Papers

We study the K-moduli space of Fano threefolds obtained by first blowing up $\mathbb{P}^3$ along an elliptic quartic curve $C$ and then blowing up a fiber $\ell_p$ of the exceptional divisor $E \to C$. We prove that this K-moduli space is isomorphic to a VGIT quotient parametrizing pairs $(C,p)$, with linearization induced by the CM line bundle. In particular, we classify all K-(semi/poly)stable members of this deformation family. The main new ingredients in the proof include the geometry of the Hilbert scheme of elliptic quartic curves, deformation theory of Fano–K3 pairs, and optimal bounds on the local volumes of threefold singularities.

Let $\{Y_n; n\ge 1\}$ be a sequence of independent and identically distributed random variables with mean zero in Peng's framework of the sub-linear expectation space $(\Omega,\mathscr{H},\widehat{\mathbb E})$, and $S_n=\sum_{i=1}^nY_i$. In this paper, we establish a limit law of \begin{align*}\lim_{n\to \infty}\max_{k\le n}\frac{S_k}{\sqrt{2k \log\log n}}. \end{align*} Different from the result obtained by Chen (2015) in which the limit is a constant, it is shown that under the upper capacity the limit may be prescribed as a given function of $Y_1,Y_2,\ldots$, taking values in the standard deviation interval. As a result, it is also shown that the set of limit points in the compact law of the iterated logarithm can be a symmetric random interval. This paper (Chinese version) has been submitted to Special Issue of Science in China-Mathematics in Celebration of Professor Peng Shige's 80th Birthday. In Theorem 2.2 of the original paper, an additional condition (2.6) is needed.

For a nonconstant planar $p$-harmonic function $u$, with $1 < p < \infty$, and according to previous results, most notably Aronsson's fundamental analysis of the hodograph representation in a neighborhood of an isolated critical point, we introduce the flux-normalized frequency $$ N_*(x_0,r) = \frac{r\displaystyle\int_{B_r(x_0)} |D u|^p\,dx} {\displaystyle\int_{\partial B_r(x_0)} |D u|^{p-2}(u-u(x_0))^2\,dS}. $$ This ratio constitutes the natural counterpart of Almgren's frequency function in the nonlinear setting, as it recovers precisely the degree of homogeneity for every homogeneous $p$-harmonic profile. We further establish the corresponding gauge-corrected version of Almgren-type monotonicity and reinterpret, in terms of the frequency function, the classical planar unique continuation property. In addition, we derive a quantitative pinching estimate for the flux-normalized frequency and identify the specific obstruction that prevents a straightforward extension of the method to higher dimensions.

The real stability radius problem asks for the smallest real perturbation that makes a given matrix lose rank, and it is a structured counterpart of the corresponding complex stability radius problem. In this paper, we consider an angular version of this question for sectorial matrices, where the size of a perturbation is measured by its largest phase instead of its largest singular value. More precisely, for a sectorial matrix $M$, we investigate the minimum required largest phase of a real and sectorial perturbation $\Delta$ for which $I+M\Delta$ loses rank. We show that for some matrices, the complex and the real angular stability radius are the same, but that they are in general different, such as when $M$ is diagonal or complex symmetric. For these cases, we derive a formula for the real angular stability radius, and this formula depends on the two largest phases of $M$. Moreover, in these cases we show that the result for $n \times n$ matrices can be reduced to that for $2 \times 2$ matrices by compressing an arbitrary real destabilizing perturbation to the real two-dimensional subspace spanned by the real and imaginary parts of a destabilizing vector.

Magnitude gives an effective size of a finite metric space and is now used in several data-analysis settings. For such applications, it is natural to ask how magnitude behaves under Gromov-Hausdorff perturbations, including collisions of points. However, magnitude is nowhere continuous on finite metric spaces with the Gromov–Hausdorff topology. We show that this failure is nongeneric in a precise measure-theoretic sense. After fixing the number of points in each collapsing cluster, magnitude is approximately continuous. More strongly, when clusters of points collapse to the points of a limit space, the total weighting of each cluster approximately converges to the weighting of the corresponding limit point. The main tool is a general result on approximate limits of inverses near singular matrices.

