Papers

Moduli spaces of $\epsilon$-stable quasimaps exhibit a wall and chamber structure, interpolating between the moduli of stable quasimaps and Kontsevich's moduli spaces of stable maps. In the case where the target is projective space, we develop an intrinsic framework for the wall crossing phenomenon by constructing an algebraic stack where strictly semistable objects appear at a wall in the space of $\epsilon$-stability conditions. We show this algebraic stack admits a proper good moduli space and analyze the variation of $\Theta$-stratification on this stack. This yields a new approach to developing a K-theoretic wall crossing formula for $\epsilon$-stable quasimap invariants via non-abelian localization.

We investigate the small circle, or high temperature, limit of the supersymmetric index identities for the three-dimensional abelian mirror symmetry of SQED. There exist two possible limits, depending on how the parameters of the theory are scaled with the radius of the circle. In both cases the result is qualitatively similar. One side always reduces to the sphere partition function of a two-dimensional $\mathcal{N}=(2,2)$ gauged linear sigma model (GLSM). The opposite side has a two-fold interpretation, either as the sphere partition function of the Landau–Ginzburg (LG) model that is Hori–Vafa dual to the GLSM, or as a Coulomb gas integral for a correlation function of Liouville or Toda CFT. This approach thus provides a systematic way to generate integral identities between partition functions of GLSMs on the one hand, and partition functions of LG models or CFT Coulomb gas integrals on the other. The latter perspective finds useful applications in the recently proposed 2d/2d correspondence, which relates sphere partition functions of unitary 2d $\mathcal{N}=(2,2)$ theories and correlation functions of non-unitary 2d CFTs that both descend from compactifications of a unitary 4d $\mathcal{N}=2$ SCFT. We present an example based on the $(A_{k-1},A_{N-1})$ Argyres–Douglas theories, where the CFT is a non-unitary minimal model. We also give a purely two-dimensional derivation of the identities obtained in the small circle limit which is inspired by the Kapustin–Strassler piecewise derivation of 3d abelian mirror symmetry.

Construction of physically realistic theories admitting regular black hole solutions remains an important open problem in gravitational physics. While theories with electromagnetic fields minimally coupled to gravity have been extensively studied over the past two decades, theories with nonminimal couplings remain comparatively unexplored. We investigate theories containing the interaction terms $R F_{ab}F^{ab}$, $R_{ab} F^a_{\ \, c} \, F^{bc}$ and $R_{abcd} F^{ab} F^{cd}$, which generically arise in low-energy effective Lagrangians. We prove that magnetically charged regular black holes are excluded, except possibly for finely tuned choices of coupling constants, and argue that a similar conclusion applies to electrically charged regular black holes. We further show that similar conclusions hold for Lagrangian terms of the form $f(R,F_{ab}{\star F}^{ab})$.

We derive spectral identities involving the Laplace spectrum and the integrals of products of eigenfunctions on flat tori and orbifolds. Using semidefinite programming, we derive upper bounds on sums of squares of triple products from these spectral identities, following the approach of the conformal bootstrap. Physically, these upper bounds give constraints on sums of squares of cubic coupling constants in toroidal compactifications of higher-dimensional field theories. For Euclidean lattices, one of the bootstrap bounds leads to an upper bound on the mean square number of minimal vectors that are minimal distance from each minimal vector, divided by the kissing number of the lattice. By finding exact functionals for the semidefinite programming problems, we prove that this bound is saturated in 2, 4, 8, and 24 dimensions by the hexagonal lattice, the $D_4$ lattice, the $E_8$ lattice, and the Leech lattice, respectively.

