Artificial intelligence assistants deployed in online learning environments create new opportunities to collect large volumes of learner interaction data and generate insights to improve student outcomes. Architecture for AI-Augmented Learning (A4L) is a modular data architecture that enables the collection, integration, and analysis of learner interaction data from educational AI systems, supporting the generation of instructional insights that facilitate personalized learning and reinforce the bidirectional feedback loop between instructors and learners. This study examines the modular design of the A4L Data Analytics Pipeline, an extensible data infrastructure that enables the ingestion, processing, and analysis of heterogeneous datasets generated by educational AI assistants. We describe the design principles and development process used to extend the pipeline's analytical capabilities while preserving flexibility across domains. We evaluate the pipeline through case studies spanning three research domains corresponding to three educational AI assistants deployed in online learning environments at Georgia Tech. Results show that a common set of statistical analysis methods can be consistently applied across datasets with differing structures and instructional contexts, enabling the pipeline to reproduce key analytical findings across domains. We demonstrate how analytical capabilities initially developed for one domain can be extended to support richer analyses in another, illustrating the pipeline's extensibility. These findings suggest that the A4L Analytics Pipeline can serve as reusable infrastructure for analyzing data generated by future educational AI assistants. By enabling analytics that can be systematically extended to new domains, the pipeline provides a foundation for deriving insights that inform the design and evaluation of educational AI systems.

We establish improved sample-complexity bounds for sample-based Lindbladian simulation based on the Wave Matrix Lindbladization (WML) algorithm. For a jump operator $L$ with dimension $d$, we derive an explicit non-asymptotic sample complexity bound $n_d^*(t,\varepsilon) \le \left( \frac{2d+3}{8} \right) \|L\|_\infty^2 \left( \frac{t^2}{\varepsilon} \right)$, holding for simulation time $t$ and error $\varepsilon$. This refines the dimension dependence of the best previously known bound, $O(d^2 t^2/\varepsilon)$, from [Go et al., Quantum Sci. Tech. 10, 045058 (2025)]. Remarkably, we show that this dimensional overhead can be entirely avoided when $\| L\|_\infty^2 = O(1/d)$, a condition satisfied with high probability for random Lindblad operators, yielding a typical-case sample complexity of $O(t^2/\varepsilon)$. On the other hand, in the worst case, we show that WML necessarily requires $Ω(dt^2/\varepsilon)$ samples by constructing an explicit example with a rank-one Lindblad operator. Our results reveal a sharp dichotomy between typical and adversarial sample complexities in Lindbladian simulation, thereby strengthening the theoretical foundations of sample-based quantum algorithms.

We study invariant statistical connections on the space $\mathcal{N}_0^n$ of zero-mean multivariate normal distributions (the multivariate centered Gaussian model) equipped with the Fisher metric $g^F$. We introduce moduli spaces of invariant statistical connections on homogeneous Riemannian manifolds via two natural equivalence relations arising from a categorical viewpoint, and apply this framework to $(\mathcal{N}_0^n, g^F)$. We explicitly determine the $GL(n,\mathbb{R})$-invariant and $\mathrm{Isom}(\mathcal{N}_0^n, g^F)$-invariant statistical connections, with particular emphasis on the dually flat case, and describe the corresponding moduli spaces.

We study Bernoulli percolation on $\mathbb Z^d$ in dimensions ${d>6}$. We prove that a classical consequence of the van den Berg-Kesten inequality, often referred to as the Simon-Lieb inequality in the context of the Ising model, admits a partial reversal. As a main application, we show that the quantity $φ_{p_c}(S)$, introduced by Duminil-Copin and Tassion (Comm.\ Math.\ Phys., 2016), is uniformly bounded over all $S\subset \mathbb Z^d$. This partial reversal further yields a short and self-contained route to several key results, including near-critical estimates on the two-point function and sharp bounds on the critical one-arm probability.

We study the moduli stacks of real vector bundles of fixed rank and degree on a type I real algebraic curve and determine its mod 2 cohomology algebra in terms of characteristic classes.

