Papers

In this article, we define the parabolic Grassmann bundle associated to a parabolic vector bundle over a smooth projective variety, generalizing the construction of parabolic projective bundles developed in \cite{BL}. We determine its Néron–Severi group and compute its nef, pseudoeffective, and Mori cones over smooth projective curves. We also compute the corresponding cones for the fiber product of two parabolic Grassmann bundles over a smooth projective curve.

We prove an anisotropic analogue of the higher regularity theorem of Huisken and Ilmanen for inverse mean curvature flow. For an arbitrary smooth Minkowski norm, we first prove a Huisken–Ilmanen type Harnack estimate for smooth closed strictly star-shaped solutions. We then construct global smooth solutions starting from $C^1$ strictly star-shaped hypersurfaces with bounded nonnegative weak anisotropic mean curvature. Combining this construction with the asymptotic theory for weak inverse anisotropic mean curvature flow, we show that weak solutions starting from bounded smooth initial sets become smooth outside a compact set.

We prove that $|6K_X|$ defines a generically finite map for all minimal 3-folds $X$ of general type with $P_2(X)\geq 2$, which is optimal. We also prove that $|nK_X|$ defines a generically finite map for all minimal 3-folds $X$ of general type when $n\geq 38$. The essential technical ingredients of this paper are a new generic finiteness criterion for surfaces and an effective comparison inequality under a special resolution.

For a $C_2$-commutative ring spectrum $R$, a twisted $R$-algebra is an $R$-module with a multiplication whose order is switched by the $C_2$-action. In this paper, we construct various quotients of $R$ as twisted $R$-algebras, when $R$ is an even real commutative ring spectrum. These are constructed as Thom spectra of maps out of suitable $C_2$-actions on $S^1$ and $U(n)$. One such example is given by $K\mathbb{R}$ which is endowed with a twisted $K\mathbb{R}$-algebra structure. Other examples include quotients such as $M\mathbb{R}/(2,x_1,\dots, x_{n-1})$ over the real bordism spectrum $M\mathbb{R}$, and the real $2$-periodic Morava $K$-theories as modules over the real Morava $E$-theory spectra. In the context of twisted $R$-algebras, one may consider the real topological Hochschild homology, and for Thom spectra, one has a nice formula again as a Thom spectrum. We use this to obtain computations for the real topological Hochschild homology of $K\mathbb{R}/2$ as a twisted $K\mathbb{R}$-algebra. The computation also involves a splitting of the units spectrum $gl_1K\mathbb{R}$, which is an analogue of the classical splitting of the units of $K$-theory.

Recent work suggests that the convergence rate of gradient descent (GD) in smooth convex optimization can be significantly improved by employing large stepsizes that may violate the descent property. In particular, if the total number of iterations $n$ is given, an $O(n^{-1.271})$ convergence rate can be achieved for both function value and squared gradient norm minimization. On the other hand, in the setting of anytime convergence, where $n$ is not known in advance, the best known rates of GD are much slower: $O(n^{-1.119})$ for function value minimization and $O(n^{-1})$ for squared gradient norm minimization. It remains open whether any of these upper bounds can be improved, as they are far from the classical $\Omega(n^{-2})$ lower bound for any first-order method. In this work, we establish two lower bounds on the anytime convergence of GD. We show that no positive stepsize schedule can achieve an $o(n^{-1.334})$ anytime rate for function value minimization, nor an $o(n^{-1})$ anytime rate for squared gradient norm minimization. The key ingredients of our analysis are novel upper bounds on the number and the magnitude of large stepsizes, derived by analyzing GD on quadratic functions and variants of Huber functions. Our work provides the first lower bounds for the COLT 2024 open problem posed by Kornowski and Shamir regarding the optimal anytime convergence rates of GD.

