Sm_{\rm rcf}$ and $\Rm_{\rm rcf}$, and showed that there are towers \[ \Sm_{\rm rcf}\longrightarrow\tSm_q\longrightarrow\Sm_q, \qquad \Rm_{\rm rcf}\longrightarrow\tRm_q\longrightarrow\Rm_q . \] Computations for small values of $q$ suggested that the first map in each tower is always an isomorphism, and the general case was left open. We prove that \[ \Sm_{\rm rcf}=\tSm_q,\qquad \Rm_{\rm rcf}=\tRm_q \] for every admissible $q$, the comparison being made over $\F_{q^4}$ in the Suzuki case and over $\F_{q^6}$ in the Ree case. The proof combines a Kummer normal form of the ray class extension, allowing a constant twist, with the centrality of its Galois group among lifted automorphisms, the standard involution of the base curve, and the first positive non-gap at the rational point at infinity.

In this work, we recast two widely used acceleration methods for fixed-point iterations, Anderson acceleration (AA) and the nonlinear generalized minimal residual method (NGMRES), as two-grid methods. By explicitly deriving the error propagation matrices for AA and NGMRES on linear problems, we show that both methods, together with several of their existing variants, can be reformulated as two-grid methods with a single pre- or postsmoothing step and a coarse-grid correction given by a projection with respect to a suitable inner product. This reformulation not only unifies AA, NGMRES, and their variants within a single algorithmic framework, but also enables the design of new variants by independently adjusting the components of the two-grid method, such as the coarse space and the smoothing steps. In particular, we propose a new variant that enlarges the coarse space by incorporating the most recently updated residual, an improvement that is naturally revealed by the two-grid formulation but is hidden in the standard formulations of the AA and NGMRES algorithms. Numerical experiments on both an SPD Poisson problem and a family of non-SPD convection-diffusion problems show that this new variant often outperforms both AA and NGMRES, as well as their variants.

We show that every finite premaniplex of rank $3$ or $4$ is the symmetry type graph of a finite maniplex, settling the finite rank $3$ and $4$ case of the maniplex version of the symmetry type graph problem. The proof uses the fact that the universal string Coxeter groups of rank $3$ and $4$ are amalgams (of finite groups), hence they act on a tree, which allows us to use a lifting theorem of Potočnik and Spiga.

Let $K$ be an imaginary quadratic field and let $p \ge 5$ be a prime that is unramified in $K$. Let $\mathcal{A}_f/\mathbb{Q}$ be an abelian variety of $\mathrm{GL}_2$-type associated with a weight-two modular form $f$, with good non-ordinary reduction at $p$, and suppose that $(f,K)$ satisfies the generalized Heegner hypothesis. In the case where $p$ is inert in $K$, we further assume that $\mathcal{A}_f$ is an elliptic curve. We develop an Euler-characteristic formula for signed Selmer groups over anticyclotomic $\mathbb{Z}_p$-extensions that applies when the corresponding Selmer modules have arbitrary $\Lambda$-rank. Assuming one inclusion in the signed Iwasawa main conjecture, we apply this formula to prove Kolyvagin's conjecture on the non-vanishing of the Kolyvagin system attached to Heegner points. Our results extend to the non-ordinary setting the results of Wei Zhang, Burungale–Castella–Grossi–Skinner, Castella–Sano and Kim in the ordinary case, and complement the works of Sweeting and Kim in the non-ordinary case under different hypotheses. We also study the effect of the exceptional zero phenomenon on the Iwasawa main conjecture in the inert case.

We study dimension-free Bohnenblust–Hille inequalities for local operators on systems of $K$-level qudits. For the operator space of support at most $d$, uniformly over all tensor-product orthonormal operator bases, we prove a Bohnenblust–Hille inequality with the optimal exponent $2d/(d+1)$ whose asymptotic exponential base in the interaction order is of order $O(\sqrt K)$. Our proof is intrinsic, based on a one-site scalarization and block-transversal decoupling, and does not rely on a scalarization to the cyclic group or on a Remez-type argument. We complement the upper bound by showing that the asymptotic exponential Bohnenblust–Hille base satisfies $cK^{1/4}\le \beta(K)\le C\sqrt K$ with universal constants $c,C>0$. In this sense, the present work may be viewed as continuing the line of work of Slote–Volberg–Zhang from a complementary intrinsic viewpoint. We also revisit their Gell–Mann and Heisenberg–Weyl scalarization procedures. In the Heisenberg–Weyl setting, the prime-dimensional case already yields the optimal interaction exponent, whereas for composite $K$ the scalar total degree used in their reduction leads to a larger exponent. A support-sensitive formulation recovers the optimal interaction exponent for every $K$, while the intrinsic argument gives the stronger dependence on the local dimension at the level of the asymptotic exponential base. As further consequences, we determine the sharp scale, up to factors exponential in $d$ with base depending only on $K$, of coefficient $\ell_1$-normalization and unconditionality on the exact-support spaces. This yields dimension-free coefficient sparsification, including sparse generalized-Pauli approximation in Heisenberg–Weyl coordinates, as well as normalization and query bounds for canonical LCU/qubitization constructions.

