Papers

Everything takes place in the category of commutative unitary rings. For a fixed ring $R$, $\alg{R}$ is the class of $R$-algebras and $\igr{R}$ the subclass of idempotent generated $R$-algebras. Following Bezhanishvili et al and their study of Specker and locally Specker $R$-algebras, this paper studies the interplay of properties of $R$ and $A\in \igr{R}$ (both as rings and as $R$-modules). Examples: (1) If $R\sbq A\in \igr{R}$ and $R$ is weak Baer (aka p.p.\ ring) and $A$ is ring essential over $R$, then $A$ is weak Baer and locally Specker. (2) If $R$ is semiprime and all the idempotents of the complete ring of quotients are adjoined to $R$ to form $A$, then $A_R$ is flat iff $R$ is weak Baer, in which case $A$ is locally Specker. The Pierce sheaf is often used since it is based on idempotents. Properties are examined, old and new, that are true for $R$ iff they are true for all the Pierce stalks. Among the new is the result for f-rings (pure ideals are generated by idempotents): $R$ is an f-ring iff each of its Pierce stalks has no non-trivial pure ideals. This allows the expansion of the known classes of f-rings; f-rings play important roles in $\igr{R}$.

In the last few years, a number of authors have proved divisibility properties satisfied by various functions which count the number of overcubic partition $k$–tuples of weight $n$ for small values of $k$. In this work, we use generating functions to prove some of their results as well as multiple infinite families of new congruences for overcubic partition $k$-tuples which do not yet appear in the literature. In particular, we focus on a new perspective which provides insights as to why these functions are often divisible by powers of 2, and we also prove families of congruences whose moduli are odd. For example, we prove that, for all $m\geq 0$, $\OL{b}_{4}(22m + 11) \equiv 0 \pmod{11}$ and we also prove infinite families such as $\OL{b}_{9l+2}(9m+3) \equiv 0 \pmod{3}$ for all $m,l\geq0$.

Motivated by the classical stochastic maximum principle, random matrices and free stochastic differential equations we, develop an analog maximum principle for control problems driven by non-commutative random variables, e.g. random matrices. We formulate an optimal control problem in the setting of free probability, consisting of the controlled forward equation, a free backward stochastic differential equation. For both, we give global existence theorems. Due to the non-commutative Itô-formula, the definition of the Hamiltonian differs to the commutative case. Our strategy is to stay as close as possible to the commutative case. Finally we formulate and proof the maximum principle in the context of free probability. Several examples show the application of the maximum principle, where explicit solutions can be found.

Large areas on all continents except Antarctica are covered by dryland vegetation patterns, with wavelengths typically in the range of tens of meters. In models, many wavelengths are simultaneously stable, and we argue that this multi-stability also holds for real world patterns. We then study the shape of the Busse balloon representation of multi-stability for a previously introduced Klausmeier model with non-local grazing. For this we first extend application of singular perturbation to Klausmeier/Gray-Scott models to include non-local interaction, providing analytical control near the homoclinic limit, and then use numerical continuation to demonstrate deformation of Busse balloons away from the typical banana shape, in four non-local grazing regimes. Since the Busse balloon has been invoked to support "evasion of tipping", we underscore the importance of the shape of the Busse balloon when inferring ecosystem response to, e.g., climate change.

The success of chemotherapy depends on the effectiveness of the drug delivery strategy and its ability to destroy cancer cells while minimizing damage to healthy tissues. The main objective of this work is to minimize the density of invasive tumour cells by controlling the chemotherapeutic agents. For this, we address an optimal control problem with mixed control and state constraints. The concentration of chemotherapeutic drugs is represented as a control variable. We use a nonlinear reaction-diffusion equation to describe the effect of drugs on the progression of invasive tumours. We start with the mathematical analysis of this initial boundary value problem. Then, we formulate the optimal control problem, explain the role of different constraints, and derive first-order necessary conditions of optimality. Finally, in order to demonstrate the efficiency of the proposed strategy, numerical simulations in the case of the eradication of malignant lung tumours, are presented and analysed.

We develop a general framework for understanding and computing both the groupoid homology of an ample groupoid and the topological K-theory of its reduced C*-algebra, based on two main ideas: discretisation and independent resolutions. Discretisation shows that a special class of ample groupoids we term independent groupoids are homologically and K-theoretically equivalent to discrete groupoids. We introduce the notion of a resolution by independent groupoids and provide a recipe for building a controlled independent resolution of a given ample groupoid of interest, leading to a systematic way of studying its homology and K-theory. In order to illustrate our general ideas and methods, we work out several concrete examples and applications. Garside categories provide a wide range of examples, including higher rank graphs, self-similar groups and spherical Artin-Tits groups. We also present an application to the homology of Stein's groups.

