Papers

Martingale posteriors and related predictive resampling methods replace the likelihood–prior pair used within Bayesian inference with a predictive model for future observations. These methods are simple to implement and increasingly popular due to their computational efficiency, but little is known about their ability to accurately quantify uncertainty. In this work, we study the concentration and calibration properties of the martingale posterior for general functionals, and show that credible sets can systematically undercover if the predictive algorithms are not carefully tuned. We propose a simple remedy: the bagged martingale posterior. Rather than starting every predictive path from the observed sample, we start the paths from random bootstrap resamples of the data and then amalgamate the resulting draws. Critically, this approach incurs no additional {simulation} cost compared to standard predictive resampling, and delivers conservatively calibrated credible sets. This scalability enables application to a range of challenging examples, including sparse high-dimensional regression and nonparametric conditional quantile models.

The para–Krawtchouk polynomials define, through their recurrence relation, a finite oscillator model that carries out fractional revival. We determine the continuum limit of this Hamiltonian, both from its recurrence relation and directly on the polynomials. The limit is the singular (isotonic) oscillator $-\partial_\eta^2+\eta^2+\tfrac{\gamma^2-1}{4\eta^2}$: the deformation parameter $\gamma$ survives as a central impurity of the couplings, decaying as the inverse square of the distance — the marginal rate, which the continuum scaling turns into a finite $1/\eta^2$ term. Using the decomposition of the para–Krawtchouk polynomials into two Hahn families, we prove the confluence, for every degree, of the eigenvectors on the two sublattices to generalized Hermite functions. The bispectral pair of the discrete model, which generates the Hahn algebra, becomes in the limit a pair of generators of $\su$, the dynamical algebra of the singular oscillator.

We prove the norm principle for the extended Clifford group $\Omega(q)$ of every nondegenerate quadratic form $q$ of even dimension at least $4$ over an arbitrary field $F$ of characteristic different from $2$, for all finite separable field extensions $L/F$. The main new ingredient is a factorization theorem for proper similitudes: for a quadratic extension $L/F$ and $f\in \mathrm{GO}^+(q)(L)$, there exist $u\in \mathrm{SO}(q)(L)$, a maximal $F$-torus $S\subset \mathrm{GO}^+(q)$, and $s\in S(L)$ such that $f=us$.

We study representations of compact oriented surface groups into $\mathrm U(p)$ with elliptic-unipotent boundary holonomies, meaning that no boundary holonomy has eigenvalue $1$. We prove that the signature of the associated flat Hermitian bundle completely determines the connected component, and that every component is path-connected. For genus $g\geq1$ and $n\geq1$ boundary components, there are $np-1$ components; for $g=0$ and $n\geq2$, there are $(n-2)p+1$. The disk case is empty, while the closed-surface representation spaces are connected. The same component classification holds after taking the quotient by conjugation. Our proofs use the boundary rho invariant and explicit matrix deformations. For a three-holed sphere, the representation components have the homotopy types of complex Grassmannians, and their conjugation quotients are contractible.

The paper contains the study of noncommutative Burkholder inequalities. We prove that the constant involved is of order $O(p/\log p)$ when $p\to \infty$, which is known to be optimal even in the commutative setting. This answers an open problem raised by Randrianantoanina in 2007, and by Junge and Xu in 2008.

We prove that the g-polynomial of every paving matroid has nonnegative coefficients. Our proof combines a deletion-contraction argument with an elementary coefficient inequality for the g-polynomials of uniform matroids.

Let $G/P$ be a flag manifold with $G$ simple. We determine the rational homotopy groups $\pi _{\ast }(G/P)\otimes \mathbb{Q}$ and describe the rational cohomology ring $H^{\ast }(G/P;\mathbb{Q})$ solely in terms of the rational homotopy types of $G$ and $P$, respectively. The proof is based on the restrictions of basic Weyl invariants of $G$ to suitable simple factors of $P$, which also address a key nonvanishing issue not settled by earlier approaches of Meier, Shiga-Tezuka, Kotschick, and Terzic.