The total character $\tau_G$ of a finite group $G$ is the sum of all irreducible complex characters of $G$, and the total degree of $G$ is $T(G) := \tau_G(1)$. A proper subgroup $H$ of $G$ is rich if $\tau_G$ is ”contained” in the permutation character $(1_H)^G$. In the first part of this paper, we investigate rich subgroups whose index is a product of two primes. We also consider rich subgroups of symmetric and alternating groups. In the second part we establish a formula for $T(G)$ in the case where the order of $G$ is a prime power. This result is analogous to a formula for the class number of $G$ proved by P. Hall, and it confirms a conjecture by Heffernan and MacHale from 2008. In the last part of the paper, we investigate finite groups $G$ where $T(G)$ is small, in a certain sense.

We present NLF (Nonlinear Laplacian Flow), a unified framework and linear-time solver for convex network-flow equilibria. Congestion routing, minimum-delay routing, and maximum flow share one form: the nonlinear graph Laplacian $B\rho(B^T\phi)=\alpha d$, where a monotone edge law $\rho_e$ encodes the physics (undirected graphs; directed variants are future work). NLF solves it by a damped chord-Newton iteration whose frozen linearization – a weighted graph Laplacian – is inverted by a near-linear Laplacian solver (default: approximate Cholesky, LAMG+ interchangeable). The nonlinear solve costs $2$–$4$ linear Laplacian solves, making the wall-clock empirically $O(m)$ in the edge count $m$ (not a proved bound). On single-commodity congestion (BPR cost), NLF converges on all 2,003 SuiteSparse corpus graphs up to $1.8\times10^7$ edges. Against a state-of-the-art interior-point method, NLF is a median $2.6\times$ faster where both converge and $>45\times$ on poorly-separable graphs where the IPM's direct core is superlinear; against L-BFGS, a median $4.2\times$ faster and the only solver to finish on the 90 hardest instances. A multicommodity extension routes $K$ commodities through one shared hierarchy at $O(Km)$ per step. The same machinery recovers the exact max-flow as a short sequence of Laplacian solves, with the cut potential as a by-product. Code: https://github.com/orenlivne/nlf

We provide a characterization of the connected subgraphs of the graphs with vertex set the non-isotropic points in a quadratic space $(V,Q)$, two points adjacent if and only if they span a tangent line. Here $(V,Q)$ is a quadratic space $V$ over a finite field $\mathbb{F}_q$ of order $q$, where $q>3$ is odd, equipped with a non-degenerate quadratic form $Q$. The local structure of the graph (i.e. the graph induced on the neighbors of a point) determines the structure of the full graph. This characterization helps to determine the automorphism group of these graphs.

We consider a mathematical model of a poro-visco-elastic medium subject to frictional contact with a rigid obstacle, and study its numerical approximation. This model couples the Biot equations and contact conditions in the form of normal compliance and Coulomb friction. The resulting variational problem consists of a linear partial differential equation coupled to a nonlinear variational inequality. We propose and analyze a fully discrete numerical scheme for this problem, using conformal finite elements in space and the implicit Euler method in time. Existence and uniqueness of the discrete solution is established, and stability and a priori error estimates are derived. A numerical experiment is performed in which numerical error estimates are computed and compared to the theoretical results.

For knot surgery in $S^3$, Heegaard Floer homology provides an obstruction due to Hom–Karakurt–Lidman. We extend this obstruction to all integer homology spheres $Y$, for both positive and negative 1/m surgeries. This is used to test infinitely many small Seifert fibered examples and hyperbolic examples. Moreover, we deduce a lower bound on the $b_2(W)$ of smooth cobordism between a pair of integer homology spheres.

We study branching processes in an i.i.d.\ random environment with immigration in the critical regime, where the underlying offspring mechanism satisfies the critical condition that the log of the average population growth, across environments, and before immigration, is zero. In this setting environmental fluctuations are balanced on average, and the long-term behaviour is determined by the interaction between these fluctuations and the immigration sequence. While recurrence and transience criteria for critical BPREI were established by Bauernschubert (2014), the possibility of null recurrence remained unresolved. We show that, under natural integrability assumptions on the offspring and immigration distributions, a critical BPREI is null recurrent. In particular, the process returns to zero infinitely often but admits no stationary distribution. Our results close a gap in the classification of the critical regime and provide a structural understanding of the balance between environmental variability and immigration. As an application, we resolve the open critical case of the Randomised Reproducing Graph (`RRG') model introduced by Jordan (2011), showing that in the critical regime the proportion of vertices of a fixed degree admits no limiting distribution.