The bilayer Ruddlesden-Popper nickelate $\mathrm{La_3Ni_2O_7}$ has emerged as a promising platform for exploring and understanding high-temperature superconductivities. While most prior doping studies have focused on hole doping via strontium (Sr) substitution or by tuning oxygen content, electron doping remains largely unexplored. In this work,we systematically investigate electron doping in $\mathrm{La_3Ni_2O_7}$ thin films through tetravalent element substitution, employing first-principles density functional theory calculations. Our results reveal that, unlike in cuprates, $\mathrm{cerium}$ (Ce) doping is difficult to effectively introduce electron carriers into the low-energy bands. In contrast, zirconium (Zr), hafnium (Hf), and thorium (Th) can act as efficient electron dopants. These element substitutions can significantly increase the interlayer hopping $t_{\perp}$ between $d_{z^2}$ orbitals, which may lead to enhanced superexchange coupling $J_{\perp}$ , and thereby potentially elevated superconducting $T_c$. We evaluate the interaction parameters using constrained random phase approximation. Our results identify candidate dopants for achieving electron-doped $\mathrm{La_3Ni_2O_7}$, offering a route to clarify the ongoing debate on pairing mechanisms in this system.

We present RaFI, a CUDA and MPI based software framework that simplifies the task of building GPU-enabled data-parallel software where rays or similar work items need to migrate between different GPUs. RaFI provides a simple interface for CUDA kernels to forward such work items to other GPUs, while under the hood managing all the CUDA and MPI related work required to make this happen. We describe RaFI's motivation and implementation, and show its potential in several example applications.

A Kondo insulator (KI) is a prototypical example of a highly entangled phase of matter, where many-body interactions between local moments and delocalized electrons engender the non-magnetic insulating ground state. Conventionally, the local moments arise from atomic multiplet states with a narrow bandwidth, limiting Kondo coherence to low temperatures. Here, we realize a new paradigm for constructing the KI state with hybridized molecular orbitals in FeSb2. Resonant inelastic X-ray scattering (RIXS) at the Fe L-edge reveals distinct signatures of band-like continuum states and localized states. Comparisons with first-principles calculations establish a mixed-configuration ground state with hybridized Fe d-Sb p molecular orbitals as basis states. By systematically investigating the RIXS momentum, temperature, and doping dependences, we find propagating collective modes commensurate with many-body charge and spin excitations. Our results pave the way for understanding the emerging class of unconventional d electron insulators and engineering high temperature Kondo many-body states.

Objectives: We captured a fine-grained dataset of organic socializing with socially meaningful group labels to fill a gap in the study of face-to-face interaction. Prior interaction data from conferences, classrooms, hospitals, and workplaces exhibit network signatures such as heterogeneous contact rates, clustering, and bursty dynamics. However, schedules, room assignments, and authority roles in these settings may obscure organic social group dynamics. Studies on group mixing often rely on demographic proxies like gender, or assigned categories like school classes, rather than relationship-based groups. We aim to test if temporal network signatures from institutionally structured settings generalize to informal social interaction. Data description: We present the first, to our knowledge, public temporal proximity network dataset of a privately hosted social event, with contextual relationship-based group membership. At the outdoor cocktail hour of a wedding, 95 participants wore proximity sensor badges that detected other badge-wearers within approximately 1.5 m in 5 s intervals. The public dataset, coarsened to 10 s temporal bins, contains 7,213 contact events over 2,760 observed dyads. Participants self-reported their relationship category with respect to the wedding couple, enabling group mixing analysis. Beyond testing the generalizability of interaction patterns, this dataset supports modeling of social events for applications such as contact tracing policy and social-space design.

Text-to-Visualization (Text-to-Vis) translates natural language queries into visualization query languages, enabling non-expert users to perform data analysis. However, most existing methods follow a one-shot paradigm that requires users to specify all visualization details in a single round, often leading to cognitive overload and incorrect visualizations. In this paper, we propose PMVis, a progressive multi-turn paradigm for text-to-vis, where users' intents are refined through multi-turn interactions. To support research in this paradigm, we construct PMVisBench, the first dataset designed to capture the progressive and iterative nature of real-world user queries. It is built through VQL simplification and NLQ reconstruction, with explicit rule constraints to ensure each intermediate VQL remains valid and meaningful. Building upon PMVis, we further introduce PMVisAgent, an agent-based framework that simulates realistic user-system dialogues. PMVisAgent consists of a User, a System, and a Validation Agent that performs verification and repair via a ReAct-style tool-use loop to mitigate error accumulation across rounds, with explicit interaction and verification rules to ensure reliability of the multi-agent system. Extensive experiments on PMVisBench demonstrate that PMVisAgent significantly outperforms state-of-the-art text-to-vis baselines. It achieves up to 17.57\% and 23.21\% improvements in execution accuracy in single-table and multi-table settings, respectively, while ablation studies confirm the importance of combining progressive interaction with clarification. The code is available at https://github.com/wxxv/PMVis.