For $N\ge3$, let $C(N)$ be the optimal constant in the nearest-neighbour Hardy inequality on $\mathbb Z^N$ with $u(0)=0$, and set $A_N=(N-2)^2/4$. We prove that the continuum coefficient remains an upper bound, $C(N)\le A_N$, in every dimension, and determine the exact values $C(3)=A_3=1/4$ and $C(4)=A_4=1$. In higher dimensions we show $C(N)<A_N$ for $N=9,10$ and obtain the explicit bound \[ C(N)\le 3N-\sqrt{N^2+8N-8}<2N\qquad(N\ge3), \] which lies below $A_N$ from dimension eleven onward. The low-dimensional equalities follow from shifted radial supersolutions, angular convexity, and a discrete ground-state representation. We also show that this shifted-power mechanism cannot work at the continuum coefficient from dimension five onward. Dimension nine is treated by a Gaussian Rayleigh–Ritz construction combined with exact Jacobi theta-function estimates, while the higher-dimensional bounds are obtained through finite-dimensional orbit compressions. Finally, we derive positive spatial remainders in dimensions three and four and a spectral consequence for the associated discrete Schrödinger operators in the high-dimensional regime.

We study the two-dimensional directed polymer model in a Gaussian environment which is independent in time and spatially correlated, with covariances $h(x)$ either summable or with a critical decay, satisfying $h(x) \sim (\log |x|)^a/|x|^2$ as $|x|\to\infty$ for some $a>-1$. We determine the precise high-temperature asymptotics of the free energy, confirming a conjecture of Lacoin (Ann. Probab. 2011), later refined by Cosco, Cottini and Donadini (2025). We also establish a phase transition for the diffusively rescaled partition functions: below some critical point they converge to the Lebesgue measure, while above it they converge to zero. A key feature of our approach is that both results are obtained using only second-moment estimates and are based on a simplified change-of-measure argument in the supercritical regime, that may prove useful for other disordered models.

We investigate decidability of properties in the lattice $\mathop{\mathsf{NExt}} \mathsf{Grz}_t$ of extensions of the Grzegorczyk tense logic $\mathsf{Grz}_t$ and the lattice $\mathop{\mathsf{NExt}} \mathsf{S4}_t$ of reflexive and transitive tense logics, with applications to the lattice $\mathop{\mathsf{Ext}} \mathsf{biIPC}$ of bi-superintuitionistic logics. We prove that a broad class of properties is undecidable in $\mathop{\mathsf{NExt}} \mathsf{Grz}_t$, including tabularity, Kripke completeness, the finite model property, and decidability, which also yields their undecidability in $\mathop{\mathsf{NExt}} \mathsf{S4}_t$. We also construct infinitely many tabular extensions of $\mathsf{Grz}_t$ (and thus of $\mathsf{S4}_t$) whose coincidence problems are undecidable, while presenting one tabular extension of $\mathsf{Grz}_t$ and infinitely many ones of $\mathsf{S4}_t$ with a decidable coincidence problem. As a consequence, we obtain that the finite model property and tabularity are undecidable in $\mathop{\mathsf{Ext}} \mathsf{biIPC}$, and that there are infinitely many tabular extensions of $\mathsf{biIPC}$ whose coincidence problems are undecidable. These results clarify some similarities and differences between $\mathop{\mathsf{NExt}} \mathsf{Grz}_t$ and $\mathop{\mathsf{NExt}} \mathsf{Grz}$, $\mathop{\mathsf{NExt}} \mathsf{S4}_t$ and $\mathop{\mathsf{NExt}} \mathsf{S4}$, as well as $\mathop{\mathsf{Ext}} \mathsf{biIPC}$ and $\mathop{\mathsf{Ext}} \mathsf{IPC}$. The proofs adapt Chagrov's method of reducing from an undecidable problem for Minsky machines. We isolate and explicitly formulate the method of good valuations, a recurring technique underlying several proofs in the literature that use large frames, making it available for further applications.

