Papers

Classical Ehrhart theory measures the discrete capacity of a convex rational (or integral) polytope $P$ by counting the number of lattice points in the $t$-th dilate $tP$ of $P$. In this paper, we extend this paradigm by replacing a lattice point with a geometric object $Q$ of dimension at most $\dim P$. We show that the counting function $\mathrm{ehr}(P,Q;t)$ of such valid translations of $Q$ into $tP$ inherits eventual quasi-polynomiality (or eventual polynomiality) with leading term $\mathrm{vol}(P)t^d$, where $d = \dim P$. This result is naturally derived by induction on the dimension, based on the classical quasi-polynomiality (or polynomiality) of Ehrhart functions. Similarly, replacing $P$ with its relative interior $P^\circ$ defines $\mathrm{ehr}^\circ(P,Q;t)$, which is also shown to be an eventual quasi-polynomial (or eventual polynomial). Furthermore, we prove several formulas for relative Ehrhart functions under the condition that $P = kP_0$ and $Q = lQ_0$ for some integers $k$, $l > 0$, and polytopes $P_0$ and $Q_0$ with $\dim Q_0 > 0$ such that $Q_0$ is inscribed in $P_0$. In particular, we prove that under this condition, $\mathrm{ehr}(P,Q;-t) = (-1)^{d}\mathrm{ehr}^\circ (P,Q;t+\rho )$ $(t \gg 0)$ holds for some integer $\rho > 0$ if and only if $k \mid 2l$ via the classical Ehrhart–Macdonald reciprocity law. We also prove that under the same condition, $\mathrm{ehr}(P,Q;t) = \mathrm{ehr}^\circ (P,Q;t+\sigma )$ $(t \gg 0)$ holds for some integer $\sigma > 0$ if and only if $\mathrm{ehr}^\circ (P_0;\mathrm{codeg}\,P_0) = 1$ and $k \mid \mathrm{codeg}\,P_0$.

Let $p$ be an odd prime, let $\varnothing\neq A\subseteq\mathbb F_p$ have cardinality $N$, and let $f\in\mathbb F_p[x,y]$ be a non-degenerate quadratic polynomial. Writing $S=|A+A|$ and $M=|f(A,A)|$, we prove the full-range trade-off $S^8M^6\gtrsim N^{17}(1+N^3/p^2)^{-3}$. Consequently, $\max\{|A+A|,|f(A,A)|\}\gtrsim \min\{N^{17/14},p^{3/7}N^{4/7}\}$, and in particular the exponent $17/14$ holds throughout $N\le p^{2/3}$. The proof combines a centered collision estimate for $F(u,v,w)=f(u+v,w)$, a mixed fourth-energy bound, and a popular-sum amplification. Two complementary incidence estimates enter the argument: a centered spectral bound in the dense collision regime and a point–plane bound in the sparse regime.

A thin shell carries load through its shape and, at finite deflections, its stiffness changes with the load itself, so a shape optimum found with a linear model can be far from optimal. We present an automated framework that embeds the fully geometrically nonlinear shell response in the shape-optimisation loop. The forward model, a five-parameter nonlinear Naghdi shell stabilised against locking by partial selective reduced integration, operates directly on a discrete (faceted) triangulation with a numerically recovered director field – dispensing with the exact mid-surface parameterisation of isogeometric approaches, a chart that ceases to exist once the geometry itself is the design variable. Implemented in Firedrake, the model generates its residual, consistent tangent and adjoint automatically; shape derivatives, computed by algorithmic differentiation through the full load-continuation solve, drive the Fireshape/ROL trust-region optimiser. The forward solver reproduces the Sze/Abaqus benchmark for a clamped semi-cylindrical shell under a point load, capturing the progressive stiffening that a geometrically linear model cannot reproduce. The optimisation is validated against the COMSOL benchmark, a sheet-metal bracket under bending: the framework develops the same off-mid-plane corrugation mechanism and attains an 87% reduction of elastic strain energy within the prescribed displacement budget, matching the benchmark's magnitude and area change. Applied to the curved semi-cylinder, it forms a smooth stiffening crease that reduces the shell's average deflection under load by 78%.