We study nonlinear concentration problems for time-frequency distributions in the Cohen class. Using recent techniques from quantum harmonic analysis (QHA) we provide both positive and negative results, such as sufficient conditions for the existence of optimizers in terms of the ``window operator'' and explicit examples where the supremum is never attained. We also study the structural properties of window operators, in particular operators that yield weakly continuous concentration functionals and operators for which the nonlinear concentration problem admits an optimizer, also beyond the Heisenberg representation. We then consider generalizations to the study of concentration problems for phase space representations of operators. We consider generalized Husimi distributions via quantum convolution, and their optimization problem when optimizing over Hilbert--Schmidt and density operators. Lastly, we consider representations of operators on double phase space, in the spirit of quantum time-frequency analysis, and give a full solution in terms of the Weyl symbols.

The recent discovery of altermagnetism has sparked growing interest in compensated magnetic systems as promising platforms for highly scalable spintronics. Altermagnetism is a distinct magnetic order where opposite spin sublattices are connected by rotation, yielding zero net magnetization but momentum-dependent spin splitting. To date, experimental verification of altermagnetic order has been achieved predominantly through bulk-sensitive techniques, including spin-dependent electronic spectra and transport responses. However, direct atomic-scale evidence that explicitly correlates crystal symmetry, local structural distortions, and magnetic ordering has remained unexplored. Here, we report the direct atomic-scale observation of coexisting polar distortions and altermagnetic order in MnTe, combining atomic resolution scanning transmission electron microscopy (STEM) imaging with electron magnetic chiral dichroism (EMCD) measurements. We reveal that MnTe is not an ideal uniform P63/mmc g-wave altermagnet at the atomic scale. Instead, it hosts ubiquitous inversion-symmetry-breaking distortions that lower the spin-space-group (SSG) symmetry, admits d-wave altermagnetic components, and in lower-symmetry regimes, even allow s-wave spin splitting (net magnetization). The coexistence of ferroelectric signatures and altermagnetic order establishes local lattice symmetry in MnTe as a control knob for altermagnetic spin splitting, spin current generation, and multiferroic memory applications.

We introduce a geometric formulation of quantum indeterminacy from which the standard uncertainty inequalities emerge as necessary consequences. Our approach is based on convex geometry in phase space and on methods from symplectic topology, and does not rely on statistical descriptors such as variances or covariances. Instead, we associate to empirical position and momentum data with convex bodies whose mutual relations encode the fundamental constraints of quantum mechanics. The central tools are h-polar duality and symplectic capacities, which provide intrinsic, coordinate-free bounds on admissible phase-space configurations. Within this framework, the Robertson-Schrodinger inequalities arise naturally as manifestations of deeper geometric and topological principles. This perspective suggests that quantum indeterminacy is not primarily a statistical phenomenon, but rather a structural property of phase space governed by symplectic covariance. The results thus provide a unified and conceptually transparent foundation for the uncertainty principle.

The artificial intelligence industry is not an isolated economic phenomenon; it is the current physical substrate for a broader, multi-billion-year process: the evolution of an abstract intelligence on Earth. As the scale of computation accelerates planetary-wide, the academic and popular discourse must address the physics aspect of this phase transition, beyond algorithmic architectures, alignment, and silicon supply chains. From first principles, it is clear that if the current exponential trajectory of computation holds, the ultimate bottleneck of the coming decades will be neither data nor capital, but the laws of thermodynamics and the finite heat capacity of the Earth. The evolution of intelligence is fundamentally a problem of non-equilibrium thermodynamics, bound by strict hardware limitations, and ultimately, an absolute ecological boundary condition. Civilization itself is an exceedingly rare and highly expensive thermodynamic algorithm actively fighting the default settings of the cosmos. To better understand AI's future and our future, we must rigorously examine the physical laws governing computation, complexity theory, and the narrow thermodynamic tightrope our civilization must walk to survive it