Low-rank adaptation (LoRA) and its variants provide a memory- and compute-efficient alternative to full fine-tuning of pre-trained models. However, questions remain about the comparative generalizability of these approaches and how the structural restrictions on low-rank updates preserve effective adaptation performance. We present a historical framing, covering the past (full fine-tuning and original LoRA), the present (different variants of LoRA), and propose simpler, cheaper, parameter-efficient extensions by inducing sparsity within existing LoRA variants: Cheap LoRA (cLA), training a single low-rank factor with the other fixed (deterministically or, in its randomized variant, stochastically), and the chained circulant variant, ${c}^3$LA. We frame cLA as a structured instance of asymmetric LoRA, serving as a controlled column-subspace restriction of full fine-tuning. We derive information-theoretic generalization error bounds for these variants, marking one of the first endeavors in this area. Empirically, we evaluate 11 fine-tuning methods across 10 pre-trained models and 14 datasets, analyzing the fine-tuned models' performance and generalization using tools such as loss landscapes and spectral analysis. Despite the sensitivity of fine-tuned models to the pre-trained model, datasets, and other factors, our study suggests that restricting LoRA-based PEFT methods' adaptation to a sparse, structured column space remains competitive across tasks with their parameter-matched baselines while reducing up to 10% training time and peak GPU memory up to 15%, even with a naïve, non-optimized, sparse implementation. Our theoretical and empirical generalization measures provide a more consistent and principled approach to their cost-effective adaptation than commonly used analytical tools. Overview and code are available at: https://elicaden.github.io/Beyond_LoRA/.

Titchener et al. (2005) asked whether non-simple T-prescriptions can achieve larger T-complexity than simple T-prescriptions for infinitely many finite values of the maximum codeword length. We answer this question affirmatively for every fixed alphabet of size at least two. In fact, we prove a stronger upper-envelope statement: for infinitely many lengths $N$, there exists a valid non-simple T-prescription of exact maximum codeword length $N$ whose T-complexity exceeds the best value attainable by any simple prescription with maximum codeword length at most $N$. Hence the unrestricted exact-length maximum over prescriptions is strictly larger than the corresponding simple exact-length maximum for infinitely many $N$. The proof is elementary. Once a copy pattern has been used, it cannot be selected again. Thus simple prescriptions with more and more steps must use ever more distinct words, which forces the minimal simple length thresholds to have infinitely many strict upward jumps. At each such jump, changing the last copy factor of a minimal simple prescription from $1$ to $2$ yields a valid non-simple prescription, thereby increasing T-complexity by $\log_2 3-1$ while staying below the next simple threshold.

Let \(G\) be a finite abelian group, and let \(\mu(G)\) denote the least size of a subset \(S\subseteq G\) with \(3\mid |S|\), total sum zero, and no partition into zero-sum triples; put \(\mu(G)=\infty\) if no such subset exists. We prove the exact classification \(\mu(G)=\infty\) precisely for groups of order at most \(8\) and for \(G\cong C_3^2\), while \(\mu(G)=6\) for every other finite abelian group. The cyclic special case gives \(\mu(\mathbb Z_n)=\infty\) for \(1\le n\le 8\) and \(\mu(\mathbb Z_n)=6\) for \(n\ge 9\), answering the corresponding non-partite cyclic question. We also record a higher-uniformity interval construction which explains the large cyclic witnesses as the case \(k=3\) of a general \(k\)-tuple obstruction.

The IEEE 802.11bn amendment defines significant modifications to the standard by establishing Ultra High Reliability (UHR) targets in Wireless Local Area Networks (WLANs). This is expected to deliver substantial enhancements over previous standards, including modes of operation that increase throughput, reduce the 95th percentile of the latency distribution, and decrease MAC Protocol Data Unit (MPDU) loss (all by at least 25%) compared to Extremely High Throughput (EHT) operations defined in the 802.11be amendment. A fundamental innovation for achieving these ambitious goals is the introduction of Multi-Access Point Coordination (MAPC), an unprecedented feature whereby APs will be able to coordinate among themselves to enhance spectrum utilization and advance towards reliability. This paper provides a comprehensive overview and analysis of this key framework. We begin by reviewing existing AP coordination solutions that precede the 802.11bn standard, which serve as a foundation for understanding the transition to the current framework. We then describe the technical 802.11bn MAPC framework as defined by the task group. A detailed overview of each candidate MAPC feature is provided, contextualized with the relevant state-of-the-art. Furthermore, we introduce Kom8ndor, an open-source Wi-Fi 8 simulation tool, to evaluate these candidate MAPC features and showcase their potential to achieve UHR goals. Finally, we outline the future of MAPC beyond 802.11bn, exploring promising directions such as coordination schemes beyond 802.11bn (e.g., Joint Transmission (JT)) and new ideas.