We study equilibrium computation in simulation-based games motivated by competition in natural-hazard insurance markets. The motivating application considers a multi-region market in which each insurer selects regional pricing and reinsurance decisions to maximize expected financial performance subject to multi-period insolvency constraints. Competitors' decisions affect both profitability and insolvency feasibility through the insured portfolio. Consequently, the strategic interaction defines a generalized Nash game with discontinuous player objectives induced by discrete household purchase decisions and potentially nonconvex feasible sets generated by the insolvency constraints. Discontinuity, nonconvex feasibility, and simulation-defined objectives and constraints complicate the direct use of standard optimality-based equilibrium formulations. Moreover, repeated evaluation of the multi-period simulator makes equilibrium search computationally expensive. To address these computational challenges, we develop an evaluation-efficient framework for computing approximate local equilibria, which combines a response optimizer tailored to the distinct structures of pricing and reinsurance with damped better-response dynamics to mitigate cycling. In a case study of the North Carolina hurricane insurance market, the framework identifies multiple approximate local equilibrium candidates both with and without a household affordability cap and finds no profitable sampled unilateral deviations within the tested neighborhoods under the original simulation model. A representative equilibrium candidate is computed in approximately 5–10 minutes on a single GPU, making repeated equilibrium analyses computationally practical.

Let the center of a $d$-dimensional Euclidean ball be uniformly distributed modulo $\mathbb Z^d$. We study the resulting number of integer lattice points as the dimension and radius tend to infinity. The critical scale is $4\pi^2R_d^2/(d+2)=\log d$. In a logarithmic window around this scale, the logarithm of the lattice-point count divided by the volume satisfies a central limit theorem. At a fixed offset from the critical scale this yields a genuine lognormal limit. Above the critical scale, the count-to-volume ratio converges to $1$ in measure; its normalized error has a standard normal limit, and its variance satisfies an asymptotic formula.

We construct infinite families of reduced almost perfect nonlinear (APN) permutations of $\mathbb{F}_{2^m}$ that admit coefficientwise permutation lifts to $\mathrm{GR}(2^k,m)$ for every $k>1$. These families disprove Conjecture~2 of Rønjom and Sandrib (Cryptogr. Commun., 2026), even for reduced polynomial representatives. For every odd $m\ge5$ with $3\nmid m$, we obtain explicit seven-term polynomials over $\mathbb{F}_2$ with formal derivative identically one. We derive them from a complete criterion for the reduced representative of $A(x)^{2^s+1}$ to have no critical point in $\mathbb{F}_{2^m}$, where $A$ is an invertible linearized polynomial and $\gcd(s,m)=1$. We then apply the cube case to construct a second family of degree $2^{m-1}+2$, whose nonconstant derivatives avoid zero by an explicit trace identity. A fixed parameter in $\mathbb{F}_{32}$ gives this family in every dimension $m=5r$ with $r$ odd, including a degree-$18$ example over $\mathbb{F}_{32}$. Both constructions remain in Gold linear equivalence classes, so liftability of reduced representatives is not invariant under invertible $\mathbb{F}_2$-linear input changes. We obtain the lifting conclusions from the standard Galois-ring permutation criterion. We also verify a degree-$24$ example supplied by Zhang and Haobo, prove a trace-adjoint criterion for reduced quadratic functions, and interpret the derivative condition through ramification at rational points.

Let $n\geq 2$. We establish isodiametric-type upper bounds for Steklov eigenvalues of bounded Lipschitz domains $\Omega$ in Euclidean space $\mathbb{R}^n$ and hyperbolic space $\mathbb{H}^n$. In Euclidean space, we prove \[ D(\Omega)\sigma_k(\Omega)\leq C(n)k, \] with optimal dependence on $k$. For domains in either $\mathbb{R}^n$ or $\mathbb{H}^n$, we establish \[ D(\Omega)\sigma_k(\Omega)\leq C(n)k^2\left(\frac{|\Omega|^{\frac{1}{n}}}{D(\Omega)}\right)^{\frac{n}{n-1}}. \] The exponent of the volume-diameter ratio is optimal in both settings, and with this exponent fixed, the quadratic dependence on $k$ is also optimal. Interpolating the preceding volume-diameter estimate with a volume-normalized Steklov bound, we further obtain \[ D(\Omega)\sigma_k(\Omega)\leq C(n)k^{1+\frac{1}{2n-1}} \] in hyperbolic space. These results answer Open Question 4.37 affirmatively and resolve the hyperbolic part of Open Question 4.35, both posed in [5].