We study attainment boundaries for linear optimization over unbounded convex sets. For epigraphs of finite convex functions, convex conjugacy separates recession-cone copositivity, boundedness below, and attainment through three nested subsets of conjugate space. At a finite but unattained boundary value, we establish a facewise asymptotic selection theorem for inward perturbations of the objective. The theorem identifies the escaping directions of the perturbed minimizers, their precise blow-up scale, and the leading asymptotics of the optimal value; the critical set may be multidimensional, and no radial symmetry is assumed. For radial asymptotically conic epigraphs, we obtain a complete boundary trichotomy and universal scaling laws for objective tilting, hard truncation, and power regularization. For shifted ellipsoidal second-order cone programs, we derive explicit primal–dual formulas together with sharp escape, conditioning, and regularization rates. These models also admit an exact robust-optimization representation. At the attainment boundary, strict primal feasibility, zero duality gap, and dual attainment can coexist with failure of primal attainment.

In this work, we show that if $h(x)$ is an irreducible monic integer polynomial of degree $2q$ (with $q$ prime), whose defining field extension of $\mathbb{Q}$ contains a Galois extension of degree $q$, then there is a positive density of integers $n$ such that $h(n)$ is squarefree; in particular, $h(n)$ is squarefree for infinitely many integers $n$. As an application, we prove that the family of exceptional cubic fields contains an infinite subfamily whose unit shapes converge to the hexagonal lattice. To the best of our knowledge, this is the first example of a family of non-Galois totally real cubic fields whose unit shapes converge to the hexagonal lattice.

In this paper, we prove that a one-dimensional Peano continuum $X$ admits a weak homotopy equivalence $X\to C$ to a countable space $C$ if and only if the wild set $\mathbf{w}(X)$ is countable.

Let U be uniformly distributed on the unit sphere. We develop a self-contained forward and inverse theory for the random width w_K(U) and brightness b_K(U) of three-dimensional convex bodies. For every full-dimensional polytope, a global spherical co-area formula expresses the width density as a finite sum of angular apertures determined by the normal fan of its difference body; in particular, the density is piecewise real analytic with a finite geometrically determined critical set. This theory yields exact densities for the width of the regular tetrahedron, resolving a question of Finch, and for the regular truncated octahedron, together with the tetrahedral brightness law and the equivalent rhombic-dodecahedral width law. On the inverse side, second- and third-order polarized cosine-transform moments reconstruct finite labelled direction systems whenever the observed triangles span the cycle space of the correlation graph; signed-graph switching describes the unavoidable ambiguity. In contrast, equal three-dimensional intrinsic volumes do not determine either the width law or the brightness law, even for centrally symmetric bodies. Removing the spatial rank constraint gives a dimension-free identifiability theorem for centered multivariate folded-normal vectors: pairwise absolute moments and an anchored family of triple absolute moments, comprising |m - 1|^2 labelled observations for a complete correlation graph, determine the correlation matrix up to diagonal sign conjugacy without fourth-order moments. A harmonic decomposition further identifies the degree-two variance contribution as a constant multiple of the squared Frobenius norm of the traceless part of the weighted frame operator and explains why this contribution vanishes under irreducible symmetry.

Let $\lambda_k(\Omega)$ denote the $k$th Dirichlet Laplacian eigenvalue of a bounded planar domain $\Omega$, with eigenvalues counted with multiplicity. We disprove a conjectured higher-index extension of the Payne–Pólya–Weinberger inequality that was recorded as an open problem by Ashbaugh. For every integer $m\geq3$, we construct a bounded planar domain $\Omega_m$ with $C^\infty$ boundary such that $$ \frac{\lambda_{2m}(\Omega_m)}{\lambda_m(\Omega_m)} >\frac{13}{5} >\frac{j_{1,1}^2}{j_{0,1}^2}, $$ where $j_{\nu,1}$ is the first positive zero of $J_\nu$, the Bessel function of the first kind of order $\nu$. For $m=3$, the annulus $\{x\in\mathbb{R}^2:1/10<|x|<1\}$ already provides such a counterexample.