Recent angle-dependent magnetoresistance measurements in underdoped cuprates have revealed compelling evidence for small hole pockets in the pseudogap regime, including observation of the Yamaji effect in HgBa$_2$CuO$_{4+δ}$ (Chan et al., Nature Physics 10.1038/s41567-025-03032-2 (2025)). A key distinction between theories is their predicted Fermi volumes, measured as fractions of the square lattice Brillouin zone: $p/4$ per pocket for spin density wave (SDW) versus $p/8$ for fractionalized Fermi liquid (FL*), where $p$ is the hole doping. We calculate the $c$-axis magnetoresistance $ρ_{zz}(θ, ϕ)$ within the semiclassical Boltzmann formalism for both states, and using the ancilla layer model (ALM) for FL* in a single-band Hamiltonian. The results from the $\text{FL}^*$ phase show good consistency with current experimental data. Conversely, the results for the SDW phase are highly sensitive to the ordering momentum along the $z$-direction. An ordering vector of $Q = (π, π, π)$ yields predictions that starkly disagree with the experiment. The only possibility for agreement within the SDW scenario is to assume an ordering momentum of $Q = (π, π, 0)$. However, even in this specific case, the SDW scenario predicts a marginally smaller Yamaji angle at $ϕ=0$ than the FL* theory, and a second Yamaji peak near in-plane angle $ϕ= 45^\circ$, which was not observed in the experiment. In reality, the Néel ordering vector is likely uncorrelated between adjacent layers, so that there is no coherent interlayer transport of hole-pocket quasiparticles in the SDW scenario, and consequently no Yamaji effect. Our results support the FL* interpretation of Fermi arcs in the pseudogap phase, and establish Yamaji angle measurements as a discriminatory tool between theoretical models.

Consider a non-relativistic quantum particle with wave function $ψ$ in a bounded $C^2$ region $Ω\subset \mathbb{R}^n$, and suppose detectors are placed along the boundary $\partial Ω$. Assume the detection process is irreversible, its mechanism is time independent and also hard, i.e., detections occur only along the boundary $\partial Ω$. Under these conditions Tumulka informally argued that the dynamics of $ψ$ must be governed by a $C_0$ contraction semigroup that weakly solves the Schrödinger equation and proposed modeling the detector by a time-independent local absorbing boundary condition at $\partial Ω$. In this paper, we apply the newly discovered theory of boundary quadruples to parameterize all $C_0$ contraction semigroups whose generators extend the Schrödinger Hamiltonian, and prove a variant of Tumulka's claim: all such evolutions are generated by the placement of a linear absorbing boundary condition on $ψ$ along $\partial Ω$. We combine this result with the work of Werner to show that each $C_0$ contraction semigroup naturally admits a Born rule for the time of detection along $\partial Ω$, and we prove that a detection will almost surely occur in finite time if detectors have been placed everywhere along $\partial Ω$.

We introduce a new version of the KL-divergence for Gaussian distributions which is based on Wasserstein geometry and referred to as WKL-divergence. We show that this version is consistent with the geometry of the sample space ${\Bbb R}^n$. In particular, we can evaluate the WKL-divergence of the Dirac measures concentrated in two points which turns out to be proportional to the squared distance between these points.

In automata theory, while determinisation provides a standard route to solving many common problems in automata theory, some weak forms of nondeterminism can be dealt with in some problems without costly determinisation. For example, the handling of specifications given by nondeterministic automata over infinite words for the problems of reactive synthesis or runtime verification requires resolving nondeterministic choices without knowing the future of the input word. We define and study classes of $ω$-regular automata for which the nondeterminism can be resolved by a policy that uses a combination of memory and randomness on any input word, based solely on the prefix read so far. We examine two settings for providing the input word to an automaton. In the first setting, called adversarial resolvability, the input word is constructed letter-by-letter by an adversary, dependent on the resolver's previous decisions. In the second setting, called stochastic resolvability, the adversary pre-commits to an infinite word and reveals it letter-by-letter. In each setting, we require the existence of an almost-sure resolver, i.e., a policy that ensures that as long as the adversary provides a word in the language of the underlying nondeterministic automaton, the run constructed by the policy is accepting with probability 1. The class of automata that are adversarially resolvable is the well-studied class of history-deterministic automata. The case of stochastically resolvable automata, on the other hand, defines a novel class. Restricting the class of resolvers in both settings to stochastic policies without memory introduces two additional new classes of automata. We show that the new automaton classes offer interesting trade-offs between succinctness, expressivity, and computational complexity, providing a fine gradation between deterministic automata and nondeterministic automata.