We give a polynomial-time algorithm to sample from the Gibbs measure of the Sherrington-Kirkpatrick (SK) model with $o_n(1)$ error in total-variation distance (TVD) at any inverse-temperature $\beta < 1$. The algorithm combines algorithmic stochastic localization (ASL) with rejection sampling over path-space via Jarzynski's equality (JE). The analysis extends the authors' prior $\beta < 1/2$ result [arXiv:2605.03718] by replacing all global regularity requirements in the stochastic differential equation (SDE) error analysis with local regularity around likely trajectories. The relaxed regularity is established using Celentano's proof of the local strong convexity of the TAP free energy [arXiv:2208.09550]. The analysis utilizes the cavity interpolation theory and free probability toolkit developed in the authors' previous result, where the former applies nearly verbatim and the latter applies supplemented with Lipschitz and $C^2$ extensions of various functions. The ASL and JE analysis arises from using the TAP free energy as an efficiently computable proxy for the actual free energy of the stochastically localized Gibbs measure [$ \S $ 3, arXiv:2605.03718]. We give a list of \vocab{desiderata} encapsulating the approximation and regularity properties required of the TAP free energy, relaxing those of [$ \S $ 2.5, arXiv:2605.03718] to only require local regularity. These generic desiderata are potentially applicable in other settings where a free energy surrogate exists, giving algorithmic sampling guarantees while bypassing the usual functional inequalities based approach.

Let $C(d,2)$ be the optimal dimension-free ratio between the polytorus and Boolean-cube norms of complex multiaffine polynomials of degree at most $d$. For every $d$, $C(d,2)$ equals the unrestricted complexification constant of the real cube for complex polynomials of degree at most $d$. Moreover, $\lim_{d\to\infty}C(d,2)^{1/d}=1+\sqrt2$, and for every $d\ge3$, $\beta(1-6/(5d))(1+\sqrt2)^d\le C(d,2)\le(1+\sqrt2)^d$, where $\beta=15\pi/[2(5\sqrt{26}+\log(5+\sqrt{26}))]=0.8473222863\ldots$.

We prove that fiberwise-connected finite locally free commutative group schemes over a ring $R$ are equivalent to an explicit full subcategory of the derived category of modules over the Cartier ring of $R$.

We establish linear time precutoff for oriented kinetically constrained models (KCMs) on a $d$-dimensional box of side length $n$, whenever the finite-volume spectral gap is bounded away from zero uniformly in $n$. A typical example is the North-East model, a $0$-$1$ spin system on the two-dimensional integer lattice that evolves according to the following rule: whenever a site's southerly and westerly nearest neighbours have spin $0$, with rate one it resets its own spin by tossing a $(1-q)$-coin; at all other times its spin remains frozen. This settles, in the oriented case, a conjecture which states that there is linear time precutoff for all KCMs in the (open) ergodic parameter regime ($q>q_c$). The result has been shown recently, for general update families, in a perturbative regime ($q$ close to $1$). The previous general upper bound on the mixing time for oriented update families, in the full ergodic regime, was $O(n \log n)$. The proof combines a spectral estimate for a certain killed process with a graphical coupling to a stationary process and a backward witness path argument, exploiting the orientation of the constraints, to control the probability of failing to couple.

Let $\Gamma$ be a finitely generated hyperbolic group, and let $Q$ denote the conformal dimension of its Gromov boundary $\partial\Gamma$. We prove that for every $p < \frac{Q}{Q-1}$, the group $\Gamma$ admits a proper uniformly Lipschitz affine action on $\ell^p$ with compression exponent $1/p$. The $\ell^p$-space hosting the action is designed from a Markov chain on a specialized hyperbolic filling of $\partial\Gamma$.