This paper studies the modularity problem for elliptic curves from the elementary geometry of peels of decorated equilateral triangulations. A peel is a finite developing domain together with the boundary identifications that reconstruct the triangulated surface. In genus one it recovers the universal cover and its rank-two deck lattice. For a fixed prime $\ell$, the finite unramified coverings $[\ell^n]:E\to E$ form a compatible tower whose deck groups are $E[\ell^n]$; their inverse limit is the Tate module $T_\ell(E)$, and over $\Q$ the compatible Galois action gives the usual $\ell$-adic representation. Cyclic isogenies, by contrast, give the index-$q$ lattice neighbors that form the local model for Hecke and Brandt correspondences. For a semistable elliptic curve $E/\Q$ of prime conductor $p$, the missing reciprocity is the construction, without using modularity, of a nonzero class $c_E$ in the degree-zero supersingular Brandt module such that $B_qc_E=a_q(E)c_E$ at every good prime. I make this obstruction explicit and study oriented CM packets attached to the Frobenius discriminants $D_q=a_q(E)^2-4q$. New elementary results show that $D_q$ is a nonsquare modulo $p$ exactly when the residual Frobenius polynomial is irreducible, that any residual image containing such an element gives a positive-density set of primes for which $p$ is inert in the quadratic field determined by $D_q$, and that the corresponding discriminants are necessarily unbounded.

High-order shock-capturing schemes often face severe computational bottlenecks due to wide stencils, stagewise nonlinear weights, and expensive local characteristic decompositions. To address this, we propose a compact finite-average-moment (FAM) framework for hyperbolic conservation laws that completely decouples the formal spatial order from the number of evolved local degrees of freedom. By evolving only a $\mathbb{P}^1$ moment state (i.e., cell averages and scaled first-order moments), we achieve up to sixth-order accuracy ($k=3, 4, 5, 6$) via compact, polynomially exact linear moment reconstructions. A key algorithmic innovation is consolidating the nonlinear stabilization into a single, derivative-free oscillation-elimination (OE) procedure applied only after the final Runge–Kutta stage. This OE procedure exactly preserves cell averages while applying an explicit exponential correction to the first-order moments, avoiding repeated stagewise nonlinear weights or limiting. Fourier analysis of the linear backbone demonstrates $k$th-order accuracy, strong spurious mode damping, and $(k+1)$th-order cell-average superconvergence. Extensive numerical experiments on scalar laws and the Euler equations verify the expected smooth accuracy and robust, nonoscillatory shock resolution. Notably, by performing componentwise reconstruction directly in conservative variables and streamlining stabilization, the FAM schemes deliver significantly lower complete-run wall-clock times compared to the multi-resolution weighted essentially non-oscillatory (MR-WENO), unified-stencil Hermite WENO (HWENO-U), and OE-HWENO methods.

Let $(\mathfrak g,J,g)$ be a unimodular Hermitian Lie algebra containing an abelian ideal $\mathfrak a$ of real codimension two. We study the curvature behaviour of $g$ and show that, if $g$ has constant Chern holomorphic sectional curvature, then it must be Chern flat. We also show that, for every canonical metric connection $D_s^r$ of $g$ other than the Chern connection, if $D_s^r$ has constant holomorphic sectional curvature, then $g$ is Kähler flat, and in this case $\mathfrak g/\mathfrak a$ is abelian.

