Papers

We present the first rigorous analysis of decentralized geometric routing in Geometric Inhomogeneous Random Graphs (GIRGs), a weight-agnostic variant of the greedy routing protocol. While greedy routing in GIRGs is known to explain the algorithmic small-world phenomenon by finding ultra-short paths of length $\Theta (\log \log n)$, it assumes additional knowledge of vertex weights beyond geometry, an assumption that is often restrictive or unavailable. We investigate whether the underlying geometry alone is sufficient for efficient navigation. We prove that for power-law weight exponent $\tau \in (2,3)$ and geometric decay parameter $\alpha > \tau - 1$, geometric routing succeeds with constant probability and finds ultra-short paths of length $\Theta (\log \log n)$, matching the optimal asymptotic guarantees for greedy routing. Our analysis further reveals that, upon success, both protocols follow a similar two-phase trajectory, consisting of a rapid ascent to the heavy vertices, followed by efficient navigation to the target. These results demonstrate that, in the appropriate regime, the network's geometry alone implicitly guides the path to the target through its high-weight core.

The generalized Turán number $ex(n, K_{t, t}, K_{2, t+1})$ is the maximum number of copies of $K_{t, t}$ that a $K_{2, t+1}$-free graph on $n$ vertices can contain. Recently, Pohoata, Tidor, and Yu established that $ex(n, K_{t, t}, K_{2, t+1}) = \Theta_t(n^2)$ for all integers $t \geq 3$. In this short note, we use an explicit construction to establish that when $t$ is a prime power and $n = t^{2e - 1}$, then $$ ex(n, K_{t, t}, K_{2, t+1}) = (1 + o(1))\frac{n^2}{2t(t-1)}. $$

Native compilation changes how a program represents values and performs operations from processor to processor. In our view, proofs that justify source-level decisions must remain connected to those changes. We develop fixed-point scaffolding for this continuity in Clef and our Composer compiler. Our Program Semantic Graph retains types, joint constraints and proof dependencies. A thin MLIR middle end witnesses their settled consequences for native CPU, GPU, NPU and FPGA realization. The account connects nanopass traversal with preservation contracts, structural theorems and adjoint interfaces between reasoning modes. Four verification tiers organize automatic inference and reusable domain and system proofs, with probabilistic relations admitted at Tier 4. Established lower-tier facts can directly support richer derivations. We state conditions for composing lowering contracts and assembling compatible program segments, then show how their dependencies govern differential compilation after an edit. A worked foreign-call boundary follows source ranges through argument marshaling, the actual call and return conversion. Current FFI mechanisms bind checking evidence to those witnessed operations. A recent paper from Urschel provides elimination-growth bounds that supply a quantitative example. Reusable premises inform representation and error contracts that retain their sampling, arithmetic and exceptional-path assumptions. A further example admits value-indexed pairing through ordinary MLIR operations, connecting this scaffold to negative and fractional types. Together these constructions explain how a thin middle end can carry the consequences of rich source proofs into unboxed computation graphs. Integrity and efficiency share the same retained evidence. It justifies native realizations and identifies which analyses, proofs and lowered fragments remain reusable during development.

Peskine and Szpiro's Intersection Theorem for finitely generated modules was generalized by Foxby to all modules and bounded complexes of such. We strengthen Foxby's result and prove a dual result, which for complexes with finitely generated homology recovers Iversen's Amplitude Inequality but also applies to derived complete complexes.

Let $\mathbb Z_n$ be the cyclic group equipped with the uniform probability measure $\pi$, and let $A_{\psi_n}$ be the Laplacian with word length $$ \psi_n(k) = \min(k,n-k). $$ For every $n\ge4$, we prove the sharp log-Sobolev inequality $$ \text{Ent}_{\pi}(|f|^2) \le 2\pi(\bar{f}A_{\psi_n} f), \qquad f:\mathbb Z_n \to \mathbb{C}, $$ where $\text{Ent}_{\pi}$ is the relative entropy with respect to $\pi$. Equivalently, the Poisson semigroup $P_t=e^{-tA_{\psi_n}}$ satisfies the optimal hypercontractivity. The proof is inspired by the recent work of Frank and Ivanisvili [FI26] on a sharp log-Sobolev inequality for the nearest-neighbor simple random walk. Similar arguments yield a simple proof of Weissler's sharp log-Sobolev inequality for the Poisson semigroup on the circle \cite{Weissler1980}. The same inequalities were independently obtained by Yao~\cite{Yao2026} using a different method. We also prove quantitative stability estimates. For $n\ge 4$ and $f:\mathbb Z_n\to[0,\infty)$ with $\pi(f^2)=1$, $$ 2\pi(fA_{\psi_n}f)-\operatorname{Ent}_{\pi}(f^2) \ge \frac1{12}\|f-1\|_{L^2(\pi)}^4, $$ with coefficient $1/(12d)$ on products $(\mathbb Z_n)^d$. The quartic order and the $d^{-1}$ dependence are optimal.