Let $R$ be a Henselian local ring, let $\kappa$ be the residue field of $R$, let $C$ be a smooth projective curve over $R$ with geometrically connected fibers, let $G$ be a reductive $C$-group with isotrivial radical torus $\mathrm{rad}(G)$, and let $E\to C$ be a $G$-torsor. We show that, if either the kernel of the central isogeny $G^{\mathrm{sc}}\times_C \mathrm{rad}(G)\to G$ is étale over $C$ or $\kappa$ is large, the Zariski-local triviality of $E_\kappa\to C_\kappa$ implies the Zariski-local triviality of $E\to C$. We also prove an averaged form of this result, assuming only that $\mathrm{rad}(G)$ is isotrivial, as well as a variant for projective homogeneous spaces under no restrictions on $G$. As consequences, we obtain a local-global principle for torsors over function fields of curves over Henselian discrete valuation rings, strengthening work of Gille-Parimala-Suresh and a Henselian version of a theorem of Drinfeld-Simpson. Our proofs are geometric and rely on compactifications of torsors and on a relative and arithmetic version of the comb smoothing technique, which we develop in detail, building on work of Kollár and Graber-Harris-Starr.

A random geometric graph (RGG) is generated by first sampling latent points $x_1,\ldots,x_n$ independently and uniformly from the unit sphere in $\mathbb{R}^d$, and then connecting each pair $(i,j)$ if $\langle x_i,x_j\rangle$ exceeds some threshold $\tau$. We study the sharp detection threshold – the largest dimension at which the RGG can be statistically distinguished from the Erdős–Rényi graph with the same edge density $p$. This threshold is conjectured to be $d \asymp (nh(p))^3$, where $h(p)=p \log \frac{1}{p} + (1-p) \log \frac{1}{1-p}$ is the binary entropy function. Previous works proved this conjecture for dense graphs with constant $p$ and, up to polylogarithmic factors, very sparse graphs with $p=\Theta(1/n)$. In this paper, we prove that detection is impossible when $d\gg (nh(p))^3$ and $d\ge (1+\epsilon) n$ for any constant $\epsilon>0$, thereby resolving the conjecture in the regime $p\gtrsim n^{-2/3}/\log n$ and improving upon the state of the art in the regime $1/n \ll p \ll n^{-2/3}/\log n$. The key to our proof is a sharp analysis of the posterior distribution of the latent points given the observed graph, obtained through an information-theoretic comparison argument combined with strong log-concavity.

Let $X_0\leftarrow X_1\leftarrow\cdots$ be a $\mathbb Z_p^d$-tower of finite graphs, and let $U_n$ be the Grover transition matrix on $X_n$. We study Iwasawa-type $p$-adic growth laws for the polynomial spectral quantities \[ \det P(U_n), \] where $P(A)$ is a monic polynomial. The basic object is the spectral resultant \[ \mathcal R_{X,P}(T)=\operatorname{Res}_A(\mathcal F_X(A,T),P(A)), \] where $\mathcal F_X(A,T)$ is the universal Grover–Ihara spectral polynomial of the tower. In the integral setting, this resultant generates the zeroth Fitting ideal of a natural finite module over the Iwasawa algebra; when the resultant is nonzero, this module is torsion. The polynomial $P$ packages prescribed spectral values into a single spectral packet. If $P$ is coprime to the Bass factor $A^2-1$ and $\mathcal R_{X,P}$ does not vanish at torsion characters, then $\det P(U_n)$ is nonzero for all $n$ and we prove a Cuoco–Monsky type leading asymptotic formula for $v_p(\det P(U_n))$. The leading terms are given explicitly by the $\mu$- and $\lambda$-invariants of $\mathcal R_{X,P}$, with a separate correction coming from the Bass factor. For $P(A)=A-a$, with $a\ne\pm1$ and $a$ not an eigenvalue at any level, this recovers the leading invariants in the fixed non-eigenvalue formula for Grover characteristic polynomials. We also prove an equivariant factorization of spectral resultants for finite connected $p$-group covers. As a consequence, we obtain an unramified equivariant Kida formula under explicit integrality and nonzero-resultant assumptions. Finally, when $\gcd(P,A^2-1)=1$, we show that torsion zeros of $\mathcal R_{X,P}$ correspond exactly to occurrences of roots of $P$ as Grover eigenvalues at finite levels. The examples include the $K_3$-tower, non-abelian Heisenberg $5$-group covers, and an explicit torsion-zero spectral packet.