Plasmode simulation - resampling covariates, and often treatment, from a real cohort and regenerating the outcome from a model with an injected effect - is a standard tool for comparing causal-inference estimators. We argue that its validity turns on two separable design choices, and that separating them clarifies a recent debate over resampling versus regenerating the treatment. Our central message limits what such a benchmark can show: a plasmode can refute a universal claim about an estimator but cannot certify one. Refuting needs only that the simulated data genuinely belong to the class over which a guarantee is quoted; certifying also needs the benchmark's difficulty to carry over to the target population, which the analyst cannot verify. Two choices set that difficulty. Regularity is whether the simulated law preserves the overlap that inverse-weighting and targeted-learning theory require. Resampling the real treatment with fine-grained covariates destroys it, and the resulting coverage failure falls on flexible, machine-learning-based estimators, while parametric-nuisance estimators - including inverse-probability weighting - stay near nominal; it disappears once the treatment is regenerated from a bounded propensity or the covariates are coarsened, and cross-fitting removes only part of it. Projection is that the injected outcome model hands a matching estimator the right answer by construction; this survives the regularity fix and can reverse estimator rankings even when overlap is intact. We give a regular-plasmode algorithm, a confirming real-data simulation, and practical recommendations. A plasmode is a filter against estimators, not a warrant for them.

Leaks such as the Panama and Paradise Papers expose large networks of offshore entities, and they invite an obvious question for network learning. Can the relations these structures are built to hide—above all, who beneficially owns what—be recovered from the public part of the leak by link prediction? We argue that the answer is mostly no, and that the analyses which suggest otherwise are measuring the wrong thing. Our main result is a distribution-free identifiability bound. For any recovery rule that respects graph isomorphism, and that covers every topological link-prediction score together with every message-passing graph neural network, a concealed endpoint left isolated in the observed graph is interchangeable with its structural twins, so its hidden edge cannot be recovered above chance. Isolation is only the sharpest case. In general the ceiling on recovery is set by the size of a node's structural-indistinguishability class, for which the Weisfeiler–Leman colour class is a computable stand-in, and node degree is at best a loose proxy. On the full ICIJ Offshore Leaks graph (814K entities and 84K labelled beneficial-owner edges) a classifier-free, degree-controlled probe reproduces an exact $0.5$ floor for isolated owners, who make up $20.6\%$ of all owners, and a trained graph neural network lands on the same floor. Recovery climbs only as structural distinctiveness grows, and the floor reappears in every one of the five leaks. Along the way we document five evaluation traps. Each one makes a bound that cannot be beaten look beaten, and we give a short rule that avoids them. The practical upshot is to redirect effort from the hidden principal, which is close to unrecoverable, toward the machinery of concealment, and to spell out why fusing external data helps far less than one would hope.

We prove a Kolmogorov–Riesz compactness theorem for Banach lattices over spaces of homogeneous type. Especially, we consider the one for $L^1$. Our characterization is formulated in terms of boundedness, tightness, and approximation by averaging operators, which replace translations in the absence of a group structure. The result requires neither the Vitali covering property nor continuity assumptions on the measure of balls, lower volume bounds, or finiteness of the underlying measure space, thereby extending several recent compactness criteria in the literature. Our approach begins with a new proof of the Kolmogorov–Riesz theorem in $L^1$, which avoids reflexivity and yields a unified treatment of $L^p$-spaces for $1\le p<\infty$. We then establish an approximation theorem based on Christ's dyadic cubes and use it to characterize the closure of compactly supported continuous functions in general ball Banach function spaces. Applications include compactness criteria for rearrangement-invariant Banach function spaces and Morrey spaces. In the Euclidean setting, we also identify the resulting approximation space with the classical heat-semigroup (equivalently, mollifier) closure.