We introduce a notion of topological complexity of a group with respect to a family of subgroups, with the case of the trivial and diagonal families recovering the classical category and topological complexity, respectively. These invariants come equipped with a battery of upper and lower bounds derived from functoriality and Bredon cohomology. We introduce a new family of subgroups, the permutational family, and show that the corresponding invariant is a lower bound for the distributional topological complexity of Dranishnikov-Jauhari and Knudsen-Weinberger. As a first application, we show that the permutational and diagonal families coincide, and thus that classical and distributional topological complexity are equal, for a large class of torsion-free groups, extending work of Dranishnikov. Second, we show that the lower bounds of Grant-Lupton-Oprea are in fact lower bounds on distributional topological complexity; in particular, it follows that Farber's conjecture holds distributionally.

Using relations established by Bringmann and Kane in a recent preprint based on the function $\omega_{k+1,D}$ studied by Mono, Rolen, and the second author, we introduce two further functions naturally associated with a construction by Bringmann and Mono. We exploit results by Löbrich–Schwagenscheidt, Mono, and the second author on representations of these functions as traces of cycles integrals in order to provide more identities of the same kind.

Let $E/\mathbb{Q}$ be an elliptic curve. Zywina's refinement of Koblitz's conjecture predicts that there are infinitely many primes $p$ of good reduction for which $\#E_p(\mathbb{F}_p)$ is prime precisely when $E$ has no congruence obstruction. For non-CM elliptic curves over $\mathbb{Q}$, we classify all primitive congruence obstructions. In particular, we show that a primitive obstruction of composite level can occur only at $6$, $10$, $14$, $15$, or $30$ and determine the corresponding Galois images. As a consequence, for non-CM elliptic curves, the existence of any congruence obstruction is detected by the mod $210$ Galois image, so the positivity of the Koblitz–Zywina constant is determined at level $210$. We also determine which primitive obstruction levels can occur simultaneously. Finally, we prove, assuming GRH, that if $E$ has no congruence obstruction, then for every $0<\kappa<1/8$ there are infinitely many primes $p$ of good reduction for which the least prime divisor of $\#E_p(\mathbb{F}_p)$ is greater than $\kappa\log p$.

Many real-world high-dimensional time series exhibit long-memory, but Gaussian graphical model testing in this regime remains understudied. We develop a direct, data-adaptive test statistic for assessing conditional independence in the graph structure of stationary Gaussian time series. We establish a finite-sample, Berry–Esseen type Gaussian approximation bound for the statistic, which applies to both short-memory and long-memory time series. The testing procedure is fully data-adaptive using block bootstrap method, on which we provide a finite-sample validity result including in the ultra-high-dimensional scenario, and can be extended to comparing graphical structures in two-sample tests. We also develop a consistency-empowered correction to the statistic and show that such tests attain asymptotic consistency in both size and power. Our proposed method is applied to a real-world fMRI data to understand functional connectivities within brain in different periods.

Let $S$ be a closed translation surface of genus $g\ge 2$ with $\mathop{area}(S)= 4\pi g$. We show that for any $\lambda \in (0, 1)$ there exist $\lfloor \lambda \cdot g\rfloor$ homologically independent simple closed curves of length at most $ C(\lambda) \cdot \log(g)$, where $C(\lambda)$ is a constant that depends only on $\lambda$. This result is obtained from a mixed graph construction based on a Voronoi graph of the surface with the cone points as seeds and its dual graph. We also give a complementary result using only the Voronoi graph that produces $2g$ short loops that form a homology basis. This construction, however, depends on the length of a shortest saddle connection of the surface $S$.