Various classes of Gaussian random fields associated with transient Markov processes $Y$ have been introduced in the probability and mathematical physics literature. The present paper is based on a natural class of Gaussian random fields, termed universal Gaussian random fields (UGRF), for an underlying Markov processes $X$, on a state space $(S,\mathcal{S})$ and having an ergodic invariant initial distribution $\pi$, via a central limit theorem of Rabi Bhattacharya for appropriately scaled additive integral functionals $\int_0^{nt}f(X(s))ds = \sum_{j=1}^n\int_{(j-1)t}^{jt}f(X(s))ds$ for $f\in1_\pi^\perp\equiv \{f\in L^2(S,\pi):\langle f,1\rangle_\pi=0\}$. A Lamperti-type time change is introduced to obtain an infinite dimensional stationary Ornstein-Uhlenbeck evolution within a framework introduced in a classic paper of K. Itô. In particular it is shown that the Itô's deterministic component vanishes under this time change, and Itô's continuous regularity theory is applied. Connections with GRFs associated with Markov processes $Y$ in a sense of Dynkin, and a sense of Diaconis and Evans, respectively, are established under additional conditions on the infinitesimal generator $(A,\mathcal{D}(A))$ of the underlying Markov process $X$.

Inferring directed interactions between neural systems from EEG and MEG remains challenging due to noise, nonstationarity, and the high sample complexity of information-theoretic estimators. Transfer Entropy (TE) provides a principled and model-free measure of directed information flow; however, its practical estimation is not stable in finite data regimes (particularly as embedding dimension increases). This work introduces Embedded Polygon Symbolic Transfer Entropy (EPSTE), a framework that reframes TE estimation as a learnable problem operating on structured symbolic representations of local temporal morphology rather than raw signal amplitudes. Neural time series are decomposed into sequences of geometric primitives derived from local triplets of samples encoding complementary aspects of waveform structure such as magnitude, curvature and directional change. These primitives are discretised into symbolic tokens, yielding a compact but expressive state space over which symbolic TE is estimated. A recurrent neural network with attention-based multiple-instance learning is trained to predict surrogate-validated TE values from bags of symbolic temporal windows. The method is evaluated on source-reconstructed MEG data parcellated using the AAL90 atlas and compared against a standard symbolic baseline using identical architectures and supervision. The results demonstrate that while local window-level predictions are noisy, aggregation across trials and channel pairs yields stable directed dependencies. At the pair level, EPSTE achieves near-perfect recovery of ground-truth directed structure and significantly lower absolute error than the baseline, indicating that representational geometry plays a critical role in enabling practical learnability of information-theoretic dependencies.

We prove that the projective Clifford group $C(A)$ associated with a finite abelian group $A$ is isocategorical to the affine symplectic group $\mathrm{ASp}(A)=\mathrm{Sp}(V_A)\ltimes\widehat{V_A}$, where $V_A=A\oplus\widehat{A}$. This explicit isomorphism of tensor categories gives common parametrizations of their irreducible characters and conjugacy classes, under which their character tables agree entry by entry. It also gives an explicit basis for every tensor-power commutant of the projective Weil representation, indexed by symplectic orbits of tuples in $V_A$ whose entries sum to zero. For cyclic $A$, we classify these orbits by oriented two-generated submodules and obtain exact commutant dimensions in every tensor power.

We develop multilevel adaptive-rank iterative methods for the solution of linear and nonlinear systems arising from high-dimensional partial differential equations. Our contributions are threefold. First, we extend the projection method of Ballani and Grasedyck [6] to enable flexible preconditioning of high-dimensional linear systems in low-rank tensor formats. Second, we construct multilevel preconditioning strategies by adapting geometric multigrid methods to the low-rank setting. In contrast to prior work, which primarily employs multigrid as a standalone solver, we emphasize its role as an efficient and robust preconditioner. Third, we integrate these techniques within an inexact Newton framework for the solution of nonlinear systems. The proposed methods are evaluated on a range of model problems, including both linear and nonlinear equations, to assess their convergence behavior and computational efficiency. The results demonstrate that multilevel adaptive-rank strategies yield robust and scalable preconditioners, providing effective solvers for high-dimensional problems in low-rank formats.