Large-scale disasters, such as pandemics and climate-related events, place extraordinary pressure on healthcare providers due to extreme demand surges. Managing these surges is essential to sustaining healthcare resilience. Although numerous studies on healthcare resilience, far less attention has been given to physicians and to how patterns of patient movement can help redistribute demand and alleviate stress on overburdened providers. In this study, we analyzed billions of electronic medical records documenting patient visits to primary care physicians (PCPs) to construct inter-regional patient flow networks across the U.S. During the COVID-19 pandemic, we observed that cross-regional flow rose to 2.81%, compared to the pre-pandemic level of 2.08%. This redistribution absorbed, on average, 58% of the excess stress on PCPs, meaning more than half of the surging demand was handled by patients' moves to less burdened regions, an absolute 43 percentage point improvement from the pre-pandemic baseline of 15%. Further analysis suggests that strengthening cross-regional patient flow could allow the healthcare system to absorb even more stress and reduce the demand for PCPs. These findings provide structural insights for the healthcare system to enhance its pandemic preparedness and disaster responses, and to improve patient care during crises.

Hierarchical multiplex imaging approaches generate spatially resolved single-cell measurements across multiple, spatially organized fields of view (FOVs) within patient tumor specimens, thereby enabling systematic investigation of how the organization of the tumor microenvironment varies along biologically meaningful intratumoral gradients. Existing approaches fail to jointly address this multi-resolution data structure needed to recover true biological signals. We propose MoSAIC: multi-resolution spatial regression analysis of cell colocalizations, a hierarchical Bayesian spatial regression model designed for multi-resolution spatial data. MoSAIC decomposes the joint variation into three model components: (i) global tumor-gradient effects, (ii) patient-specific effects to capture inter-patient variability, and (iii) Gaussian process models to account for spatial dependence between FOVs within each patient tumor tissue. Simulations demonstrate MoSAIC has improved prediction and model fit compared to existing spatial and non-spatial model alternatives. Our method is motivated by and applied to a renal cell carcinoma multiplex imaging cohort to investigate immune-tumor colocalization patterns across the epithelial-to-mesenchymal transition (EMT) gradient. MoSAIC identifies increased macrophage-tumor colocalization and decreased cytotoxic T-tumor colocalization progressing across the increasing EMT gradient, consistent with EMT-associated immune suppression and spatially varying immune engagement. Overall, MoSAIC provides an interpretable, multi-resolution framework for quantifying spatial tumor-gradient effects in cancer imaging studies. Software is available on GitHub at jcaldous/MoSAIC.

In this paper, we are dealing with constrained vector optimisation problems where the objective function acts between real linear-topological spaces. Our aim is to study the relationships between the sets of properly efficient solutions to constrained and unconstrained vector optimisation problems under certain cone convexity assumptions on the objective function using a vectorial penalisation approach.

Skipper Charge-Coupled Devices (Skipper-CCDs) are a leading technology in the search for sub-GeV dark matter and coherent elastic neutrino-nucleus scattering. A key background for rare-event searches with these detectors arises from "spurious charge" -- single-electron events generated when charges are transferred through the active region to the serial register, and across the serial register to the readout stage. We present a characterization of spurious charge in both the active region and the serial register of SENSEI Skipper-CCDs, and show that, in a well-shielded low-background environment, the dominant contribution originates in the serial register during Skipper readout, when horizontal clocks are held at constant voltage between pixel transfers. Motivated by this finding, we develop a "tri-level" clocking scheme in which the held-low phase is raised to an intermediate voltage during readout to suppress trap-mediated charge generation. Using the SENSEI detector near the MINOS cavern, we measure a serial-register single-electron density of $(2.9 \pm 0.1) \times 10^{-5}$ electrons/pixel/image under standard SENSEI readout conditions, reduced to $(4.0 \pm 0.4) \times 10^{-6}$ electrons/pixel/image with tri-level clocking -- a factor of $\sim$7 improvement. This technique offers a promising path to lower backgrounds in current and future Skipper-CCD experiments.

The crystalline differential operators on a smooth variety X give rise to a non-split Azumaya algebra over the cotangent bundle of the Frobenius twist X'. In some cases, this Azumaya algebra splits when restricted to finite covers of X'. In this short note, we show that, whenever X has a non-closed global one-form, there is a degree one cover of X' on which the Azumaya algebra does not split, answering a question of Sasha Petrov.

Self-dual theories, being UV-finite, should have their own AdS/CFT dualities. Higher-spin extensions of self-dual theories are attractive to simplify the CFT duals. The maximal self-dual theory is Chiral higher-spin gravity, which should be dual to a subsector of Chern--Simons matter theories. As a step toward establishing self-dual holography, we develop the Fefferman--Graham expansion and holographic dictionary for arbitrary spin self-dual theories. As an application, we derive bulk-to-bulk propagators and compute three- and four-point AdS/CFT correlators in a contraction of Chiral higher-spin gravity, which is a higher-spin extension of self-dual Yang--Mills theory.