Let $\overline{Z}$ be a smooth projective variety with an simple normal crossing divisor $D$ such that $\Omega_{\overline{Z}}^1(\log D)$ is nef. We prove that if $Y\subset Z:=\overline Z\setminus D$ is a smooth closed subvariety with nonzero Euler characteristic, and $P$ is a perverse sheaf on $Y$ with full support, then $\chi(Y,P)>0$. This is a strict version of an inequality obtained in arXiv:2408.15788. Applying this to the trace-zero part of a finite direct image yields a logarithmic Riemann–Hurwitz inequality: if $Y$ has dimension $n$, any finite surjective morphism $f\colon X\to Y$ of degree $d$ with $X$ smooth satisfies $(-1)^n\chi(X)\ge d\,(-1)^n\chi(Y)$, the difference being an explicit sum of nonnegative intersection numbers. When $(-1)^n\chi(Y)>0$ this forces any such $f$ with $\chi(X)=\chi(Y)$ to be an isomorphism. We verify the nef hypothesis for subvarieties of semiabelian varieties, and for $\overline{\mathscr M}_{g,n}$—the moduli of curves, obtaining in particular that every finite surjective self-morphism of a moduli space of curves with level structure is an isomorphism.

Becker et al. produced a striking trace formula for the sum of fourth powers of generalized magic angles, $ \theta $, for the chiral model of twisted bilayer graphene (TBG) with the exact Bistritzer–MacDonald potential: $\sum \theta^{4} = 8 \pi / \sqrt 3 $. The purpose of this note is to generalize this formula to a larger class of potentials satisfying the symmetries of the model.

In this paper, we give a negative answer to Wang's conjecture. For every $n\geq 3$, there exist $\varepsilon>0$ and $\delta \in (0,1)$ such that, for all $q$ and $\lambda$ satisfying \[ \frac{n}{n-2}-\varepsilon<q\le \frac{n}{n-2}, \qquad \frac{1}{q-1}\cdot \delta<\lambda\le \frac{1}{q-1}, \] there exists a bounded Euclidean domain \(\Omega\subset\mathbb{R}^n\) with principal curvatures bigger than \(1\) for which the following nonlinear Robin problem admits a nonconstant positive solution: \begin{align*} \begin{cases} \Delta u=0, & \text{in}\ \Omega,\\[2mm] \dfrac{\partial u}{\partial\nu}+\lambda u=u^q, & \text{on}\ \partial\Omega. \end{cases} \end{align*}

We establish one form of the vortex filament conjecture for the three-dimensional Gross-Pitaevskii equation. Given any smooth closed embedded binormal flow of curves on a compact time interval, we construct, in the small-core limit $\varepsilon\to0$, a family of exact solutions whose degree-one vortex filaments converge uniformly to the prescribed flow. Near the filament the solutions have the standard planar vortex profile, while away from it their phase gradients converge to the associated Biot-Savart field. We also derive a refined modulation law for the vortex filament.

The maintenance of continuous variable information in working memory is thought to rely on persistent patterns of cortical activity. In delayed-estimation tasks, neural activity can form localized activity peaks, or “bumps,” whose positions track the remembered variable. Such activity is well described by continuous-attractor neural field models, but most existing models collapse cortical inhibition into a single homogeneous population. Here, we introduce a stochastic neural field model with distinct excitatory, parvalbumin-expressing (PV), and somatostatin-expressing (SST) populations to examine how inhibitory subtype structure shapes persistent activity. Using a Heaviside firing-rate approximation, we derive stationary bump solutions and reduce their linear stability to separate shifting and scaling modes. We show that population thresholds and inhibitory timescales determine both bump stability and the mechanism by which stability is lost, while inhibitory connection strengths and spatial scales substantially reshape the stable parameter region. In particular, broader SST connectivity promotes stable bump states. Finally, we derive an effective diffusion coefficient for noise-driven bump wandering and show that increasing the SST spatial footprint reduces the rate of memory diffusion. Together, these results demonstrate how inhibitory subtype structure can shape both the deterministic stability and stochastic precision of continuous-attractor memories.