We study the subalgebra structure of Krichever-Novikov algebras, which are Lie algebras of vector fields on smooth affine curves. Our main result is that every infinite-dimensional subalgebra of a Krichever-Novikov algebra is isomorphic to a finite-codimensional subalgebra of another Krichever-Novikov algebra. We then present some applications of our main result. First, we show that the universal enveloping algebra of any such infinite-dimensional subalgebra is not noetherian. We then prove that all Krichever-Novikov algebras satisfy the Dixmier property that all their nonzero endomorphisms are automorphisms, except for the Witt algebra of vector fields on the once-punctured affine line. Finally, we provide an explicit classification of the infinite-dimensional subalgebras of the Witt algebra.

We introduce notions of path indicators and path correspondences for finite quantum graphs, study their basic properties, and compute them explicitly for classical graphs, trivial graphs, and complete quantum graphs.

We answer some questions regarding the mean value theorem and related results in constructive mathematics. The answers to these questions reveal interesting properties of the Mean Value Theorem, Law of Bounded Change, and Constancy Principle. We see that, in contrast to the Intermediate Value Theorem whose approximate analogue was shown to hold constructively by Frank, the natural approximate versions of these theorems fail to hold in neutral constructive mathematics. Our proof of this makes use of the existence of a topos in which the real numbers are in bijection with the naturals, which was shown by Bauer and the second author. Using Booij's notion of locators, we show that the aforementioned approximate analogues do hold in the presence of a small amount of countable choice, and also under suitable locator lifting hypotheses. We also show that an even weaker approximate analogue of the Mean Value Theorem holds in neutral constructive mathematics.

The Morse index of a free boundary minimal surface is encoded in its Jacobi-Steklov spectrum, and we test how faithfully a physics-informed neural network (PINN) reproduces that spectrum on a problem whose answer is already known in closed form. The benchmark is the critical catenoid in the unit ball $\mathbb{B}^3$, where it is well known that the Morse index equals $4$ and the nullity equals $2$. Separating the angular variable reduces the eigenvalue problem to a family of one-dimensional Robin problems on $[-T,T]$, one for each Fourier mode. A network that enforces the parity of each mode by construction, and carries the eigenvalue as a trainable parameter, returns the three eigenvalues below the stability threshold to within $10^{-6}$ to $10^{-4}$ of their exact values, with PDE residuals of order $10^{-4}$; assembling them recovers the index $4$ and the nullity $2$. We then track the spectrum along a one-parameter homotopy joining a flat reference operator to the catenoid Jacobi operator and identify the crossings at which the index changes. Since the critical catenoid is rigid, a fact we prove, this homotopy deforms operators rather than surfaces. We close by explaining how the same pipeline, with its one-dimensional solver replaced by a two-dimensional one, is poised to address genuinely geometric families in ellipsoidal balls, where the boundary curvature is no longer constant, and the Morse index is not yet known.

In this work, we study the fluctuations of a Mixture of Experts (MoE) generated by a quantum neural network trained via gradient flow on supervised learning problems. Our main results establish the Central Limit Theorem (CLT), and Sanov's principle for an MoE as the number of experts diverges. We demonstrate that the fluctuations of the empirical measure of its parameters close to its corresponding limit probability measure solve a linear transport equation. As a byproduct, we show that the MoE converges to a limit function which solves an evolution equation governed by the neural tangent kernel associated with the quantum neural network.

The scientific utility of large single-dish radio telescopes depends critically on the stability and fidelity of their beam patterns, which govern angular resolution, sensitivity, and polarimetric accuracy. For the 32-m Ghana Radio Astronomy Observatory (GRAO) antenna, electromagnetic simulations reveal residual sidelobes, structural diffraction, and cross-polar leakage that limit performance in high-dynamic-range and polarisation-sensitive observations. To address these limitations, we develop a finite-impulse-response (FIR) spatial filtering framework that reformulates beam optimisation as a digital signal processing problem. By exploiting the equivalence between angular displacement and spatial frequency, classical FIR design methods, window-based and Parks-McClellan algorithms are adapted to operate directly on simulated Jones fields. This approach enables controlled suppression of high spatial frequency artefacts responsible for sidelobes and polarisation mixing, while preserving the telescope's diffraction-limited resolution. Applied to the GRAO 5 GHz beam model, the method achieves substantial reductions in near-in sidelobe ripple, improves beam smoothness, and lowers cross-polar leakage below -30 dB at boresight. These improvements translate into enhanced calibration stability and polarimetric precision, strengthening the telescope's capacity for Very Long Baseline Interferometry, spectral-line surveys, and pulsar timing. Beyond GRAO, the method provides a generalisable, non-invasive, and computationally efficient pathway for beam control applicable to other single-dish and phased-array instruments. The results establish digital spatial filtering as a practical complement to conventional optical or mechanical optimisation, advancing the integration of electromagnetic modelling and signal processing in next-generation radio astronomical instrumentation.