In this paper, we introduce the notions of uniform modulo-recurrence and strong modulo-recurrence. We show that Sturmian words and maximal complexity words are strongly modulo-recurrent. Next, a relationship between the window complexity and the classical complexity of the Thue-Morse word is established. We provide an aperiodic recurrent word such that the window complexity is bounded. We also construct a family of aperiodic recurrent words such that their window complexity $P^w(n)$ is in $O(n^\alpha)$, while at least $n^\alpha$ for infinitely many $n$, where $0<\alpha<1$. Finally, we establish that the window complexity of a uniformly recurrent aperiodic word is unbounded.

Let $\mathscr{H}$ be a complex Hilbert space and $\mathscr{B}(\mathscr{H})$ be the algebra of all bounded linear operators on $\mathscr{H}$. For $T \in \mathscr{B}(\mathscr{H})$, let $|T| := (T^*T)^{\frac{1}{2}}$. We refer to the sequence $\big\{|T^n|^{\frac{1}{n}}\big\}_{n \in \mathbb{N}}$ as the NPS (normalized power sequence) of $T$. In this article, we show that the NPS of a Riesz operator $R \in \mathscr{B}(\mathscr{H})$ converges in norm to a positive operator $H$, and provide an explicit description of the spectral resolution of $H$ in terms of the Riesz idempotents associated with the non-zero eigenvalues of $R$. Since every compact operator is a Riesz operator, this gives us a stronger, spatial generalization of the Yamamoto-Davis theorem, which asserts that $\lim_{n \to \infty} s_j(K^n)^{\frac{1}{n}}$ is equal to the $j^{\textrm{th}}$-largest eigenvalue-modulus of the compact operator $K$, where $s_j(\cdot)$ denotes the $j^{\textrm{th}}$-largest singular value. In recent work, the present authors have established the norm convergence of the NPS for spectral operators. Using a rank-one perturbation of a unitary operator, we demonstrate that this fails in general for essentially spectral operators (of which Riesz operators form a subclass).

Multi-agent systems (MAS) were expected to overcome the limitation of single-agent systems (SAS) through collaboration. However, under typicality conditions on the task's constraint graph and bounded inter-agent communication, we prove that the success probability of a MAS is closely tied to the connectivity of task constraints, where each agent has limited information-processing capacity. Specifically, the success probability decays exponentially with an information bottleneck that emerges from partitioning the task's constraint graph among agents. We define this quantity as the minimum cut cost $C_{\min}$ of the potential constraint graph of each task. This information-theoretic bound applies to both open systems with external feedback and closed systems without. We validate our theory on both synthetic experiments and real-world empirical data from SWE-bench submissions. From our framework, effective MAS design should incorporate task-inherent constraints alongside engineering optimization, and when $\Cmin$ is high, practitioners should restructure tasks rather than simply scaling agents or communication.

Sunflowerability, or the infinite sunflower property, was introduced and studied by Ackerman, Karker and Mirabi as a structural generalization of the well-known \Delta-system lemma for sets. It turns out that for relational Fra\”issé limits with strong amalgamation, this property is equivalent to the so-called galah property, which was introduced by Sullivan and Winkel as an asymmetric variation of indivisibility. This paper is about these two properties and is divided into three parts. In the first part, we show that the conjecture proposed by Ackerman, Karker and Mirabi about the infinite sunflower property in higher dimensions is far from being true by proving that no infinite structure has the infinite n-sunflower property in dimension k for any n, k \geq 2. In the second part, we give a complete characterization of the galah property for Henson directed graphs, homogeneous metric spaces and homogeneous ultrametric spaces, thereby answering the second question asked by Sullivan and Winkel. The third part contains several additional results about the finite sunflower property, including a strengthening of recent results about indivisibility for some classes of undirected graphs obtained by Guingona et al..