The enumeration of planar maps with control on the boundary metric, i.e. the pseudometric induced on the outer face of the map by its bulk graph distance metric, is a difficult problem in general. However, we show that for a family of bipartite planar map models with special q-deformed face weights that arise in the physics context of the double-scaled Sachdev-Ye-Kitaev model (DSSYK) the enumeration admits a very simple answer. Encoding the boundary metric of a bipartite planar map by its so-called geodesic chord diagram, we prove that the weighted enumeration depends only on the crossing number of the chord diagram. At fixed perimeter, the induced law of the geodesic chord diagram in these planar map models coincides exactly with the chord diagram representation of the DSSYK model.

In this paper, we show that if the link of an isolated complex surface singularity is either a $Sol^3$-manifold or an $\widetilde{SL}(2;\mathbb{R})$-manifold with its canonical contact structure, then it admits infinitely many strong symplectic fillings that are pairwise non-diffeomorphic and not related by a sequence of blow-ups or blow-downs. As a consequence, the link of any cusp singularity, exceptional unimodal singularity, or hyperbolic Brieskorn singularity admits infinitely many pairwise non-diffeomorphic minimal strong symplectic fillings.

We construct two arrangements of nine lines in the complex projective plane with isomorphic intersection lattices but with different minimal degrees of Jacobian relations. The common weak combinatorics is \[ (n_2,n_3,n_4)=(9,7,1), \] so the example is not the classical Ziegler-Yuzvinsky pair, whose weak combinatorics is $(n_{2},n_{3}) = (18,6)$. For the two defining equations $f$ and $g$ we prove \[ {\rm mdr}(f)=4,\qquad {\rm mdr}(g)=5. \] Since the degree is $d=9$, the first equality gives ${\rm mdr}(f)<d/2$. Hence the pair gives a counterexample to the Generalized Terao Conjecture.

In 2020, Panda, Verma, and Keerti asked whether the central graph of every graph satisfies the AVD-total coloring conjecture (AVD-TCC). In this paper, we use idempotent commutative Latin squares and a theorem of Borodin, Kostochka, and Woodall to prove the AVD-TCC for central graphs of all connected graphs with order at least five and minimum degree at least two. We further determine the AVD-total chromatic number of subdivision graphs of connected graphs and establish new results on the AVD-total coloring of joins of graphs, central graphs, and deleted lexicographic products.

This paper investigates the logical strength of two theorems in modal propositional logic - the Hennessy-Milner theorem and the van Benthem characterization theorem - within the framework of second-order arithmetic. We demonstrate that the Hennessy-Milner theorem is equivalent to $\mathrm{ACA}_0$ over $\mathrm{RCA}_0$. For the van Benthem characterization theorem, we introduce three variants: the semantic, syntactic, and hybrid forms. We show that the semantic form is provable in $\mathrm{RCA}_0$, the syntactic form is provable in $\mathrm{PRA}$, and the hybrid form is equivalent to the weak completeness theorem for first-order logic over $\mathrm{RCA}_0$.