In classical symplectic geometry, under mild conditions, Thurston proved that one can construct a compatible symplectic form on the total space of a symplectic fibration with a connected symplectic base. Here we prove a derived symplectic analogue of this result, called the derived Thurston theorem. More precisely, we show that if a morphism $\pi: X \rightarrow S$ of derived stacks has a shifted symplectic fibration structure and the target stack $S$ admits a shifted symplectic structure, then, under certain conditions, one can construct a shifted symplectic structure on a dense Zariski-open derived substack of the source stack $X$, compatible with $\pi$ in a sense similar to the classical case. In this derived setting, we also present numerous examples of shifted symplectic fibrations, develop an affine model construction for such fibrations, and provide applications of the derived Thurston theorem.

A single geometric operation – projecting a reference onto a constrained family under a Bregman divergence – underlies a striking range of statistical methods. This tutorial develops the operation first as pure convex geometry, with no statistics attached. A strictly convex generator $G$ and its conjugate $F$ furnish two coordinate systems, a projection theorem with existence and uniqueness, and a Pythagorean {theorem}; the Pythagorean theorem itself produces {two} dual projections – the information (e-) projection onto moment-constrained families and the moment (m-) projection onto exponential families – exchanged by the conjugacy $G\leftrightarrow F$, so a single theorem governs both. Part~II reads off the statistics. The generalized linear model is treated in detail as the concrete carrier of the two projections: {under the canonical link,} the score equation is exactly the Pythagorean orthogonality, and the fit is simultaneously an e-projection in the natural coordinate and an m-projection in the mean coordinate. Maximum entropy, survey calibration, over-identified moment models, the EM algorithm, variational inference, autoencoders, and expectation propagation then fall into place as instances of the same construction – exactly where the underlying families are flat, and as controlled approximations or neighboring-divergence analogies where they are not. The mathematics of Part~I is self-contained; the statistical sections presume only familiarity with the methods being unified.

Let $M$ be a closed oriented $3$-manifold generically embedded in a complex $3$-manifold $X$. Its CR singular set is an oriented link $L\subset M$. We prove that if a sublink $L'\subset L$ bounds an oriented Seifert surface $S\subset M$ in the complement of $L\setminus L'$, then the CR singularities along $L'$ can be cancelled by an arbitrarily $\mathcal C^0$-small isotopy supported in an arbitrarily small neighbourhood of $S$.

Neuronal communication is inherently time-delayed, due to the finite speed of signal propagation. Although often considered challenging or disruptive, such time delays can also endow neural circuits with useful capabilities. Here, we show that delays in excitatory connections between excitable neurons coordinate their synchronization patterns by creating self-sustained oscillations that may be out-of-phase or in-phase. The emergence of these oscillations leads to an abrupt, explosive, transition to in-phase synchronized regimes due to small changes in connection strength or time-delay. We describe the mechanism underlying these phenomena as an interaction between the neuron's excitable dynamics and the delay in signal transmission, explaining many aspects of how the oscillations emerge. We show this phenomenon in different network connectivities, neuronal models, with and without excitation, with and without noise, highlighting the generality of the mechanism.

We establish the exact reconstruction thresholds for a class of broadcasting models on hypertrees over a symmetric binary alphabet. As a consequence, we show that the condensation threshold coincides with the Kesten-Stigum threshold for random NAE-SAT and random hypergraph bicoloring with arity at most four at any temperature, confirming a prediction of Ricci-Tersenghi et al. '19. We also determine the exact weak recovery threshold for the two-community hypergraph stochastic block model in certain parameter regimes. The proof combines an improved robust non-reconstruction analysis with a computer-assisted rigorous population dynamics algorithm. It relies heavily on information-theoretic tools, in particular the theory of binary memoryless symmetric channels and channel comparison methods.