We study Cheeger-type inequalities for persistent Laplacians associated with inclusions of simplicial complexes $\mathcal{K}\hookrightarrow \mathcal{L}$. We introduce a persistent up $p$-Laplacian $\Delta_{q,p,\mathrm{up}}^{\mathcal{K},\mathcal{L}}$ for $p\geq 1$. For $p=2$, this recovers the usual persistent up Laplacian, while for $p=1$ it yields a nonzero persistent Cheeger constant $\varphi_q^{\mathcal{K},\mathcal{L}}$. We prove a Cheeger-type inequality relating $\varphi_q^{\mathcal{K},\mathcal{L}}$ to the smallest nonzero eigenvalue of $\Delta_{q,\mathrm{up}}^{\mathcal{K},\mathcal{L}}$. This gives a persistent extension of recent work by Jost and Zhang (arXiv:2302.01069). We then study two more structured settings. Under a locally complete $q$-skeleton assumption on $\mathcal{K}$, we extend the complete-skeleton isoperimetric inequality of Parzanchevski–Rosenthal–Tessler (arXiv:1207.0638) to the persistent setting. For orientable $(q+1)$-dimensional pseudomanifolds, we prove a Kron-type reduction of the persistent up Laplacian to a vertex- and edge-weighted graph Laplacian, possibly with Dirichlet boundary terms, and obtain two-sided Cheeger inequalities; this is related to the dual-graph perspective in the work of Steenbergen–Klivans–Mukherjee (arXiv:1209.5091). We also describe the nonzero persistent Cheeger constant $\varphi_q^{\mathcal{K},\mathcal{L}}$ explicitly in terms of the dual graph in the non-branching pseudomanifold case. Finally, we specialize our two constructions to graph inclusions and compare them with the graph-pair theory of Mémoli–Wan–Wang (arXiv:2012.02808). We establish precise relationships between the two persistent Cheeger constants arising from our constructions and the corresponding constants derived via Kron reduction.

The classical no-$k$-in-line problem asks for the largest number of points that can be placed on an $n \times n$ grid without having $k$ of them collinear. A natural extension, motivated by the analogous question by Erde for $k\in \mathbb{Z}$, is the \emph{extensible no-$(k(n)+1)$-in-line problem}, which seeks a subset of points in $\mathbb{Z}^2$ with maximal possible density such that at most $k(n)$ points are collinear within the subgrid $[1,n]^2$. We construct optimal sets for linear functions and positive-density sets for power functions. We prove that any configuration achieving $\liminf\frac{S_n}{n k(n)} \ge 0.897$ must satisfy $k(n) = \Omega( n^c)$ for some $c>0$ constant; therefore, the extensible no-$k$-in-line problem has no configuration with this property when $k$ is a constant. Finally, we reduce the problem to the extensible no-$k$-in-line problem, showing that if a positive-density point-set exists for a constant limiter function, then one also exists for any sufficiently regular function $k(n)$.

Deep neural networks learn representations where individual features often lack interpretable meaning; a single neuron may activate for scattered, unrelated inputs. We introduce coherence, a geometric property inspired by neural coding in the brain, where neurons like grid cells and head direction cells respond to contiguous regions of state space. A non-negative matrix is coherent if each row (sample) attends to geometrically clustered columns (features) and vice versa, and in addition every sample is well described by some feature and every feature is needed by some sample. We prove that coherent matrices induce a bounded interleaving between the Vietoris-Rips filtrations of samples and features, guaranteeing that both spaces share compatible topological structure. This geometric constraint facilitates interpretability. For example, if data lies on a circle, coherent features must tile that circle into contiguous arcs. We introduce Coh, a differentiable objective function based on Fréchet variance that enforces coherence during training. Unlike sparsity, which bounds how many samples a feature activates on, coherence bounds which samples, requiring geometric connectivity rather than only rarity. This yields not just interpretable features but an interpretable feature space. We validate Coh in an auto-encoder using synthetic and rotated MNIST datasets and in a token embedding of BERT using language data.