We discuss dominant representations (or Young diagrams), in thermal matrix models with gauge symmetry from the perspective of partial deconfinement. We propose a prescription for defining the dominant representations for thermal matrix models with interaction terms. As an explicit example, we consider the large-$N$ Gaussian matrix model. We obtain the Vershik–Kerov–Logan–Shepp (VKLS) shape of the dominant Young diagrams through a new analytic saddle-point analysis based on a mapping of the representation theory of U($\infty$) to free fermions in two spacetime dimensions. This computation provides a direct derivation of the previously observed functional relation between the shape of the dominant Young diagrams and the eigenvalue distribution of the thermal holonomy: the position of the complex saddle point is naturally identified with the eigenvalue. The dominant Young diagrams admit a natural interpretation in terms of partial deconfinement: the number of rows in the dominant Young diagrams matches the size of the submatrix corresponding to the deconfined subsector.

We study a class of random aggregation trees that generalizes the discrete stick-breaking construction of the uniform labelled tree (Aldous, 1991). To sample the tree of size $n$, vertices are added sequentially, with the $i$th vertex starting a new branch with a prescribed probability $f(n,i)$; otherwise, it extends the current branch. We also define a continuum analogue by replacing the Poisson point process of intensity $tdt$ in the construction of the Brownian continuum random tree by one of intensity $f(t)dt$. We establish scaling limits for two families of discrete aggregation trees in the Gromov-Hausdorff-Prokhorov topology. When $f(n,i)=(i/n)^\beta$, with $\beta>0$, after rescaling the graph distance by $n^{-\beta/(\beta+1)}$, the random tree converges to the compact aggregation tree with $f(t)=t^\beta$. This recovers convergence to the Brownian continuum random tree when $\beta=1$ (Aldous, 1991), as well as scaling limits of choice spanning trees for integer $\beta$ (Archer and Shalev, 2024). We also prove convergence under rescaling to the compact aggregation tree with $f(t)=\log^\gamma(1+t)$ for every $\gamma>1$. Finally, we identify necessary conditions for compactness. Consequently, the threshold $\gamma>1$ in the logarithmic family is sharp. These results provide insight into an open problem on compactness criteria for random aggregation trees (Curien and Haas, 2014).

We develop a plus and minus Iwasawa theory for CM elliptic curves with potentially supersingular reduction at a prime $p\ge5$. We construct local points that satisfy suitable "jumping trace" relations using the Honda–Demchenko theory applied to height-two formal groups over local fields with small ramification degree. These local points allow us to study plus and minus Selmer groups of CM elliptic curves over the cyclotomic $\mathbb{Z}_p$-extension, to construct the corresponding plus and minus $p$-adic $L$-functions, and to prove an Iwasawa main conjecture relating these objects. As an application, we obtain asymptotic growth formulae for the $p$-primary part of the Tate–Shafarevich groups.

We present an extended study of the gravitationally localized soliton-like solutions to the minimally-coupled Einstein-Dirac-Higgs equations, embedded in asymptotically Minkowski spacetimes. The equations of motion describing these Yukawa-coupled Dirac stars are generalized to any even number of constituent fermions. We first expand the discussion of two-fermion ground-state solutions initially analyzed in Leith et al. [Phys. Rev. D. 107, 106020 (2023)]. Upon extending our analysis to excited states and many-fermion states, we find that both exhibit different behaviors to their two-fermion ground-state counterparts. In these excited states and many-fermion states, the Higgs field may exhibit stepwise increases and at times decrease within the soliton, which has not been seen for the two-fermion ground states. We also propose the potential cause of a significant degree of ADM-to-fermion mass disparity, which is present in states with strong Yukawa-coupling.