We develop the theory of strongly $n$-genuine polynomials $F(Y,X_1,\ldots,X_n)$, which have the property that the number of specializations $F(Y,X_1,\mathbf{x}')$ with $\mathbf{x}'=(x_2,\ldots,x_n) \in \mathbb{Z}^{n-1}$ (respectively $\mathbf{x}' \in \mathbb{F}_p^{n-1}$) such that $F(Y,X_1,\mathbf{x}')$ is reducible over $\overline{\mathbb{Q}}$ (respectively over $\overline{\mathbb{F}}_p$) can be well-controlled quantitatively. We also develop the theory of a larger class of $n$-genuine polynomials $F(Y,X_1,\ldots,X_n)$, which have the property that the number of specializations $F(Y,X_1,\mathbf{x}')$ with $\mathbf{x}' \in \mathbb{Z}^{n-1}$ (respectively $\mathbf{x}' \in \mathbb{F}_p^{n-1}$) such that $F(Y,X_1,\mathbf{x}')$ splits completely over $\overline{\mathbb{Q}}$ (respectively over $\overline{\mathbb{F}}_p$) into factors that are linear in $Y$ can be well-controlled quantitatively. For each of these classes, we prove that there are four equivalent characterizations. As an application, we demonstrate that $n$-genuine and strongly $n$-genuine polynomials can be used to prove, for any polynomial $F(Y,X_1,\ldots,X_n)$, an upper bound for the number of specializations $F(Y,\mathbf{x})$ with $\mathbf{x}=(x_1,\ldots,x_n) \in \mathbb{Z}^n$ such that the Galois group of the splitting field of $F(Y,\mathbf{x})$ over $\mathbb{Q}$ is not isomorphic to the Galois group of the splitting field of $F(Y,X_1,\ldots,X_n)$ over $\mathbb{Q}(X_1,\ldots,X_n)$. We simultaneously prove analogous results over any number field.

The Ramsey community number $r_\kappa$ is the minimum network size at which a graph's connectivity is better described by a partition into communities than by no partition, under a prescribed community-detection rule. It was introduced through numerical simulations of networks grown by local rules, which suggested that community structure can emerge without any node heterogeneity. Here I compute $r_\kappa$ analytically for the simplest homogeneous, locally wired graph: the circulant ring lattice $C_n(1,\dots,c)$. Using a Bernoulli stochastic block model with symmetric $\mathrm{Beta}$ priors as the detection rule, the Bayesian evidence for a balanced two-community partition and for the unpartitioned network are both obtained in closed form, so the transition between them can be located exactly. The result is a sharp dependence on the interaction range: the plain cycle ($c=1$) is never partitioned, its two-community posterior decaying as $n^{-(2\alpha+3)}$, so $r_\kappa=\infty$; but the next-nearest-neighbour ring ($c=2$) acquires a finite $r_\kappa\simeq 35$ nodes, above which the partition is preferred with a log-evidence growing as $(\ln 2)\,n$. This provides an exactly solvable instance of community emergence in a network with no built-in communities, and shows that a minimal amount of local connectivity is enough to break the ring.

Motivated by Su's construction of Cohen-type Eisenstein series of half-integral weight over totally real number fields \cite{Su16}, we introduce a generalization of Hurwitz class numbers to totally real number fields. Using these generalized Hurwitz class numbers, we establish an Eichler–Selberg trace formula for the space of holomorphic Hilbert cusp forms over real quadratic fields of narrow class number one. While the classical Hurwitz class numbers are defined in terms of class numbers of imaginary quadratic fields, the generalized Hurwitz class numbers appearing in our Eichler–Selberg trace formula are defined in terms of class numbers of quartic CM fields. For applications of this Eichler–Selberg trace formula, we study the distribution of the generalized Hurwitz class numbers, prove class number relations, and carry out numerical computations of traces of Hecke operators for $\mathbb{Q}(\sqrt{5})$ and $\mathbb{Q}(\sqrt{29})$.