Real-world datasets often contain spurious correlations that are not causally related to the target label. When such correlations dominate the majority of training samples, models tend to rely on them, leading to misclassification of minority samples that do not exhibit the same spurious patterns. While a potential approach is to select subsets of data to better represent the minority samples, this may require access to group labels, which are typically unknown. Furthermore, as we demonstrate, widely used sample scoring functions in the invariant subset or coreset selection literature largely depend on spurious features and therefore fail to accurately capture the importance or difficulty of core, causally relevant features. Accordingly, we propose to mitigate spurious correlations by developing a two-stage sample scoring function that disentangles the learning dynamics of core and spurious features and evaluates their difficulty separately. Based on our proposed metric, we introduce a new algorithm to find and prioritize informative samples both with and without spurious correlations. Extensive experiments demonstrate that a standard ERM model trained on our selected samples achieves superior performance compared to state-of-the-art debiasing techniques, while requiring as little as 10% of the original training data.

We prove a sharp Clifford-threshold focal-radius estimate and rigidity for immersed hypersurfaces. Under a $p$-form curvature condition, formulated by the Weitzenböck curvature term together with $\mathrm{Ric}_p\ge p$, any closed two-sided immersion $F:\Sigma^m\to M^{m+1}$ with $b_p(\Sigma;\mathbb R)\neq0$ and $1\le p\le m/2$ satisfies \[ r_f(F,M)\le\frac{\pi}{4}. \] The equality case is rigid: if the ambient manifold is complete, equality forces the hypersurface to be locally the Clifford hypersurface $S^p(1/\sqrt2)\times S^{m-p}(1/\sqrt2)\subset S^{m+1}(1)$; if the ambient manifold is compact and connected, it is a spherical space form. The curvature condition follows from $\sec\ge1$ for $p=1$, from normalized $\mathrm{PIC1}\ge1$ for $p=2$, and from curvature operator bounded below by one in all degrees. By quotient lifting and the Hopf fibrations, we also obtain focal-radius estimates in $\mathbb{CP}^n$ and $\mathbb{HP}^n$, with projective Clifford rigidity, without any Betti-number assumption.

We study several aspects of the semi-algebraicity of the neuromanifold $\mathcal{M}_{\mathbf{d}}$ of a feedforward ReLU neural network. We prove that $\mathcal{M}_{\mathbf{d}}$ in general is not a semi-algebraic quotient of the space of parameters of the network, but it can be approximated (in a precise sense) by semi-algebraic spaces. We then consider the notion of identifiable subsets of the space of parameters, where the network does not have any hidden symmetries. We prove that the maximal identifiable subset is semi-algebraic for any neural network with semi-algebraic activation functions. Lastly, we compute explicitly the identifiable subset of a shallow ReLU network and prove that for input dimensions $\ge 3$ and width $\ge 6$ it is not euclidean open.

Given an ordered set $B$ of a finite field, a combinatorial simplex over $B$ is defined as the set of vectors such that the positions of the entries, with respect to $B$, sum up to a fixed integer. CAP codes are Reed-Muller type codes defined over a combinatorial simplex. They were recently introduced by Kopparty et al. as a high-rate alternative to classical Reed-Muller codes, capable of achieving arbitrarily high rates close to one for any fixed minimum distance. In this paper, we use tools from commutative algebra to analyze a combinatorial simplex and its associated CAP code. We give a universal Gröbner basis for the vanishing ideal of a combinatorial simplex. We describe the generalized Hamming weights of a CAP code in terms of the footprint of the vanishing ideal. For the minimum distance case, we proved a closed formula. We give a set of polynomials whose evaluations on the combinatorial simplex generate the dual of the CAP code. We describe the affine permutations that leave invariant a combinatorial simplex and use this information to prove that, in some cases, the permutation group of a CAP code is a symmetric group.