Local constraints govern the low-energy physics of frustrated matter, but familiar ice-type rules constrain flux-like quantities and are insensitive to handedness. Here we show that handedness itself can be imposed as an exact local quantum constraint without selecting an axis in spin space. We construct positive-semidefinite, SU(2)-invariant parent Hamiltonians whose complete zero-energy space on a tetrahedron has a prescribed chirality sign, rather than selecting a particular chiral wave function. For spin-1/2 the local term is a rank-one projector onto a chiral tetrahedral singlet, while for arbitrary spin it factorizes as $B^\dagger B$ through a singlet-annihilation operator, with a completely characterized kernel given by the span of the globally rotated chiral color-ice states. For coherent states, the same zero-energy condition becomes an $S$-independent nonlinear constraint in which three spin directions determine the fourth through a Möbius transformation; compositions of these maps define constraint holonomies on extended lattices. Connecting the same local constraint in different ways produces qualitatively different collective regimes: corner-sharing lattices retain exponentially large quantum ground-state kernels, with rigorous lower bounds exceeding conventional ice benchmarks; edge-sharing lattices support subdimensional plane or line zero modes; while triangular constructions suppress nonuniform coherent deformations and contain the complete Anderson tower of tetrahedral magnetic order at exactly zero energy. Two inequivalent triangular coverings further show that harmonic zero-mode counting does not determine the size of the quantum kernel. These results establish a tractable setting in which local handedness, nonlinear constraint geometry, and quantum degeneracy can be disentangled and related directly to the connectivity of the constraint network.

A classical result of Andrásfai, Erdős, and Sós states that every $n$-vertex graph with odd girth at least $2k+1$ and minimum degree larger than $\frac{2n}{2k+1}$ is bipartite. Rather than imposing a minimum-degree condition, in this paper we investigate conditions on algebraic connectivity that force graphs of given odd girth to have a simple structure. The algebraic connectivity of a graph $G$, denoted by $\mu_2(G)$, is the second smallest eigenvalue of its Laplacian matrix. Our main results are as follows. 1. Every $n$-vertex triangle-free graph $G$ with $\mu_2(G)\geq \frac{n}{3}$ is bipartite. Moreover, the constant $\frac{1}{3}$ is asymptotically best possible. 2. For $k\geq 3$, every $n$-vertex graph $G$ of odd girth at least $2k+1$ with $\mu_2(G)>\frac{4n}{6k-1}$ is bipartite. 3. For $k\geq 22$, every $n$-vertex graph $G$ of odd girth at least $2k+1$ with $\mu_2(G)>\frac{3456n}{k^3}$ is bipartite. Moreover, the term $k^{-3}$ is asymptotically best possible.

In this paper, we explicitly determine an algebraic Belyi function on a unique smooth projective model $\widetilde{W}$ of the Wiman sextic curve $W$ and describe its complex uniformization in terms of modular functions associated with a certain noncongruence subgroup $\Gamma_{\widetilde{W}} \subset {\rm SL}_2(\mathbb{Z})$. As an application, we give a direct proof of the unbounded denominators conjecture for weight $2$ cusp forms with respect to $\Gamma_{\widetilde{W}}$. The conjecture is now known in full generality by the work of Calegari, Dimitrov, and Tang, following earlier progress including work of Dong, Lin, and Ng. Our proof, however, uses a degeneration of $\widetilde{W}$ over $\mathbb{F}_{5}$ together with explicit Puiseux series expansions and is substantially different from the methods employed in their work.

We consider a Hele–Shaw-type tumor growth model obtained from the incompressible limit of a porous-medium equation. The limiting model preserves the patch structure of the tumor density and therefore admits a single free-boundary formulation. We investigate how a pressure-feedback parameter associated with the homeostatic pressure affects boundary stability. Using asymptotic analysis, we study boundary perturbations of finite-radius tumors and planar traveling fronts and derive the corresponding stability criteria. For planar fronts, the stability thresholds are further shown to be bifurcation points from which nonsymmetric traveling waves emerge. Our results indicate that pressure feedback can induce boundary instability in parameter regimes where the model without pressure feedback remains stable.