This work is in the spirit of our previous investigation on a smooth Dirichlet divisor problem, where we now replace the Dirichlet divisor function $\tau$ by the sum-of-divisors function $\sigma$. We prove a totally explicit asymptotic formula for the sum of $\sigma(n)$ twisted by the weight $1-x/n$, which enables us to eliminate the difficult part in the classical average order of $\sigma(n)$. As a corollary, we deduce the convergence of an integral dealing with the error term in the Walfisz divisor problem. We also provide an appendix containing the necessary explicit results derived from the mean value theorem and the Euler-Maclaurin summation formula.

We study linear topologies on residuated lattices generated by systems of filters, with emphasis on the uniform structures and separation properties that they determine. A down-directed family of filters gives a natural compatible uniformity, and the associated topology makes the residuated lattice into a topological algebra. We characterize Hausdorffness by the triviality of the intersection of the underlying filter system. For compact topological residuated lattices, we prove the equivalence between topological profiniteness, residual finiteness, and representation as a closed subdirect product of finite discrete residuated lattices. We also analyze the descending chain condition ($DCC$) on filters. Under $DCC$, every filter system has a least element; hence every zero-dimensional linear topology is induced by a single filter, and the canonical map from filters to zero-dimensional linear topologies is bijective. This gives a corrected form of earlier representation arguments and identifies precisely where $DCC$ is required. Finally, working throughout in $\mathrm{ZFC}$, we give a sufficient criterion for the existence of non-discrete Hausdorff linear topologies, illustrated by the Gödel algebra.

In this paper, we study an elliptic problem involving a $p$-Laplacian operator and a potential well which is driven by a critical and singular nonlinearity. Under the limiting case of a parameter blowing up to $\infty$ yields solutions to a different problem where the effect of the potential well becomes negligible.

In a recent paper, Banderier et al. (2024) investigated the limiting behavior of component counts of random partitions induced by the Chinese restaurant process with parameters $\alpha\in(0,1)$ and $\theta>-\alpha$. Let $C_j(n)$ denote the number of components of size $j$ of a partition of $\{1,\ldots,n\}$ and consider $j=j_n\to\infty$ as $n\to\infty$. They identified a phase transition in the first-order limit behavior of $C_{j_n}(n)$, where the critical regime corresponds to $j_n\sim rn^{\alpha/(1+\alpha)}$ for some $r>0$. A natural next question is to understand the corresponding second-order fluctuations. We establish second-order limit theorems in the critical regime and, under an additional rate condition in the subcritical regime ($j_n\ll n^{\alpha/(1+\alpha)}/(\log\log n)^{1/(1+\alpha)}$), for the counting process $(C_{j_n}(n(1+t/j_n)_+))_{t\in\mathbb R}$. In the subcritical regime, after appropriate normalization, the limit is a stationary Ornstein–Uhlenbeck Gaussian process, whereas in the critical regime the limit is a stationary $M/M/\infty$ queue. We also establish a more refined point-process convergence in the critical regime. We first establish these results for the more general Karlin infinite urn model and then adapt the analysis to the Chinese restaurant process. For the latter model, most of our limit theorems are established in the quenched sense.

This paper investigates simple restricted modules over the deformed Schrödinger-Virasoro algebra $\mathcal{G}_{\lambda,\mu}$, which gives a complete classification of them for some $\lambda,\mu\in\mathbb{C}$. More precisely, we provide a systematic construction of these modules, including highest weight modules and Whittaker modules, by inducing simple modules from the positive part's quotient algebras. We prove that any simple restricted $\mathcal{G}_{\lambda,\mu}$-module satisfying certain injective conditions is isomorphic to such an induced module. As an application, we obtain some simple weak $V(c)$-modules over vertex algebras associated to $\mathcal{G}_{\lambda,\mu}$ for some $\lambda,\mu\in\mathbb{C}$. Note that our results include the Schrödinger-Virasoro algebra and the deformed $\mathfrak{bms}_3$ algebra as special cases, thereby improving upon some of the previously reported results of [5,Theorem 3.4] and [6,Theorem 2]. This work effectively classifies and generalizes the representation theory of the deformed family.