We prove a local bijection between alternating sign matrices (ASMs) and descending plane partitions (DPPs) in the refined fibres of Behrend–Di Francesco–Zinn-Justin. The map is constructed by moving a boundary line through the ASM and DPP pictures. At each step only a bounded neighbourhood of that boundary is changed. In growth-diagram language, the boundary states are ordinary ASM row frontiers and shifted DPP path frontiers, and the elementary moves are reversible Fomin-type square rules. The proof checks that overlapping local moves are confluent, that every square is invertible, and that the completed boundary again satisfies the ordinary ASM and DPP coordinate inequalities. The only terminal choice is resolved by a radius-two fan rule on the visible height contour. The same local framework gives a statistic on refined ASMs whose generating function is the DPP degree enumerator. The statistic has a Striker–Fulmek quadratic core \[ Q_{\rm SF}(A)= \sum_{1\le r<s\le n}\ \sum_{1\le b<a\le n} (n-r+1)A_{r,a}A_{s,b}, \] plus a bounded fan-boundary correction and, outside the normal boundary gauge, an explicit compact-potential endpoint term. In the normal gauge this is \[ W_{\rm ASM}^{\rm SF}(A)=Q_{\rm SF}(A)+B_{\rm fan}(A). \] Thus the paper gives both a local ASM–DPP bijection and a statistic-preserving refined \(q\)-enumeration. The supplementary files reproduce selected local checks and the displayed arithmetic examples.

At first glance, one would imagine that the energy field of the Ising model, the set of edges whose endpoints share the same spin, is stochastically monotone as a function of the coupling constants. However, this is not generally the case. In this paper, we introduce two weaker notions of stochastic domination that make this result true: $p$–weak and $p$–weak$^\dagger$ domination. Both of these notions depend on a parameter $p$ and we find the optimal values $p$ and $p^\dagger$ so that these dominations hold. One of the key ingredient to obtain some of the results is a new stochastic domination relating FK percolations with different parameters $q,\tilde{q}\geq 1$ that is of independent interest.

Question decomposition, i.e. breaking a complex query into simpler sub-queries whose answers are composed to produce a final answer, is a widely used strategy for improving LLM reasoning, yet it currently lacks a rigorous mathematical foundation. In this paper, we propose operads, mathematical structures that model many-in, one-out operations and compositions thereof, as a natural framework for describing question decomposition. We define the questions operad $Q$, in which operations correspond to question templates and composition corresponds to substitution of sub-answers, and show how QA models can be interpreted as algebras over $Q$. Beyond reframing existing practice, this operadic perspective points toward new methods, in particular a notion of operadic consistency, which measures whether a QA model's answers agree across the partial collapses of a question decomposition tree. Empirical evaluation of operadic consistency is reported in our companion paper (Bottman, Liu, and Richardson, 2026), which finds it strongly correlated with accuracy across twelve LLMs and four multi-hop QA datasets and outperforming standard temperature-based self-consistency baselines. We argue that operads are the natural mathematical home for question decomposition, and that invariants such as operadic consistency open new directions for analyzing and improving the reliability of multi-step reasoning.