Smith-Treumann localization for sheaves on the affine Grassmannian of a reductive group has previously been studied by Leslie-Lonergan (for spherical sheaves) and by Riche-Williamson (for Iwahori-Whittaker sheaves). In this paper, we show that the two versions are related by a commutative diagram that involves "convolution with the Steinberg module." As an application, we "categorify" certain formal characters of a reductive group called Steinberg quotients, previously introduced and studied by the second author. Specifically, we show that Steinberg quotients describe the stalks of spherical parity sheaves on the $\mathbb{Z}/p\mathbb{Z}$-fixed-locus of the affine Grassmannian.

The Helmholtz equation governs wave propagation in acoustics, electromagnetics, and seismology, but its indefinite nature makes it difficult to solve with iterative methods. Domain decomposition methods are a natural fit for massively parallel architectures, yet mapping efficient Helmholtz solvers onto modern GPUs remains a challenge. We address both with two key contributions: (1) a block-level domain decomposition scheme, in which each subdomain is assigned to a single thread block and all solves run concurrently in a single kernel launch, and (2) WaveHoltz as the subdomain solver. WaveHoltz is a fixed-point iteration that is uniquely well-suited to the GPU execution model due to its minimal memory footprint and no reduction operations. Together, these eliminate device-level synchronizations and replace global memory traffic with shared memory and register-level operations, keeping subdomain data largely resident in L1 and L2 cache. We explore two threading strategies: one degree of freedom per thread for small subdomains, and multiple degrees of freedom per thread for larger ones. Benchmarks of our CUDA based implementation on a NVIDIA A100 show that WaveHoltz achieves 2x-25x speedup over MINRES, with the advantage growing with subdomain size. Crucially, evaluating the subdomain solver in single rather than double precision yields an additional 2x-10x speedup–a benefit largely unattainable by MINRES due to loss of Krylov vector orthogonality under reduced precision.

Mapper is a well-known tool in topological data analysis, which visualizes and summarizes high-dimensional data. However, its output is sensitive to choices of lens functions, cover parameters, and clustering strategies, making evaluation challenging. Most works that have attempted to evaluate the Mapper algorithm have done so visually. In this paper, we review a roadmap for assessing Mapper algorithms along three complementary axes: stability, cluster quality, and topological shape preservation. We analyze Mapper and its variants on synthetic datasets and the UCI Digits dataset. These modes include topological explosion at high resolutions. Our findings indicate that these axes of evaluation are often in tension and that no single Mapper variant performs optimally across all three. This review provides practical guidelines for choosing Mapper variants and identifies open challenges toward a principled Mapper analysis.

We consider the parameter estimation problem in logistic regression with Gaussian design: the estimation of a fixed unknown parameter $\theta^*\in \mathbb{R}^d$ ($\|\theta^*\|_2\ge 1$) from $n$ i.i.d. samples $\{(x_i,y_i)\}_{i=1}^n$, where $x_i\sim N(0,I_d)$ and $y_i|x_i \sim {\rm Bernoulli}(1/(1+\exp(-x_i^\top \theta^*)))$. Our main aim is to characterize the finite-sample estimation performance and convergence behavior of gradient descent (GD) on the maximum likelihood objective (i.e., the logistic loss). Under small $O(1)$ stepsize and $0$ initialization, we show that GD linearly converges to a small neighborhood of $\theta^*$ achieving an $\ell_2$ error of order $O(\sqrt{\|\theta^*\|_2^5d/n})$. This substantially goes beyond existing theoretical results that lack non-asymptotic estimation error rate and exhibit much slower parameter convergence. We also establish a faster local linear convergence to the same statistical error under a large $\Theta(\|\theta^*\|_2)$ stepsize. The main technical component is to show that the gradient of the logistic loss satisfies a certain approximate invertibility condition (AIC). To that end, we uniformly control the deviation of the gradient from its population counterpart by covering and peeling arguments, and then show that the population GD is a contraction by a delicate analysis based on the eigenvalues of population Hessian matrices. Finally, we build upon the recent work Matsumoto and Mazumdar (2025) and devise a novel efficient estimator that attains a sharper rate in high dimensions. This indicates that the existing non-asymptotic guarantees exhibit sub-optimal dependence on $\|\theta^*\|_2$, and that in many regimes $\Theta(\sqrt{\|\theta^*\|_2d/n})$ is the tight estimation error rate. Numerical examples are provided to corroborate our theoretical results.