In a preceding article, we introduced stable symmetric series encoding simultaneously the normalized conjugacy classes of all symmetric groups. The same rational series remain stable for the Jack-deformed Goulden–Jackson product. We investigate their two-parameter Macdonald analogue. Starting from the $(q,t)$-deformed class product (dual to the coproduct diagonal on the $J$-basis), we construct the unique infinite series whose multiplication realizes any shifted Macdonald eigenvalue. In contrast with the classical and Jack cases, this transform is no longer multiplication by a fixed explicit series. We identify it with a composition of a Cauchy multiplication, the integral nabla operator, and a simple diagonal operator. We then determine the series realizing the Nazarov–Sklyanin operators $A^{(k)}$, derive a single generating series for all column partitions, and compare our construction with the Macdonald characters and Theta operators of Ben Dali and D'Adderio.

In this paper, we consider structure-preserving model reduction of port-Hamiltonian (pH) systems which extend classical Hamiltonian systems with dissipation and an input-output port. These pH systems are often used in multi-physics systems, as the interconnection of one or more \pH systems results again in a pH system. If particularly the system matrices associated with the interconnection and/or dissipation of a pH system are state-dependent, then the evaluation of standard reduced-order models (ROMs) may depend on the dimension of the original full-order model, resulting in high computational costs. To circumvent these high costs, we propose to use structure-preserving neural networks. In particular, we perform two steps: (1) we use the generalized manifold Galerkin projection to project the pH system onto the reduced space; then (2) we train a neural network to learn the map from the reduced-order state to the reduced-order interconnection and dissipation system matrices. To ensure that the resulting ROM is again a pH system, the architecture of the neural network is chosen such that the skew-symmetry and positive semi-definiteness of the reduced-order systems matrices are maintained. In a numerical example, we consider a nonlinear mass-spring-damper system with state-dependent system matrices. The numerical results show that the proposed method achieves a significant computational speed-up compared to the original \ROM with comparable accuracy.

In a recent breakthrough, Bamler-Lai proved the mean-convex neighborhood conjecture in all dimensions by showing that any nontrivial ancient asymptotically cylindrical mean curvature flow is – up to splitting Euclidean factors – either a translating bowl, or an ancient oval, or a translating oval-bowl. In this paper, we provide a short alternative argument for two of the three scenarios. Specifically, using more elementary/traditional methods, we show that if the convergence to the round cylinder is fast then the solution is a bowl times a Euclidean factor, and if the convergence is slow then the solution is an ancient oval. Moreover, the present paper also yields a new proof of the mean-convex neighborhood conjecture for neck-singularities that substantially simplifies and streamlines the approach from our prior work joint with Hershkovits (Acta '22) and Hershkovits-White (Inventiones '22).

Kashiwara–Nakashima tableaux and Gelfand–Tsetlin patterns of orthogonal type are famous as combinatorial models of the finite-dimensional irreducible representations of the special orthogonal Lie algebras $\mathfrak{so}_N$. The former is constructed based on the representation theory of quantum groups, especially the theory of crystals, while the latter based on the branching rule for $(\mathfrak{so}_N,\mathfrak{so}_{N-1})$. In the present paper, we construct a natural bijection between them by means of the representation theory of quantum symmetric pairs corresponding to $(\mathfrak{so}_N,\mathfrak{so}_{N-1})$.

In this paper, we establish several uniform estimates for Kähler–Ricci shrinkers without imposing any curvature assumptions. In particular, we prove: 1. a uniform lower bound for the entropy; 2. a uniform lower bound for the asymptotic volume ratio of Kähler–shrinkers with maximal volume growth; 3. a uniform lower bound for the scalar curvature on balls centered at a minimum point of the soliton potential for non-Gaussian shrinkers.