Let $F_n$ be uniform on all forests of the simple complete graph $K_n$, with isolated vertices allowed. A conjecture of Kahn and of Winkler, studied by Grimmett and Winkler, asserts that any two distinct edges of any finite graph are negatively correlated under the uniform forest measure. Stark proved this for $G=K_n$ once $n$ is sufficiently large, but did not furnish an explicit threshold. We prove it for every $n\geq 2$, strictly whenever two distinct edges exist. The difficulty is concentrated in the disjoint-edge orbit, whose correlation ratio tends to one. We remove this cancellation before estimating anything: the desired inequality becomes an exact comparison among the first two moments of the component count and the expected sum of squared degrees. When the component count fluctuates, this comparison contains a variance term absent from the fixed-component identities of Tang and Zhang. Component marking, tail elimination, and effective Stirling bounds control the three moments for $n\geq 651$; exact integer recurrences cover the remaining values.

Let $M$ be chosen uniformly from all matchings of a finite linear $k$-uniform hypergraph $H$, and let $\overline{q}(H)$ be the average probability that a vertex is left uncovered. If $H$ has maximum degree $D$ and normalized average degree $\beta=k|E(H)|/(|V(H)|D)$, then, for every fixed $k\geq 2$ and uniformly in the order, $\overline{q}(H) \leq 1-\beta+(\beta k+o_D(1))\log\log D/\log D$. The underlying estimate is the order-uniform count $\log Z(H)\geq (|E(H)|/D)(\log D-(k+o_D(1))\log\log D)$, and it also counts matchings of size $(1-o_D(1))|E(H)|/D$. We prove this by sampling edges, deleting those incident with unusually large sampled degrees, and controlling only the total deleted mass before applying the Molloy-Reed list edge-colouring theorem. For $d$-regular $H$, a quantitative version of the earlier Asratian-Kuzjurin sampling-to-counting route, using the near-perfect-matching theorem of Gould and Kelly, sharpens the coefficient: for every fixed $k\geq 3$, $\sup_H \overline{q}(H) \leq (\max\{3,k-1\}+o_d(1))\log\log d/\log d$. Kahn's stronger pointwise prediction was disproved by Lee for $k\geq 3$. The qualitative averaged conclusion was already recorded by Kahn and Kim, crediting Anders Johansson, and also follows from the Grable-Asratian-Kuzjurin enumeration; neither source states a rate.

Adaptive stochastic quantization (ASQ) is a recently introduced quantization approach that optimizes the Mean Squared Error (MSE) for a given input while preserving unbiasedness. It is designed to alleviate the communication and memory bottlenecks of modern data and machine learning workloads, including model, gradient, and KV-cache compression and nearest-neighbor search. Further, practical systems can then compress quantized data with a lossless entropy encoder. However, existing unbiased methods, including ASQ, choose their quantization values without considering this later encoding stage, leaving accuracy on the table. We formulate the Entropy Constrained Adaptive Stochastic Quantization (ECASQ) problem, which jointly selects adaptive quantization values to minimize MSE under an entropy budget and an unbiasedness constraint. We give an optimal dynamic program with $O(sd^2)$ time and $O(d^2)$ space for a length-d vector and at most s quantization values, and a GPU-friendly approximate dynamic program with $O(sd^2)$ time and $O(d)$ space. The approximation guarantees that the solution has an MSE no larger than the optimal solution that uses one fewer bit of entropy per entry. We also provide an iterative refinement procedure for the approximation solution that, in our experiments, yields near-optimal results while retaining a substantial speed advantage over our solver for the optimal solution.

We develop a diffeological framework for the geometry of Euler–Reynolds subsolutions of the incompressible Euler equations. Passing to a lifted formulation in which the velocity, quadratic flux, and trace-free Reynolds stress are treated as independent variables, we construct a diffeological limit space obtained as the weak closure of smooth strict subsolutions. Its internal tangent spaces provide an intrinsic notion of infinitesimal deformation despite the absence of any underlying manifold structure. We prove that ambient realizations of internal tangent vectors satisfy the linearized Euler–Reynolds equations in the sense of distributions, thereby establishing a natural first-order theory for lifted weak limit spaces. We further describe the tangent directions compatible with the Euler locus and identify the kernels of the natural velocity, flux, and full-state observables. These results distinguish observable perturbations from hidden stress-gauge directions that encode infinitesimal variations of the Reynolds stress. Finally, we introduce a finite-mixture model for lifted Euler–Reynolds states whose differential realizes explicit tangent directions and relates Reynolds stress to infinitesimal phase splitting. Under a genericity assumption, every deviatoric stress tensor is realized by such a first-order mixture defect, yielding phase-counting bounds and a minimality result for the observable hierarchy.