We prove a growth dichotomy for infinite approximate groups, and more generally approximate semigroups, in hyperbolic groups. If \(G\) is a finitely generated hyperbolic group and \(A\subseteq G\) is infinite with \[ A^2\subseteq AX \] for some finite \(X\subseteq G\), then either \(\langle A\rangle\) is virtually cyclic, or \(A\) has positive exponential growth in the ambient word metric. We also introduce a product-growth criterion for the existence of growth rates of approximate semigroups. The criterion applies to hyperbolic groups: if \(G\) is hyperbolic with finite generating set \(S\), then there is a constant \(c_{G,S}>0\) such that \[ |UV| \geq c_{G,S}\,\frac{|U||V|}{n+k+1}, \qquad U\subseteq B_n,\; V\subseteq B_k. \] The linear loss is optimal in order whenever \(G\) contains an element of infinite order. In the free group with its standard generating set one may take \(c_{G,S}=1/4\). We also prove that, in a free group, if \(U\subseteq S_n\) and \(V\subseteq S_k\), then \[ |UV|\geq \left(\frac{2}{3}+\frac{1}{3\cdot 4^{\min\{n,k\}}}\right)|U||V|, \] and this constant is sharp for all \(n,k\).

We construct a graphical coaction for Friedmann-Robertson-Walker (FRW) integrals at all loop orders in conformally-coupled scalar theories with non-conformal polynomial interactions. Our construction makes use of intersection theory in the context of (partial/relative) twisted (co)homology, which we use to decompose FRW integrals (and their discontinuities and derivatives) into building blocks that can be represented as decorations of the original Feynman diagram. This facilitates a purely graphical description of the coaction, up to rational prefactors that can be read off from the graph. Our construction provides a comprehensive combinatorial framework for dissecting the analytic properties of cosmological observables; in particular, we demonstrate that the combinatorics of the differential equations that govern FRW integrals – their so-called kinematic flow – is a natural consequence of our coaction. We have also developed a user-friendly web application that computes the graphical coaction of any graph: https://frwcoaction.ca. Whenever possible, the web application also computes the differentials and discontinuities. A Mathematica notebook with the same functionality is also hosted at on a public GitHub repository.

There exist an absolute constant $c>0$ and arbitrarily large finite sets $A\subset \mathbb{R}$ with $$\left| \left\{x+y+(x-y)^2:\ x, y \in A\right\}\right| \le|A|^{2-c}.$$ Since $x+y+(x-y)^2 \in \mathbb{R}[x,y]$ is a polynomial which is neither additive nor multiplicative, this provides a counterexample for the Elekes-Rónyai problem. The proof combines two amplifications of the same local congruence defect: horizontal amplification over squarefree products of rational primes, and vertical amplification through bounded root-discriminant towers in which those primes split completely. In this way a fixed local density defect becomes macroscopic, producing a power saving. This phenomenon also suggests a broader mechanism for producing similar extremal constructions throughout combinatorics and number theory.

Let $p \colon \mathcal{A} \to G$ be a Fell bundle over a locally compact Hausdorff second countable groupoid $G$ equipped with a Haar system, and let $\Gamma$ be a discrete group. Given a continuous $1$-cocycle $c \colon G \to \Gamma$, we show that the $\mathrm{C}^*$-algebra of the restricted Fell bundle $\mathcal{A}|_{G_e}$ embeds isometrically into $\mathrm{C}^*(G;\mathcal{A})$, where $G_e = c^{-1}(e)$ is the clopen subgroupoid corresponding to the identity element. We exploit this embedding to show that $\mathrm{C}^*(G;\mathcal{A})$ admits a natural structure of a topologically graded $\mathrm{C}^*$-algebra in the sense of Exel. As a consequence, we obtain a canonical coaction $\delta$ of $\Gamma$ on $\mathrm{C}^*(G; \mathcal{A})$. We further show that the associated coaction crossed product $\mathrm{C}^*(G; \mathcal{A})\rtimes_\delta \Gamma$ is naturally isomorphic to the $\mathrm{C}^*$-algebra of a Fell bundle constructed from the cocycle data.