The purpose of this work is to provide a broader framework for analyzing the 1D mass-subcritical inhomogeneous nonlinear Schrödinger equation (INLS) \[iu_t + u_{xx} \pm |x|^{-b}|u|^{\alpha-1}u=0, \quad 1<\alpha< 5-2b\; \text{and}\; 0< b\leq 1/4.\] in Sobolev spaces of negative order, reaching almost critical Sobolev spaces by working in Fourier-Lebesgue and modulation spaces. More precisely, we establish low-regularity well-posedness in the Fourier-Lebesgue $\widehat{L}^{p}$ spaces for $4/3<p<8$. Primarily, we prove global well-posedness for arbitrarily large initial data via the data decomposition method adapted for the Fourier-Lebesgue spaces, with generalized Strichartz estimates for $p<2$ and classical Strichartz estimates for $p>2$. Furthermore, we obtain analogous results for (INLS) in modulation spaces $M^{p,p'}$ for $4/3<p<2$.

Vasudeva Murthy's relaxation approach [A.S. Vasudeva Murthy, J. Comput. Appl. Math., 203(2), pp. 437-443, 2007], originally proposed for the Jin-Xin relaxation model, provides an alternative formulation with invariant properties that is consistent and retains the semilinear structure incomparison to the standard one. In this work, Vasudeva Murthy's relaxation approach for various relaxation systems are proposed such as the shallow water equations, the Broadwell model, the Euler equations with heat transfer and two-dimensional Jin-Xin model. For proposed relaxation models, the associated integral invariants are rigorously established at the theoretical level. The main advantage of the integral invariant is that it provides a conserved quantity for the relaxation system by incorporating the coupled contributions of the solution variables in vector form. To validate the analytical results, numerical simulations are carried out for each model using three second-order numerical schemes: CS-EBT2, a semi-implicit second-order central finite-volume scheme for hyperbolic systems with relaxation source terms [S. Sahu, E. Macca, and R. Samala, J. Comput. Phys., 563, 115100, 2026]; UCS2, a finite-volume central relaxation-type scheme [S. F. Liotta, V. Romano, and G. Russo, SIAM J. Numer. Anal., 38(4), 1337-1356, 2000]; and IMEX-RK2, a second-order Implicit-Explicit Runge-Kutta scheme [Pareschi and Russo, J. Sci. Comput., 25, 129-155, 2005]. Numerical results, compared to exact or finely resolved reference solutions, confirm that the models preserve integral invariants, remain stable under CFL restrictions, and exhibit robust and accurate behavior across all benchmark systems tested.

We study asymptotic cones of Lie groups, presenting a link between the geometry of the weights and the highly non-simply-connected nature of asymptotic cones. We show that a Lie group that admits a group of SOL-type as a quotient is such that the fundamental group of its asymptotic cone contains the fundamental group of the Hawaiian earring space. We also obtain a strong non-vanishing result for the first homology group of the asymptotic cone. The assumption on the Lie group, introduced by Abels and called the "SOL obstruction", is equivalent to an explicit geometric condition on the weights. Our work builds upon the results of Burillo, who proves the statement on asymptotic cones in the case of SOL, and on those of Cornulier and Tessera who show that the SOL obstruction implies exponential growth of the Dehn function.

Breakpoint graphs originate in comparative genomics, where their alternating cycles encode relationships between genomes. We study a constrained class of three-colored breakpoint graphs associated with permutations and develop cycle-refined generating functions for two extremal families. These families have a natural topological interpretation: their canonical surfaces are, respectively, the sphere and the projective plane. The spherical family is characterized by noncrossing configurations, while the projective-plane family admits a different decomposition involving a distinguished family of Möbius ladders. The resulting generating-function equations retain the full cycle structure but nevertheless admit substantial reductions. This leads to explicit Catalan-weighted evaluations, polynomiality results for refined cycle statistics, and a connection between a natural diagonal specialization and noncrossing trees. The two topological families exhibit markedly different combinatorial mechanisms, providing complementary examples of how local transformations of breakpoint graphs can control refined permutation enumerations. As a further application, the same Catalan-weighted sums arise in asymptotic unitary-Weingarten expansions for entanglement of random Gaussian states in linear optics. The combinatorial results determine the leading and constant-order moment polynomials entering the Rényi entropy expansion, with the projective-plane contribution giving the finite-size constant correction.