Stationary wave functions at the integer quantum Hall transition are known to be multifractal, but the exact form of the multifractality spectrum has remained a subject of debate. While conformal field theory arguments predict a parabolic law, numerical simulations show deviations from parabolicity. We resolve this discrepancy by pointing out that powers of the local wave intensity fail to obey the Gaussian Free Field and Abelian Fusion Hypothesis assumed in earlier analysis. Rather, due to the non-Abelian nature of the underlying effective field theory, wave intensity correlators are dressed by insertions of background charge distributed uniformly in space. An exact expression for the $q$-moments of point-contact generated eigenstates is presented. Numerical tests are performed for the critical Chalker-Coddington network model on a rectangular torus. Our results are in precise agreement with the predictions of conformal symmetry realized as a level-4 current algebra.

We study the overdetermined problem of inverting the n-dimensional cylindrical Radon transform, which maps a function to its integrals over cylindrical surfaces. In practice, the cylindrical Radon transform arises in photoacoustic tomography as a measurement model for integrating line detectors. By exploiting its relationship with the Funk and Radon transforms, we derive an analytic inversion formula for the cylindrical Radon transform. We also present numerical implementations of the proposed inversion algorithm, demonstrating its stability.

Topological Data Analysis uses tools from algebraic topology to study the shape and structure of data. The Mapper algorithm provides a graph-based summary of high-dimensional datasets by combining a filter function, a cover of the filter range, and clustering on the corresponding pullback sets. Several variants of Mapper have been proposed, including Conventional Mapper, F-Mapper, and Shape Fuzzy C-Means Mapper. In this article, we introduce Gustafson-Kessel Fuzzy Mapper Graphs, a geometry-adaptive extension of Shape Fuzzy C-Means Mapper. The proposed method replaces spherical fuzzy covers with ellipsoidal covers induced by the Gustafson-Kessel fuzzy clustering framework, making it more suitable for high-dimensional datasets with anisotropic and non-spherical geometry. We develop a stability framework for the graphs produced by Gustafson-Kessel Mapper and Shape Fuzzy C-Means Mapper. We prove that the membership functions depend smoothly on the fuzzifier, establish a precise condition for the existence of edges, and show that the graph is locally stable under small perturbations of the fuzzifier. We further describe the critical-event structure of graph changes in terms of threshold crossings of the membership functions and show that the graph is constant between consecutive critical events. When the threshold-crossing set is finite, this yields an eventual freezing threshold. Finally, we empirically show that Gustafson-Kessel Mapper can produce more stable graphs than Shape Fuzzy C-Means Mapper on high-dimensional and geometrically complex datasets.

We study classical operator inequalities in $\Phi$-product tensor algebras (a transformation $\Phi$-based generalization of the $t$-product framework) for third-order tensors. Although these algebras are algebraically isomorphic under different unitary transforms $\Phi$, we show that their quantitative behavior is not invariant. We prove that fundamental inequalities, including Golden–Thompson, Jensen, Klein, and Lieb, extend to the $\Phi$-product setting with the same constants as in the matrix case. However, the associated defect – the slack between the two sides of the inequality – depends explicitly on transform-domain noncommutativity. In particular, we establish a sharp characterization of the defect in terms of slice-wise commutators, revealing that inequality tightness is governed by transform-induced noncommutativity. We further demonstrate strong transform sensitivity by constructing explicit tensor pairs for which the defect vanishes under one transform (e.g., discrete Fourier transform) but grows linearly with the tensor depth under another transform (e.g., discrete Cosine transform), yielding an $\Omega(p)$ separation where $p$ is the matrix dimension of $\Phi$. Moreover, we prove that no transform is universally optimal: for any pair of transforms, there exist tensors for which each is strictly better than the other. These results show that the choice of transform defines a coordinate system in which commutativity is measured, inducing a nontrivial geometry of inequality tightness. Consequently, optimal transform selection is inherently data-dependent and can be formulated as an optimization problem over the unitary group.

We study weighted projective klt $\mathbb{Q}$-Gorenstein degenerations of projective space $\mathbb{P}^{n}$ and construct infinitely many new degenerations of projective space in any dimension. We also study in detail the deformations of arbitrary weighted projective threefolds. The rest of the paper provides several applications of the existence of degenerations of $\mathbb{P}^{n}$ and our methods apply to find many degenerations of $\mathbb{P}^{n}$ beyond just weighted projective spaces.