We study the problem of diffusion parameter estimation for stochastic differential equation (SDE) models in scenarios where data and model are compatible only on specific scales that have yet to be determined. We introduce a simple and efficient method for selecting suitable rates at which given time series data should be subsampled in order to ensure that the statistical structure of the subsampled data is consistent with the behavior of the SDE model on an infinitesimal scale. Our approach is based on analyzing the statistics of the lengths of monotonically increasing or decreasing segments in the subsampled data sequence, which we refer to as monotone runs. As an analytical foundation, we prove for a large class of SDEs with additive noise that the lengths of monotone runs at an infinitesimal scale are approximately geometrically distributed with success probability $1/2$. This universal characterization is employed to derive an automated method for selecting appropriate subsampling rates for given time series data that is directly applicable in real-world scenarios and does not rely on an asymptotic framework of multiscale diffusions. The approach is demonstrated using an application from industrial mathematics concerning surrogate models for fiber lay-down curves in production processes of nonwoven textiles.

We give a short proof that the majority vote of three independent consistent classifiers is an optimal learner in the realizable PAC setting. This proves optimality for the simplest voting scheme, while simplifying both the algorithmic structure and the probabilistic analysis of previous voting learners, including the algorithm of S. Hanneke and the analysis of bagging by K. Green Larsen.

This paper presents a distribution-agnostic robust trajectory-optimization framework based on chance-constrained reinforcement learning. The uncertainty is represented here through initial conditions and process noise, with the only requirement being that it can be sampled. A deterministic nominal trajectory is first computed offline, and reinforcement learning is then used only to robustify that baseline through a structured affine closed-loop correction law comprising a feedforward control adjustment and time-varying feedback gains. Probabilistic feasibility is enforced empirically through rollout-based upper-tail quantiles, while terminal dispersion is regulated through covariance-feasibility penalties. The framework is assessed on two materially different trajectory design problems. The flagship case study is a three-dimensional multi-impulse Earth-Mars transfer, where the learned policy is benchmarked against a recent robust trajectory-optimization reference under Gaussian uncertainty and then evaluated under bounded uniform uncertainty and under process disturbances not seen during training. The second case study is a stochastic atmospheric pinpoint rocket landing problem, used to assess portability to a short-horizon continuous-thrust setting with drag, mass depletion, and glide-slope constraints. The results show that the proposed framework can remain competitive in upper-tail fuel cost while preserving probabilistic feasibility, and that the same robustification scaffold can be carried across heterogeneous spacecraft trajectory planning problems without redesign of its core stochastic-control structure.

We discover six new families of modular Nahm sums in ranks 3 and 4. Two of them are rank three sums obtained by modifying two of Zagier's rank three examples. Three rank four families are derived by applying the lift-dual operation to the rank three tadpole Nahm sums studied by Milas and Wang, while the other rank four family is found by the constant term method. To prove modularity, we establish Rogers-Ramanujan type identities that express these Nahm sums as infinite products which are modular.

We find lower bounds on the topological complexity of fibers of PL and generic smooth maps $p:M^d\rightarrow\mathbb R^m$, where $M^d$ is a closed hyperbolic manifold of large injectivity radius. More precisely, we show that if the injectivity radius of $M$ is greater than $50\log((n+1)!)$, then for each dimension $0<k<d-m$ there is a point $z\in\mathbb R^m$ such that any cell structure on the fiber $p^{-1}(z)$ has more than $n$ cells of dimension $k$. The proof is based on a freedom theorem for ideals in group rings of hyperbolic groups proved in arXiv:2309.16791.

In this paper, we investigate a secure integrated sensing and communication (ISAC) system in which multiple communication users (CUs) coexist with multiple untrusted sensing users (SUs) that may eavesdrop on the confidential information intended for the CUs. To promote security fairness among users, we formulate a max-min secrecy rate optimization problem subject to a transmit power budget and sensing quality requirements characterized by beampattern matching error constraints. The resulting design problem is highly non-convex due to the secrecy rate expressions and non-convex sensing constraints. To address these challenges, we first reformulate the problem using semidefinite relaxation (SDR). Based on the reformulated problem, we develop a branch-and-bound (BB) framework combined with convex relaxations to obtain the globally optimal solution within a prescribed accuracy. To further reduce computational complexity, we propose a low-complexity algorithm based on successive convex approximation (SCA), which iteratively solves a sequence of convex subproblems and converges to a local solution. Numerical results demonstrate that the proposed BB algorithm achieves the global optimum and provides a benchmark for performance evaluation. Moreover, the proposed SCA-based algorithm attains near-optimal secrecy performance with significantly lower computational complexity, making it attractive for practical ISAC deployments.