Let $\mathcal H$ be a finite-dimensional complex Hilbert space, and let $w(P,\mathsf M)$ be a normalized probability weight assigned to an outcome projection $P$ as it occurs in a projective measurement $\mathsf M$. For fixed $P$, let $G_P\cong\mathcal U(P^\perp)$ be the group of unitaries acting identically on $\operatorname{ran}P$ and arbitrarily on $P^\perp$. We classify the $G_P$-orbits of projective measurements containing $P$: two measurements lie in the same orbit exactly when the multisets of ranks of their complementary outcomes agree. The orbit with profile $\lambda=(r_1,\ldots,r_k)$ is a compact homogeneous space of real dimension $(d-\operatorname{rank}P)^2-\sum_j r_j^2$. On maximal rank-one measurements there is one orbit, so context independence is equivalent to $G_P$-invariance, with equality of the corresponding uniform defects; invariance under two-level complementary unitaries already suffices. For arbitrary projective measurements, every context containing an outcome $P\neq I$ coarsens to the unique binary context $\{P,I-P\}$. Consequently, refinement consistency alone is equivalent to context independence. The finite orbit space carries a natural rank-profile refinement graph, whose diameter is $d-\operatorname{rank}P-1$ when $P\neq I$. We prove a stability theorem that compares the binary-coarsening path with a shortest path in this graph followed by one complementary unitary. In dimension at least three, Gleason's theorem converts the maximal-context invariance condition and the all-context refinement condition, on their respective domains, into the Born form $\operatorname{Tr}(\rho P)$. The results are structural characterizations, not independent physical derivations of context independence.

We develop an integrability framework for Hamiltonian dynamics on con-locally conformal symplectic (COLCS) manifolds, odd-dimensional quadruples $(M,\Omega,\theta,\eta)$ where $d\Omega=\theta\wedge\Omega$, a closed $1$-form $\eta$ determines a codimension-one distribution on which $\Omega$ is non-degenerate, and $R$ is the Reeb vector field satisfying $\iota_R\Omega=\iota_R\theta=0$, $\iota_R\eta=1$. This class simultaneously generalises LCS and cosymplectic manifolds and provides a natural arena for time-dependent Hamiltonian systems with twisted differential $d^\theta=d-\theta\wedge$. The COLCS bracket is introduced on $C^\infty(M)$ and shown to be a Lie bracket that induces Poisson structures on the subalgebras of $\theta$-strong ($\theta(X_H)=0$) and $R$-strong ($R(H)=0$) functions. A Lie-type integrability theorem is then established: given $2n-k$ functionally independent first integrals with $k$ of them $\theta$-strong and generating a solvable Lie algebra under the COLCS bracket, the flow is integrable by quadratures on the common level set. Finally, scaling symmetries of degree $(\Lambda,\beta,\gamma)$, defined by $L_X\Omega=\beta\Omega$, $L_XH=\Lambda H$, $L_X\eta=\gamma\eta$, are studied: they rescale $X_H$ by $(\Lambda-\beta)$, generate families of first integrals, and imply structural primitives for $\Omega$ and $\eta$.