Calderón preconditioning restores mesh-independent conditioning for boundary-element discretizations of the Maxwell electric field integral equation, but assembling the dual barycentric operator can dominate the computational cost. We show that this preconditioner can be assembled once at a nominal geometry and reused across connectivity-preserving shape perturbations. Exact combinatorial alignment of the RWG-BC degrees of freedom transports the duality pairing, leaving only the dual-mesh operator frozen. Under uniform discrete-stability and geometric operator-continuity assumptions, a bi-parametric operator-preconditioning analysis yields a spectral-condition-number bound uniform in the mesh size and, after tensorization, in the stochastic-Galerkin dimension. Numerical experiments on smooth and non-smooth scatterers show that the frozen preconditioner closely tracks freshly assembled Calderón preconditioners over substantial shape variations. A Monte Carlo campaign on a NASA almond, whose admissibility is screened before each shape is constructed so that no realization is rejected, separates the two strategies by less than the variation between realizations, and mesh refinement there leaves the frozen iteration count unchanged. Reuse yields roughly 30x measured speedup per new realization in our dense implementation; a cost-model estimate based on published compressed-assembly timings gives roughly 2x.

In this paper, we establish the first asymptotic convergence guarantees for the Muon algorithm through a more accurate proxy for the Newton-Schultz iteration than the typical matrix sign function. We prove that, for appropriate choices of hyperparameters, the iterates satisfy $\lim_{k\to\infty}\|\nabla f(x_k)\|=0$, and, under a global Polyak-Łojasiewicz condition, that the sequence of function values converges linearly. The key insight is that the regularization, implicit in Muon's Newton-Schulz implementation, induces a bounded preconditioner, exposing Muon as a preconditioned Polyak heavy-ball method and enabling a classical Lyapunov analysis. This observation naturally motivates applying the same preconditioning structure to the Nesterov gradient evaluation. We formalize this idea by introducing Muesterov, a Nesterov-based variant of Muon, and prove that it enjoys the same convergence guarantees, extending the theoretical framework beyond the heavy-ball setting. Numerical experiments on a scalar cross-entropy problem corroborate the theory and illuminate the joint role of the learning rate and the Newton-Schulz regularizer in controlling convergence. Preliminary numerical simulations training the nanoGPT dataset provide intuition regarding the relevance of the observations in this paper to practical applications.

We study KMS and ground states of the tame $C^*$-algebras $\mathcal{O}_{m,n}$ of Ara, Exel and Katsura under the dynamics $\sigma_t(s_i)=m^{it}s_i$, $\sigma_t(t_j)=n^{it}t_j$. Normalized restriction to the full corner $A=q\mathcal{O}_{m,n}q$, generated by the partial isometries $v_{ij}=s_i^{*}t_j$ of common energy $\log(n/m)$, is at each inverse temperature $\beta$ an affine weak-$*$ homeomorphism of the two systems. Let $m,n\ge 2$. When $m=n$ the corner dynamics is trivial, so the KMS simplices are identified with trace simplices. When $m\neq n$, KMS states occur only at $\beta=1$, and there they are not unique. Conformal branching models surject the KMS$_1$ simplex onto a transportation polytope of dimension $(m-1)(n-1)$, whose vertices lift to extremal, hence factorial, KMS$_1$ states, the fibre over each vertex being itself the KMS$_1$ simplex of a quotient of $A$.

Using recent advances in the integral canonical models of Shimura varieties using syntomic methods, we give an extension of the Langlands–Kottwitz–Scholze method from Scholze's paper on the Langlands–Kottwitz method for deformation spaces of $p$-divisible groups. Namely, we define analogues $\phi^{\mathcal{G},\mu}_{\tau,h}$ of the local test functions from op. cit. in full generality, show they satisfy reasonable harmonic-analytic properties, and give rise to trace formulae for the Galois-Hecke action on the cohomology of Shimura varieties of bad reduction. Additionally, we use these new local test functions and the work of Fargues–Scholze to state a more robust and unconditional version of the Scholze–Shin conjecture. Along the way we study the $(\mathcal{G},\mu)$-apertures of Drinfeld and Gardner–Madapusi, in particular showing that such $(\mathcal{G},\mu)$-apertures carry no more information than a Tannakian version of Fontaine–Laffaille theory, at least over nice bases, and give a very explicit description of the universal deformations of $(\mathcal{G},\mu)$-apertures.