We study heat semigroups on bounded Lipschitz domains $\Omega \subset \mathbb{R}^{d}$ with dimension $d>2$ under non-local Robin boundary conditions. The boundary operator $B \in \mathcal{L}( \mathrm{L}^{2}(\partial\Omega))$ is allowed to destroy the positivity of the semigroup. We assume that there exists a positive operator $C \in \mathcal{L}(\mathrm{L}^{2}(\partial\Omega))$ such that $$ |Bu| \leq C|u| \quad\text{for every }u\in \mathrm{L}^{2}(\partial\Omega), \qquad C{\bf 1}\in \mathrm{L}^{\infty}(\partial\Omega). $$ Under this assumption, we prove that the semigroup generated by the corresponding uniformly elliptic operator is ultracontractive. More precisely, its norm from $\mathrm{L}^{2}(\Omega)$ to $\mathrm{L}^{\infty}(\Omega)$ has short-time order $t^{-d/4}$. The main step is the construction of a positive eigenfunction $\phi$ of a dominating elliptic operator such that $$ 0 \ < \ \delta \ \leq \ \phi \ \leq \ M $$ almost everywhere in $\Omega$. The upper bound is obtained by power truncations and an iteration of Sobolev exponents. The lower bound follows by comparison with the Neumann semigroup. The two-sided estimate gives an $\mathrm{L}^{\infty}$-bound for the comparison semigroup. Nash's inequality and duality then yield ultracontractivity.

Hyperbolic length functions are among the most fundamental ones on Teichmüller space, and they are quasi-convex along Teichmüller geodesics. In this paper, we investigate the same question for energy functions of harmonic maps in two natural settings, which may be viewed as nonlinear and higher-dimensional analogs of the length functions. For a fixed domain and a varying hyperbolic target, we prove the energy is quasi-convex along Teichmüller geodesics under a filling hypothesis. Furthermore, we generalize Masur's result on asymptotic growth of the length function along the Teichmüller geodesic determined by a Jenkins-Strebel differential to the energy functions. We also prove the quasi-convexity for covering maps between closed hyperbolic surfaces with fixed target and varying domains. We derive first and second variation formulas of energy functions along Teichmüller geodesics and explain why the natural global statement is quasi-convexity rather than genuine convexity.

We study both the complex and Hermitian geometry of C-spaces, i.e., compact homogeneous spaces M=G/K admitting a G-invariant complex structure. Some results on Hodge, Bott-Chern and Aeppli cohomologies are given. We also prove classification results in the pluriclosed, parallel Bismut torsion and locally conformally Kähler (or Vaisman) cases, respectively, as well as existence results for balanced and Calabi-Yau with torsion metrics.

We establish that for a $C^r$-generic surface repeller ($1 < r \leq \infty$), its Hausdorff and box dimensions are exactly the unique zero of the sub-additive topological pressure function. As an application of our framework, we show that the attractor of a uniformly non-conformal and weakly irreducible planar non-linear iterated function system (IFS) satisfying the strong separation condition (SSC) attains its expected Hausdorff and box dimensions. Furthermore, as a direct consequence of our generic repeller theorem, we deduce that this dimension formula also holds for a generic $C^r$ planar IFS satisfying the SSC. Using this approach, we also extend the dichotomy result for graphs of Weierstrass-type functions of Ren and Shen (2021) by weakening their real-analytic requirement to arbitrary $C^r$ regularity for $r > 1$.

The cohomology of a differentiable stack can be modelled by the Bott-Shulman-Stasheff bicomplex. However, computing its cohomology is still difficult in many examples. Improving on this requires additional methods such as using spectral sequences and finding models for the cohomology of each column of the bicomplex. This paper provides a first attempt of relating and unifying two approaches of this form. Firstly, Arias Abad has generalised the Bott spectral sequence to Lie groupoids by introducing representations up to homotopy. The second approach is the recent work on a model for the cohomology of differentiable stacks associated to proper and regular Lie groupoids by this author.