We show that the Gromov–Hausdorff convergence of non-collapsed polarized constant scalar curvature Kähler (cscK) surfaces can be realized as convergence in a Hilbert scheme. We also derive uniform estimates of Bergman kernels on the effective regular set. As an application, we establish the Zariski openness of cscK metrics for certain smooth polarized families, following the approach of Donaldson.

We prove global in time regularity of solutions of the Navier-Stokes equations defined in the complex space.

We develop a framework for infinite tensor products of Hilbert spaces, operators, and semigroups tailored to questions of similarity to contraction semigroups. On the operator-theoretic side, we give a systematic treatment of incomplete infinite tensor products, including criteria for existence, non-vanishing, and continuity properties of the associated tensor product semigroups. On the semigroup-theoretic side, we prove a low-regularity similarity theorem showing that global quasi-contractive control together with local contractive information at one positive time implies similarity to a contraction semigroup, with explicit bounds on the similarity constant. These ingredients are then combined to obtain infinite analogues of the finite tensor-product splitting principle for similarity to contraction and quasi-contraction semigroups. We also clarify the role of complete tensor products and show, in particular, that whenever a complete infinite tensor product of semigroups is a \(C_0\)-semigroup, it decomposes along incomplete tensor-product components.

The nullcone membership problem, deciding whether an orbit closure contains the origin, is fundamental in computational invariant theory. For self-adjoint groups, Bürgisser, Franks, Garg, Oliveira, Walter and Wigderson gave a geodesic optimization algorithm whose complexity is controlled by the gap, a condition number of the representation. We study the gap for quiver representations under the action of the special linear group. We prove that the inverse gap is polynomially bounded in the number of vertices and the maximum dimension for type A and $\hat{A}$, as well as tree quivers with uniform dimension vectors. Consequently, the algorithm of Bürgisser et al. solves the nullcone membership problem in polynomial time for these families. In contrast, we construct families of quivers and dimension vectors where the gap is exponentially small in the number of leaves, furthermore, for every connected quiver we exhibit dimension vectors such that the weight margin (a related condition number) is exponentially small in the number of vertices. We also extend our results to $\sigma$-semistability, thereby giving a new proof of a recent result of Iwamasa, Oki, and Soma.

We prove a folklore conjecture identifying two categorical enhancements of the automorphic side of the local Langlands correspondence. Concretely, we construct an equivalence for torsion coefficients between the category considered by Zhu and the one considered by Fargues-Scholze. To achieve this, we revisit Scholze's analytification functor and apply the first author's theory of kimberlites. We discuss unconditional applications to the splitting of the semi-orthogonal decomposition on BunG, and the compatibility with Eisenstein functors. Finally, we formulate a linearity conjecture for our functor with which we can show new vanishing statements for the cohomology of local Shimura varieties, and perverse exactness statements for Hecke operators.

Let $S_{\kappa}$ be the space of cusp forms of weight $\kappa$ and level one, and let $S_{\kappa}^{\ast}$ denote its dual space. In this paper, we find explicit spanning subsets of $S_{\kappa}$ consisting of Rankin-Cohen brackets of Eisenstein series and explicit subsets of periods that span $S_{\kappa}^{\ast}$.

We propose Adaptive Accelerated Mirror Descent (AAMD), a flow-based method that combines nonlinear preconditioning, acceleration, and adaptivity in mirror geometry. The key ingredient is an accumulated Lyapunov perturbation budget: local descent failures are allowed as long as the total budget remains nonpositive, so line search is used only when stability is at risk. We prove accelerated convergence under dual relative smoothness/convexity and a mirror-geometry compatibility condition, and obtain an $O(1/k^2)$ rate for convex objectives by homotopy under a bounded-sublevel-set assumption. Experiments on relative-smoothness problems show that combining preconditioning, acceleration, and adaptivity gives substantial gains over methods using only part of this structure.

The paper is concerned with $\alpha$-stable SDEs with singular time-inhomogeneous general drift. Our drifts satisfy e.g.\,a close to minimal possible critical Morrey class condition that covers local blow-ups of the type arising in some particle systems with strong attracting interactions. We develop detailed Sobolev regularity theory for the forward and backward equations, construct Feller propagator and prove weak existence and conditional uniqueness for the $\alpha$-stable SDE.