Z/2 harmonic 1-forms arise naturally as singular limits in gauge theory and calibrated geometry, but the topology that can occur in their singular sets is poorly understood. In this paper we study the flexibility of these singular sets when the ambient Riemannian metric is allowed to vary. We prove that every compact smoothly embedded codimension-two submanifold with trivial normal bundle in a Euclidean space of dimension at least three can be realized as the singular set of a nondegenerate Z/2 harmonic 1-form for a complete metric that is Euclidean outside a compact set. As an application, we use Calabi surgery to construct Z/2-harmonic 1-forms with prescribed local singular sets on closed manifolds of positive first Betti number.

We describe how strong vorticity fluctuations appear in three-dimensional incompressible turbulence driven by a large-scale random force. By combining analytical and numerical methods, we construct a vorticity instanton: the most probable force history driving the flow to a prescribed large vorticity at a given point and time. We show how the smooth, large-scale force produces a long vortex filament with a thin, viscous-scale core. Our main finding is that the standard mechanism of amplifying a vortex tube through radial compression and axial stretching is only effective up to a threshold value. Producing vorticity above the threshold requires an additional ingredient: vorticity waves propagating along the tube. The final vorticity is the result of two colliding waves. This mechanism has striking similarities to the one recently suggested by OpenAI for creating infinite vorticity in finite-time Navier–Stokes blow-up.

We prove that any embedded genus zero free boundary minimal surface in the ball has first Steklov eigenvalue $1$. This implies the uniqueness of the critical catenoid.

We study the classical problem of sectional curvature pinching in the class of 2-step nilmanifolds, which are necessarily of mixed curvature. We show that the pinching constant of any 2-step nilmanifold lies in the compact interval $[-3, -\frac{3}{2}]$. The upper bound $-\frac{3}{2}$ is achieved by the complex Heisenberg group $\mathrm{Heis}_3(\mathbb{C})$ with a Ricci soliton metric. The upper bound exhibits rigidity: if a simply connected 2-step nilmanifold $N$ has the pinching constant $-\frac{3}{2}$, then $N$ admits a Ricci soliton complex Heisenberg group as a totally geodesic subgroup. On the other hand, the lower bound satisfies non-rigidity: any 2-step nilpotent Lie group admits a metric with pinching constant $-3$. This is derived by showing that there is an open neighborhood $U$ of $\mathrm{Heis}_3(\mathbb{R})\times \mathbb{R}^{n-3}$ in the space of $n$-dimensional 2-step nilmanifolds such that the pinching constant is $-3$ on $U$, and any 2-step nilpotent Lie group $N$ has a metric $g$ such that $(N,g)$ lies in $U$. In fact, if $N$ is not isomorphic to $\mathrm{Heis}_3(\mathbb{R})\times \mathbb{R}^{n-3}$, then there is a curve $g_t$ of metrics on $N$ with $(N,g_t)\in U$, showing that there are uncountably many left-invariant metrics on $N$ such that the pinching constant is $-3$. An algebraic characterization of a 2-step nilpotent Lie group that admits a metric with the pinching constant $-\frac{3}{2}$ is also given, and the pinching constants of various examples are computed.

We study loss-based filtering for finite-sum optimization with a subset of corrupted component functions whose gradients may be highly unreliable. Motivated by minimum-loss-based SGD (min-$k$-loss) and quantile-based methods for corrupted linear systems, we propose and analyze a general loss-filtering framework – Quantile-\(k\)-Loss SGD (Q\(k\)L-SGD) – that samples \(k\) component losses at each iteration and updates using an index chosen uniformly from the lower empirical \(q\)-quantile. We prove linear convergence of this family of methods under standard convexity assumptions, requiring the sample size to scale with the number of corruptions and a subset strong-convexity threshold. For the cases when large enough sampling is impossible or undesirable, we give a complementary small-sample probabilistic analysis that covers any sample size $k$ and the convergence behavior depends on the probability of selecting an outlier and on the curvature of the selected good step. Experiments on polynomial regression, regularized logistic regression, and regularized hinge loss show that intermediate quantiles often outperform both standard SGD and min-\(k\)-loss SGD. In particular, min-\(k\) often stalls by repeatedly selecting nearly solved components, while intermediate quantiles retain robustness and produce more informative updates.