We classify all real extremals of the sharp $L^2$ Folland-Stein inequality on H-type groups and prove a global Bianchi-Egnell stability estimate. The linearized kernel at the standard bubble consists exactly of its translation and dilation derivatives. We also obtain explicit bounds for the spectral gap above the symmetry modes and characterize the optimal coefficient in the local stability asymptotics. The results apply to arbitrary orthogonal Clifford modules.

The purpose of this article is to explicitly describe how the integration technique known as the change of variables is used in graduate-level Probability and Statistics courses. Throughout the article, we employ this integration technique seven times across various examples and results. We demonstrate that four applications of this technique are required to derive the probability density function (PDF) of the Student's t-distribution.

Given a Riemannian manifold $(M,g)$ equipped with a positive density function $f = e^{-\varphi}$, we study the weighted sectional curvature $\overline{\sec}_\varphi(U,V)$ and the $k$-th intermediate weighted Ricci curvature $\overline{\text{Ric}}_{k,\varphi}(U,V)$. We show that $\overline{\sec}_\varphi(U,V)$ and $\overline{\text{Ric}}_{k,\varphi}(U,V)$, together with the weighted second fundamental form $\widetilde{\mathrm{I\!I}}$, play central roles in controlling the topology of the manifold. Using the theory of $\widetilde{C}(k)$ functions, variational methods, and Morse theory, we prove that by imposing suitable restrictions on these quantities, an $n$-dimensional compact manifold with boundary or a proper open $n$-dimensional manifold is diffeomorphic to an $n$-ball or to a certain CW complex. This shows that such rigidity phenomena is not confined in the unweighted case, but persist in the weighted setting as well. Lastly, we prove a weighted analogue of Frankel's Theorem under positive intermediate weighted Ricci curvature. These results extend classical theorems from the unweighted setting to the weighted case, demonstrating that $\overline{\sec}_\varphi(U,V)$ and $\overline{\text{Ric}}_{k,\varphi}(U,V)$ provide effective tools in Riemannian geometry.

We prove the Sudakov minoration principle, with a universal constant, for unconditional log-concave random vectors whose coordinate magnitudes are negatively associated. The main step is a deterministic selection theorem: pairwise witnesses in convex downwardclosed sets yield, on an exponentially large subfamily, one fixed witness per label separating it from every other retained label. The selection combines a prefix counting inequality with convex separation in the space of all label-coordinate arrays. Negative association then controls the expected number of active joint-tail witnesses, while a Bernoulli argument on complementary coordinate sets retains the random signs. The proof uses joint upper-orthant probabilities and applies negative association only under the original law. Consequences include minoration for products of Orlicz-based log-concave measures and a covering estimate in the moment metric, expressed geometrically through polar Lp centroid bodies.

We study asymptotic multiplicity sequences of graded families of arbitrary ideals in Noetherian local rings. We characterize convergence of the zeroth component and prove that the first positive component not forced to vanish always converges under natural equidimensionality and catenarity assumptions. In contrast, higher components may exhibit arbitrary asymptotic behavior: even in $k[[x,y]]$, integrally closed filtrations with fixed radical, height, and analytic spread can realize any prescribed lower and upper limits. For Noetherian graded filtrations, however, all asymptotic components exist, are finite and rational, and satisfy Rees-type and valuative criteria for integral dependence. In the reduced pure-dimensional complex analytic setting, a common principalizing modification provides a second convergence mechanism without finite-generation assumptions. We also develop a diagonal mixed theory: for graded $\mathfrak m$-primary families, the top diagonal components are sums of Cutkosky's mixed multiplicities, while on a common analytic model all diagonal mixed components converge. The latter result extends the one-family common-model theorem.