The basic result in the theory of linear ODEs is that the dimension of the fundamental set of linearly independent solutions is equal to the order of equation. This view is typically extended from integer-order to fractional differential equations. The purpose of this research is to demonstrate that this expectation is incorrect and explain why. We show that linear fractional differential equations may admit additional linearly independent solutions beyond their order. This phenomenon occurs when the coefficients of the equation cross certain threshold values. This phase transition produces a hyper-dimensional fundamental set of solutions and offers new insight into fractional differential equations and leads to the revisions of the statements of initial and boundary value problems.

We introduce and study a notion of decomposition of a C$^*$-algebra over a Coxeter system based on Tits' definition of a $W$-distance. When such a decomposition comes with suitable conditional expectations, we build an associated Fock Hilbert module and reduced C$^*$-algebra $A^r$. This unifies constructions of Voiculescu [Voi85] and Caspers–Fima [CF17] and provides a noncommutative analogue of the situation of a discrete group $G$ acting on a building, in which case $A^r \cong C^*_r(G)$. We construct covariance C$^*$-algebras $\mathscr{C}(i)\supset A^r$ as a noncommutative analogue of the crossed product $C(\Omega)\rtimes_r G \supset C^*_r(G) $ where $\Omega$ is Caprace and Lécureux's minimal combinatorial compactification [CL11] of the locally finite building. Following the approach of Hasegawa [Has17] and Klisse [Kli25], we prove a universal property for the covariance algebras. This structural result yields different approximation properties, some of which are new even for the group case.

This paper studies block structured latent variable models, in which observed variables are grouped into distinct blocks based on their relationships with the underlying latent variables. These block structures are prevalent in various fields such as psychology, education, economics, and genetics. Despite their widespread applications, the fundamental statistical properties of these models remain largely unexplored. In this work, we present a comprehensive statistical analysis of the block structured latent variable models. In particular, we first derive conditions for model identifiability across various block designs. Furthermore, we investigate the maximum likelihood estimation under these identifiability constraints. To accommodate these intricate constraints associated with various block configurations, we introduce a Lagrangian-type formulation for the constrained nonconvex optimization problem and show that its optimum coincides with that of the original problem. This formulation serves as a critical tool for understanding the behavior of the constrained estimator under various block structures. Building on that, we establish sharp non-asymptotic error bounds and asymptotic distributions of the constrained maximum likelihood estimator. We also propose a computational framework to obtain the estimator and establish theoretical properties for the algorithm output. Our theoretical findings are validated through simulation studies and empirical data analyses.

We prove the existence of stability optimizers for the Yamabe inequality on closed Riemannian manifolds of dimension at least three with positive Yamabe invariant that satisfy two threshold conditions. Remarkably, the compactness threshold we uncover is different from the special case of the round sphere treated previously by the second author. More precisely, it is given by sequences blowing up in one instead of two bubbles, reflecting the compactness of Yamabe minimizers in the non-spherical case. Using the classical asymptotic analysis of Aubin–Schoen test functions, we prove that the stability constant is strictly below the one-bubble threshold in dimension at least six and when the manifold is not locally conformally flat. In the complementary case, namely in dimensions three through five or when the manifold is locally conformally flat, we find a new positive-mass-type condition which is sufficient for the strict inequality.

This paper develops a new asymptotic theory for a broad class of martingales, establishing convergence to limiting distributions that involve a functional of stochastic integrals. The proposed limit theorem substantially extends existing martingale asymptotic theory by accommodating a wider class of dependence structures. As a primary application, the theory is applied to nonlinear regression models with nonstationary time series, yielding a rigorous framework for asymptotic inference on nonlinear least square estimators.