We obtain an optimal proximity bound for integer linear programs in standard form max{cx: Ax=b, x nonnegative integer}, where A is an integer mxn matrix of rank m<n and b is an integer vector. Specifically, we show that the Euclidean distance from any optimal vertex solution of the LP relaxation to a nearest optimal integer solution is bounded by $\sqrt{\det(AA^t)}-1$ and that this estimate is asymptotically tight. We also derive bounds for the optimal integer solutions involving the product function $\prod_{i=1}^{n}(x_i+1)$ and discuss their applications in the knapsack setting.

Koopman linearization opens many possibilities for control synthesis and analysis of nonlinear systems. Whether or not any given nonlinear control system admits a finite-dimensional Koopman representation remains a crucial question to address. A related problem is to categorize the class of all Koopman linearizable nonlinear control systems. In this work, we present differential geometric conditions on the drift and control vector fields of a control-affine nonlinear system, that must be necessarily satisfied for Koopman linear transformation to exist. The same conditions are also shown to be sufficient for (a slightly weaker notion of) Koopman linearizability on control-invariant manifolds. Further, these conditions, together with an additional condition, become necessary and sufficient for Koopman linearizability to a controllable linear system. Our examples illustrate the ease of checking these conditions, and also shed light on how Koopman linearizing transformation may not exist for a control-affine system even though one can linearize the autonomous part of the system via Koopman lifting.

For even integer exponents $p$, we obtain dimension-free Markov–Bernstein inequalities for polynomials with respect to product probability measures $\mu=\mu_1\otimes\cdots\otimes\mu_n$. When the measures $\mu_j$ have unimodal densities, we prove \[ \|\nabla f\|_{L^p(\mu)} \le C(p) d^2 \|f\|_{L^p(\mu)} \] for polynomials $f$ of degree at most $d$. The dependence on $d$ is optimal already for the uniform distribution on the unit cube. For products of one-dimensional Freud measures with densities proportional to $e^{-|t|^{2m}}$, the factor $d^2$ can be replaced by $d^{1-\frac{1}{2m}}$. In the Gaussian case, for all $p\ge4$, we prove that \[ \|\nabla f\|_{L^p(\gamma^n)} \le C(p)d^{\frac12+\theta_p}\|f\|_{L^p(\gamma^n)} \] for every polynomial $f$ of degree at most $d$, where $\theta_p\le \frac{2}{3p}$ and $\theta_p=0$ whenever $p$ is an even integer. Thus, in the even-integer case, we establish the sharp dependence on the degree conjectured by Eskenazis–Ivanisvili. For general $p\ge4$, the estimate improves upon their dimension-free inequality.

This paper investigates the Kähler hyperbolicity modulus on complete Kähler manifolds, with a particular focus on hyperconvex domains and bounded strongly pseudoconvex domains. Our main result establishes a lower bound for the Kähler hyperbolicity modulus in terms of the boundary behavior of the gradient length of a plurisubharmonic function. As applications, we compute the Kähler hyperbolicity modulus for bounded symmetric domains. Furthermore, we obtain lower bounds for the Kähler hyperbolicity modulus on bounded strongly pseudoconvex domains equipped with Kähler-Einstein metrics or Bergman metrics.

Many important observables in physics and geometry are cup products of cochains. The adjusted cup product neural layer has been introduced in this paper. It is a neural primitive that hard wires the cup product with an adjustment term from higher gauge theory. This creates a readout that is gauge invariant by design. Their main theoretical result shows that on a closed cycle the output relies entirely on the adjustment coefficient. Setting this coefficient to zero removes the output completely regardless of other parameters. Thus the adjustment is the only source of gauge invariant signal. They prove this observable is a nonzero quadratic form and is exactly invariant under one and two gauge transformations.