All stabilizing PID controllers form a set in three-dimensional space. A novel viewpoint to its boundary as a ruled surface (or surfaces) being cut with 3D planes is presented. The characterization, being not too new, contributes to an understanding of the stability set as the whole, instead of the classical view as a stack of 2D slices, say, on the P-coefficient. The viewpoint gives clear insight on the structure of the PID stability region, and, in particular, splits its boundary into continuous parts. It is followed by natural 2D unwrapping of the stability set boundary. It also correctly handles pure imaginary zeros in transfer function. A wireframe 3D visualization reveals the structure of the stability set. The presentation is valid both for ideal and filtered PID controllers, as well as for time-delay systems and other linear systems. Finally, based on the viewpoint, a simple formula for stability (fragility) radius is provided.

We reinvestigate the theory of spectral sequences by studying the $(\infty,1)$-category of filtered spectra through the cofiber-of-$\tau$ formalism of Burklund-Isaksen-Pstragowski-Wang-Xu. In this framework, we define and analyze hidden extensions along arbitrary maps of filtered spectra, establishing computational principles that extend the generalized Leibniz rule and the generalized Mahowald trick of Lin-Wang-Xu, as well as Burklund's Leibniz rule for total differentials, from the Adams spectral sequence to this broader setup. Our formulation uses a more refined, layered notion of extension, which slightly sharpens these statements even for the Adams spectral sequence. As an application, we study equivariant slice spectral sequences and obtain new families of "exotic transfer" differentials in the $C_4$-slice spectral sequences for the Hill-Hopkins-Ravenel theories $\mathrm{BP}^{((C_4))}\langle m\rangle$ for every $m \ge 1$.

A variational analysis of the Bogoliubov–Hartree–Fock (BHF) energy of the translation-invariant, spinless Pauli–Fierz Hamiltonian with massless dispersion relation built up on work of the first author, Breteaux, and Tzaneteas (2013) and of the first author and Hach (2022) is presented. The main results are lower and upper bounds on the BHF energy for fixed total momentum expressed through simpler variational problems defined on the space of positive Hilbert–Schmidt operators and a new variational formulation of the upper bound for zero total momentum. Specifically, we introduce a change of variables which considerably simplifies the energy functional and the derivation of its stationarity condition.

We introduce Lie groupoid equivariant neural networks as a specialization of recently proposed topological category-equivariant neural networks to the differentiable setting. Lie groupoid equivariant neural networks are composed from Lie groupoid lifting convolutions and Lie groupoid convolution layers, and we show how for suitable Lie groupoids they are equivalent to certain Lie algebroid-equivariant neural networks. We additionally describe groupoid invariant global pooling as a generalization of group invariant global pooling. Furthermore, we show that each of the aforementioned layers is a special case of recently introduced admissible category-equivariant layers by demonstrating that they define continuous natural transformations between continuous feature functors.

Quantum transit times have a long history of inequivalent definitions, including phase times, dwell times, and quantum-clock constructions. In this context we revisit the Salecker–Wigner–Peres stationary quantum clock as a phase-sensitive scattering observable, with clock time defined by the energy derivative of the transmission phase shift across the interaction region. For real compactly supported one-dimensional potentials, we show that the raw stationary Peres clock generically contains a universal \(1/\sqrt{E}\) continuum-edge term whose coefficient is fixed by low-energy scattering data. For the attractive square well, this threshold singularity is inherited from the vanishing exterior momentum and the associated scattering matching, rather than from resonant delay itself. We derive the exact stationary clock time for the square well and introduce a new threshold-subtracted clock observable. Away from exceptional zero-energy tuning, the subtraction removes the universal low-energy term and isolates the resonant contribution. Comparison with the dwell time and the transmission Wigner phase delay shows that the threshold-subtracted clock acquires the expected local Lorentzian form near isolated transmission resonances. Near the continuum edge, if \(\varepsilon\) denotes the detuning from threshold, the resonant peak grows only as \(\varepsilon^{-1/2}\), whereas the unsubtracted threshold background grows as \(\varepsilon^{-3/2}\). A symmetric barrier–well–barrier cavity and a numerical asymmetric two-step attractive well provide complementary controls. The result is a new threshold-subtracted stationary-clock candidate that separates universal threshold kinematics from pole-sensitive resonant delay.