We calibrate credit default swaps and index tranches with elastically stopped Lévy processes: each firm defaults when the running supremum of a latent, spectrally positive distress process crosses an independent exponential barrier. This yields a Cox construction with totally inaccessible default times, while retaining the interpretability and explicit formulas of a structural approach. Adding a single common compound Poisson jump factor to every firm's latent driver gives a parsimonious multi-credit model with simultaneous defaults, which is priced by an exact Wiener–Hopf Monte Carlo scheme. Its tractability rests on a single-name result we prove: a finite partial-fraction formula for the Laplace transform of the default probability under phase-type jumps. On daily CDX North American High-Yield and Investment-Grade panels, our drivers attain the lowest out-of-sample errors in a six-model field and reproduce the inverted spread curves of names heading into default, which a Lévy subordinator provably does not. At the index level, the two-parameter dependence structure closes $73\%$ to $89\%$ of the tranche pricing gap left by independent marginals with the dependence parameters frozen, and up to $95\%$ once re-marked to tranche quotes; our framework dominates a single-factor Gaussian copula and the affine intensity benchmark of Duffie–Gârleanu on both indices.

We study the computation of static mean-field equilibria on a compact state space by formulating the equilibrium condition as a variational inequality over probability measures. We propose an entropic variant of Korpelevich's extragradient algorithm—the Kullback–Leibler Mirror-Prox method—in which Euclidean projections are replaced by relative-entropy proximal steps. Each half-step is therefore an explicit exponential reweighting of the current measure, implemented on a finite state-space discretization. Under Lasry–Lions monotonicity and continuity assumptions, we prove convergence of mesh-refined ergodic averages and obtain finite-iteration Minty-residual and approximate-equilibrium bounds that jointly quantify iteration and discretization errors. Under strong monotonicity, we derive metric convergence rates for the last, best, and averaged iterates. We also develop a KL-type Tikhonov regularization that selects the equilibrium minimizing relative entropy with respect to a reference measure. The framework applies to potential and nonpotential cost operators and does not require differentiability or convexity of the cost in the individual state.

For any finite modular tensor category A, we show that the associated derived $\infty$-category D(A) supports a topological quantum field theory in dimension 3. This TQFT takes the form of a symmetric monoidal functor from an $\infty$-category of surfaces with markings by objects in D(A), and appropriately decorated bordisms, to the $\infty$-category of dg vector spaces. We show that the state spaces in this theory are naturally identified with linearized mapping spaces for D(A). Though our TQFT requires markings from the derived $\infty$-category, we show that all markings can be removed after taking a homotopy truncation. The resulting unmarked TQFT produces projective mapping class group actions on cohomology in dimension 2, and in particular a projective SL_2(Z)-action on Hochschild cohomology. We expect these mapping class group actions to recover those of Lentner et al. arxiv:2003.06527 and Schweigert-Woike arxiv:2004.14343. In dimension 3 we obtain power-series valued knot invariants, and power-series valued invariants for modular tensor categories. We also obtain power-series invariants for closed 3-manifold, though these can already be calculated at the abelian level. Our derived field theories are proposed as mathematical formalizations for topological A-twists of certain N=4 supersymmetric QFTs, in dimension 3. Following physical principles, we discuss the possibility of deforming our TQFTs along local systems via an analogous (conjectural) deformation of quantum group representations along the Langlands dual group.

For several natural phenomena, the use of the surface area functional is a first approximation. In order to capture microstructures, numerous models in applied sciences employ directionally dependent functionals, known as anisotropic energies. Since anisotropic energies are not invariant under rigid motions, their critical points do not enjoy the same conservation laws as isotropic minimal surfaces. For instance, the monotonicity formula for the density ratio is not known to hold for minimizers of general anisotropic energies. Consequently, the study of anisotropic minimal surfaces is more challenging than the study of their isotropic counterparts. In this mini-course, we give an overview of the state of the art in the existence and regularity theory of anisotropic minimal surfaces. In particular, we first focus on solutions of the anisotropic Plateau problem and subsequently move to the investigation of the rectifiability and regularity theory for critical points of anisotropic energies. To conclude, we provide applications to the min-max theory for the construction of closed optimally regular anisotropic minimal hypersurfaces in closed Riemannian manifolds.

Recently, we announced a proof of a characteristic-free resolution of singularities. This is a survey article on the proof. We motivate the approach, describe the new ideas, and introduce the proof in the work.