In this paper, we study the pseudo-relativistic Schrödinger equation with logarithmic nonlinearity \begin{equation*} \left(\sqrt{-c^{2}\Delta+m^{2}c^{4}}-mc^{2}\right)u+\mu u=u\log|u| \quad\text{in }\mathbb{R}^{N}. \end{equation*} The existence and qualitative properties of action ground states are established, by means of a power-law approximation. We then prove that these states converge to the Gausson in $H^{1}(\mathbb{R}^{N})$ as $c\to\infty$. For sufficiently large $c$, we further obtain uniqueness of action ground states up to translation and sign change, as well as their nondegeneracy. Finally, we prove an exact correspondence between action and energy ground states, thereby extending the preceding results to energy ground states at arbitrary prescribed mass.

Influenza A persists by continually changing the antigens that its host population recognizes. How fast it evolves and how many hosts it infects are not independent, because the immunity that selects new variants is produced by the infections the pathogen has already caused. We analyze a model of Andreasen et.\ al.\ that captures this eco-evolutionary feedback, and we extend it to the case where infection may kill the host. In the model infected and recovered hosts are distributions over a one-dimensional antigenic space coupled by cross-immunity. The endemic state is a traveling pulse: the infected distribution keeps a fixed shape and advances through antigenic space at constant speed. In the limit of weak antigenic mutation we construct the pulse asymptotically and find that: (i) the speed of antigenic evolution decreases as prevalence rises, (ii) less standing antigenic variation accompanies either faster or slower antigenic evolution, depending on whether pathogens differ in immune evasion or host lifespan (iii) the total number of infected hosts is largest at an intermediate virulence when the host birth rate is sufficiently low.

The packing-covering constant of a point set $X\subseteq\mathbb{R}^d$ is $\gamma(X)=\mu(X)/\rho(X)$, the covering radius divided by the packing radius. Among lattices, its minimum $\gamma_d$ is known for $d\leq 5$, attained by $\mathsf{A}_2^*$, $\mathsf{A}_3^*$, and Horváth's lattices $\mathsf{Ho}_4$, $\mathsf{Ho}_5$; Böröczky proved that $\gamma_3$ is optimal without the lattice restriction, but for $d=4,5$ the non-lattice problem was open. We exhibit a $2$-periodic non-lattice point set of $\mathbb{R}^5$ with $\gamma = 9/\sqrt{40} = 1.423024\ldots < \gamma_5 = \sqrt{3/2+\sqrt{13}/6} = 1.4494568\ldots$, so that in dimension five the packing-covering problem is not solved by lattices.

Consider the one-dimensional Schrödinger evolution $$ i\partial_tw+\partial_x^2w=Vw,\hspace{5mm}(t,x)\in [0,T]\times\mathbb{R},$$ where $V$ is a real-valued time-dependent potential belonging to $L^1_tL^\infty_x+L^2_{t,x}$. We prove that an $L^2$ solution to the above equation is determined by its mass density $|w|^2$ modulo a global unimodular phase factor. This answers a question recently posed by Jaming for a large class of potentials, and recovers his earlier result for the free Schrödinger evolution. Our proof is simple and robust. It combines the basic local smoothing properties of the above Schrödinger evolutions, Strichartz estimates, and a stripwise version of a unique continuation theorem of Ionescu and Kenig to show that the interaction function $$ F(t,x,y)=u(t,x)v(t,y)-u(t,y)v(t,x) $$ vanishes identically on $[0,T]\times\mathbb{R}^2$ whenever $u$ and $v$ are $L^2$ solutions to a Schröinder euqation with potential $V$ with the same mass density. The method immediately extends to several important nonlinear models such as the cubic NLS equation, which is globally well-posed in $L^2(\mathbb{R})$ with $V=|u|^2$. Interestingly, we also show that the above phase recovery property fails in dimensions two and higher, even for the free equation with Schwartz initial data.