The spectral instability of traveling periodic water waves has been investigated for more than sixty years, since the seminal discovery of Benjamin and Feir. Despite an extensive literature, no rigorous theory has been available for arbitrary three-dimensional – longitudinal and transverse – perturbations. We establish the first rigorous description of the $3d $ unstable spectrum of small-amplitude gravity Stokes waves in deep water in a full neighborhood of the McLean resonant curves. Our results reveal that the Benjamin-Feir instability and the first longitudinal high-frequency isola originate from the same resonant interaction, hidden in the purely longitudinal setting. The dominant instabilities emerge for Fourier-Bloch parameters near the origin, corresponding to the $3d $ Benjamin-Feir modulational instability. Our approach provides quantitative bounds for the real parts of the unstable eigenvalues and establishes a computable necessary and sufficient criterion for the onset of instability near arbitrary high-frequency McLean curves. These results are enabled by three key innovations: ($i$) a Kato perturbative analysis allowing Lipschitz-type singularities of the linearized operator with respect to the Fourier-Bloch parameters; ($ii$) a polar-analytic KAM-type decoupling isolating the unstable eigenvalue pairs near the origin; and ($iii$) an analytic continuation argument in full neighborhoods of the McLean curves. A primary challenge is to establish fine regularity properties for the Dirichlet-Neumann operator conjugated via the Fourier-Bloch transform.

Finite element elasticity complexes of low regularity are constructed on tetrahedral Alfeld splits. In comparison with existing three-dimensional elasticity complexes on such splits, the complexes constructed here lower both the Sobolev regularity and the polynomial degrees, while ending in a hybridizable $H({\rm div};\mathbb S)$-conforming symmetric stress space with no vertex degrees of freedom. The construction is obtained from local Bernstein-Gelfand-Gelfand arguments applied to polynomial de Rham complexes on the Alfeld split. Two local polynomial elasticity complexes are proved: an $H^2$-$H^1({\rm inc})$ complex and a lower-regularity $H^1({\rm curl})$-$H({\rm inc}^+)$ complex. Their bubble subcomplexes and dimension formulas are derived. These local exact sequences lead to unisolvent finite elements for the displacement and incompatibility spaces and to global finite element subcomplexes of the corresponding elasticity sequences. In the lowest-order $H^1({\rm curl})$-$H({\rm inc}^+)$ finite element complex, the $H({\rm inc}^+;\mathbb S)$-conforming tensor space is piecewise cubic. At the same order, the terminal stress-displacement pair recovers the Johnson-Mercier-Křížek element, while the construction covers higher-order hybridizable symmetric stresses for all $k\ge1$. A second family gives a low-regularity $H^1$-$H({\rm inc})$ finite element complex for the standard elasticity sequence for all $k\ge2$. Commuting interpolation diagrams are established for both global complexes.

We show that $L_2$-perturbation theory can be used to transfer rates of convergence in the supremum norm as well as weak convergence in the space of continuous functions from covariance kernel estimators to the associated functional principle components (FPCs). As an application we obtain optimal rates of convergence in sup-norm, including minimax-lower bounds, as well as asymptotic normality for estimating the FPCs in a discrete observational model with errors under fixed, synchronous design. The sparse to dense transition which has previously been observed for mean function and covariance kernel estimators also applies to the FPCs. Surprisingly, eigenvalue estimation exhibits a discretization-dominated regime under sparse designs, too. Our results further apply to estimators of cross-covariance and long-run covariance kernels, as well as to covariance kernels of derivative processes. We also present results of numerical experiments in which we use the Nyström method to compute FPCs and eigenvalues, and give an empirical illustration to series of daily temperature curves.