Devnani and Eyyunni recently studied the maps pre$_k$ on integer partitions, which arise from applying elementary symmetric polynomials to the parts of a partition. They asked whether there exists $n \ge 1$ such that exactly one partition of $n$ lies in the image of pre$_2$. We show that this occurs only for $n$ in {1, 2, 4}, and that for all $n \ge 5$, at least two partitions of n are in the image of pre$_2$.

In this manuscript, we explore the application of neural networks to predict the natural parameter $\kappa \geq 0$ of Schramm-Loewner Evolution (SLE$_\kappa$) theory. SLE$_\kappa$ is a family of random fractal curves that has significant implications in Statistical Mechanics and Conformal Field Theory. This parameter $\kappa \geq 0$ plays an important role in the theory as there are models of Planar Statistical Physics that are proven to have SLE as scaling limits as well as others that are conjectured to have this limit for various choices of the parameter $\kappa \geq 0$. In addition, there are three different statistical behaviors of the SLE curves as the parameter $\kappa$ changes in $[0, \infty).$ Leveraging the powerful pattern recognition capabilities of neural networks, this study aims to develop a predictive model that can estimate the $\kappa$ parameter with good accuracy.

Let $f(k,s)$ denote the minimum integer $m$ such that any family $\mathcal{F}$ consisting of $k$-sized sets of cardinality at least $m$ always contain a sunflower of size $s$. The Erdős-Rado Sunflower Conjecture states that for every $s >2$, there is an constant $C=C(s)$ such that $f(k,s) \leq C^k$. In this paper, we prove the conjecture for shifted families.

External regret certifies stability only against replacing one's behavior by a fixed alternative. In a quantum game, this misses a natural physical move: a player can apply a local completely positive trace-preserving (CPTP) map to the state it actually received or prepared. We introduce coherent swap regret as the regret benchmark against all such local CPTP deviations, and give an algorithm achieving $O(\sqrt{dT\log d})$ coherent swap regret via entropic mirror ascent on the CPTP Choi slice with a fixed-point play rule. The main result is a three-level deviation-class landscape. Replacement channels recover ordinary external regret at rate $\Theta(\sqrt{T\log d})$. Unital channels, including unitary deviations and mixtures of unitaries, have zero minimax regret. Deterministic measurement-and-preparation channels already force $\Omega(\sqrt{dT\log d})$ regret in the moderate-horizon regime, and this rate is also sufficient for all CPTP deviations. Thus the hardness comes from non-unital use of the recommendation register, not from quantum coherence alone. As an application, decentralized full-information learning in finite quantum games reaches an $\varepsilon$-approximate separable quantum correlated equilibrium after $T=O(\max_i d_i\log d_i/\varepsilon^2)$ rounds. We identify these equilibria with channel-proofness of mediated quantum recommendation protocols, give an SDP audit for local CPTP exploitability applicable to arbitrary finite-dimensional states, and include a probing-bandit extension with pseudo-regret $O(d^{4/3}T^{2/3}(\log d)^{1/3})$ under Haar-random pure-state probes.

We study stochastic completeness for landmark spaces equipped with Riemannian metrics induced by right-invariant metrics on subgroups of the diffeomorphism group of the shape domain. We extend a previous stochastic completeness result, which only covers the case of exactly two landmarks, to landmark spaces with any number of landmarks. This succeeds the characterization of geodesic completeness for landmark spaces with arbitrary numbers of landmarks, and thus finishes the completeness characterization for landmark spaces by covering the stochastic case. The proof makes use of Grigor'yan's volume growth criterion for stochastic completeness, which requires a suitable upper bound for the volume of growing geodesic balls. We obtain quantitative controls for geodesic balls in the landmark space by bounding both its Euclidean size and the rate at which pairwise landmark distances can approach zero. We then combine this with a lower bound on the minimal eigenvalue of the landmark cometric in terms of the Fourier transform of the kernel to yield volume growth bounds sufficient to prove stochastic completeness of landmark spaces for wide classes of kernels, including Matérn kernels.