For the Jacobi family with parameters $(0,\beta)$, $\beta\geq-1/3$, we prove a continuous comparison theorem for the associated weighted Jacobi–Radau functions. When two consecutive functions are parametrised by the same Prüfer phase, the function of higher degree attains that phase closer to $\theta=0$ and has strictly larger amplitude. In particular, the moduli of all corresponding relative extrema increase strictly with the degree. The lower bound $-1/3$ is sharp for this continuous statement: when $-1<\beta<-1/3$, the amplitude inequality is reversed at sufficiently small positive phases. This local reversal does not determine the optimal range for the discrete extremal inequalities. For $-1/3\leq\beta\leq0$, the positive Jacobi product formula identifies the endpoint values of the Lebesgue functions with the global Lebesgue constants, so $(\Lambda_n^{(0,\beta)})_{n\geq0}$ is strictly increasing. At $\beta=0$ the weighted Radau functions reduce to $P_m^{(0,-1)}$. We thereby recover the theorem of Wong and Zhang and obtain, through an exact total-variation formula, a short proof of the Qu–Wong theorem on Legendre Lebesgue constants. The representation and the termwise comparison together realise the alternative-expression approach proposed by Qu and Wong, without asymptotic expansions, error bounds, or finite numerical verification. The proof is based on the equal-phase Prüfer architecture developed in the first author's earlier preprint arXiv:2608.01404; it applies that architecture to the problem posed by Qu and Wong.

Evidence suggests a tight connection between the existence of antipode-like maps on bialgebra-like structures, and the Frobenius property for the associated free Hopf module functor. In this paper, we prove that in any braided monoidal category satisfying few mild assumptions, a bimonoid is a one-sided Hopf monoid and the antipode is a bimonoid anti-homomorphism, if and only if the free Hopf module functor is Frobenius. Our interest in these structures stems from having found genuine examples of one-sided Hopf monoids in contexts where the braiding is non-trivial. In fact, we provide a construction of the free one-sided Hopf monoid over a comonoid in any symmetric monoidal category with some non-restrictive additional assumptions. We present several examples of this construction, and describe the resulting one-sided (sometimes two-sided) Hopf monoids.

We study nearest-neighbour distances and strict best returns of cyclic orbits on $(V\times G)/\Lambda$, where $V$ is a finite-dimensional real normed space, $G$ has a bi-invariant ultrametric, and $\Lambda$ is an arbitrary closed subgroup. For the quotient of the maximum product metric, the number of distinct nearest-neighbour distances in every finite orbit segment is at most $P(\overline V)+1$, where $P$ is an open-ball packing constant and $\overline V$ is obtained by factoring out the continuous real part of $\Lambda$. For inner-product norms the bound depends only on the span of the projected subgroup. This gives $2^d+1$ for the maximum norm on $\mathbb R^d$ and strong-kissing-number bounds for inner-product norms, extending the corresponding real and adelic distance bounds. Our second main result is an exact abelianization principle: every such cyclic orbit is isometric, with its integer indexing preserved, to a cyclic orbit on an abelian mixed quotient $(V\times U)/\Lambda_{\mathrm{ab}}$. The construction preserves the reduced and effective real factors. Thus noncommutativity and non-normality enlarge the class of ambient presentations but not the class of cyclic-orbit metrics. The record-count identity and its growth consequences require only an isometric embedding of a metric space. For mixed homogeneous spaces we prove packing recurrences and, using abelianization, the sharp three-step and five-step recurrences in effective Euclidean dimensions one and two, without a normalizer assumption. We also extend the purely real recurrence to noncompact quotients and give counterexamples showing why the stronger real recurrence can fail after an ultrametric factor is added.

Results of Fulton-Lazarsfeld and Ross-Toma show that Schur classes of ample and nef vector bundles have remarkable positivity properties. We generalize these results to a purely algebraic setting. We show that if a projective bundle ring, a ring which resembles the cohomology ring of a projective bundle over a smooth complex projective variety, has the Kähler package with respect to a suitable cone, then the associated Schur classes satisfy a version of the Hodge-Riemann relations. This result is new even for the projectivization of an ample vector bundle over a smooth complex projective variety. We apply this result to prove that Schur coefficients of matroids are nonnegative.

In this work, we establish the sharp $L^2 \to L^4$ extension inequality for a large class of quadratic surfaces in $\mathbb{F}^3$, where $\mathbb{F}$ is a finite field of odd characteristic. These surfaces include the cones without the origin and the spheres with nonzero radius $j$ such that $-j$ is not a square in $\mathbb{F}$. We also show that the maximizers of these inequalities must have constant modulus.