Physics-informed neural networks (PINNs) formulate the solution of partial differential equations as residual minimization problems over neural network parameterizations. Although highly flexible, optimization of PINNs using modern variants of Stochastic Gradient Descent algorithms is expensive. On the other hand, iterative computation of PINN parameterization using the Gauss-Newton method suffers from convergence difficulties, dense Jacobian structures, and poor conditioning that limit the effectiveness of second-order optimization methods. In this work, we introduce IGA-ODIL, a spline-based residual minimization framework combining ideas from Optimizing DIscrete Loss (ODIL), robust variational residual minimization, and Isogeometric Analysis (IGA). Instead of neural-network parameterizations of PINNs, the unknown solution is represented by smooth B-spline basis functions, leading to sparse structured Jacobians and efficient Gauss--Newton optimization. We also derive robust residual formulations based on weighted Gram operators, making the loss function related with the true error. The resulting systems inherit locality, sparsity, and approximation-theoretic properties of classical finite element and isogeometric methods while preserving the residual-learning philosophy of scientific machine learning. The proposed methodology is evaluated on several benchmark problems, including Poisson equations, convection-dominated advection--diffusion equations, Helmholtz problems with highly oscillatory solutions, nonlinear Allen--Cahn equations, and inverse Helmholtz parameter identification. Numerical experiments demonstrate orders-of-magnitude speedups compared with PINNs and CRVPINNs while maintaining high accuracy and robustness.

Synchronization, a ubiquitous phenomenon in classical systems, has recently been extended to the quantum domain. Here, we show quantum synchronization of a bosonic mode exhibiting a Fock state-like limit cycle, manifesting as a steady state with a negative Wigner function. We demonstrate that this non-classical state can be phase-locked to an external drive, achieving synchronization within an Arnold tongue regime. We argue that synchronization is a dynamical property and fundamentally tied to the suppression of phase slips, which we show to occur with exponentially decreasing probability. We introduce a novel method to extract the phase slip rate from the Lindblad time evolution of the system. This work opens new avenues for understanding and manipulating non-classical synchronization dynamics.

We examine the intrinsic physical-chemical properties of the conjugated ladder-type polymer poly(benzimidazobenzophenanthroline) (BBL) in response to electron transfer. We aim at explaining the origin of the anti-ambipolar behavior behind the observed BBL nonlinear response associated with specific device architectures. To elucidate this point, we use theory and computation based on first principles, including density functional theory optimizations, ab initio molecular dynamics, time-dependent DFT, and Marcus-theory analysis. Our results reveal that this redox response is not simply monotonic but follows an alternating odd/even pattern in which gap narrowing and reopening occur sequentially before near-gapless behavior emerges at high charging. Converging theoretical evidence in this work demonstrates that bell shaped conductivity in BBL originates in its fundamental electronic structure and supramolecular organization.

We perform a mathematical and statistical analysis of the Wasserstein least squares problem, a regression method for vector-valued covariates and distribution-valued responses. Our proposal contrasts with other distributional regression methods by having a direct interpretation in terms of random variables, as a nonparametric analogue of the classic random-effects model. On the mathematical side, we use a strategy of Lavenant (2024) to show that Wasserstein least squares is the canonical extension of Euclidean least squares to the space of probability distributions from the perspective of convex analysis; this viewpoint gives rise to multimarginal and dual formulations of the Wasserstein least squares problem, extending a similar theory for Wasserstein barycenters. We perform a statistical analysis of the Wasserstein least squares problem under the template deformation model, showing, surprisingly, that estimation is possible at the n^{-1/2} rate. As a special case, we obtain improved rates of estimation for Wasserstein barycenters, which are an exponential improvement over those established by Ahidar-Coutrix, Le Gouic and Paris (2020). Finally, we propose a heuristic particle method for Wasserstein least squares and use it to conduct a novel analysis of large-scale demographic data from the RAND Health and Retirement Study.

We analyze the effect of microscopic heterogeneity on the Lorenz curve of macroscopic observables. Lorenz curve of a response function being a cumulative and bounded quantity, is often a more stable function than the corresponding probability density. We show here that by doing an exponent spectrum analysis of the complementary Lorenz curve, it is possible to obtain a reflection of the underlying heterogeneity that causes the response function to depart from a power law behavior. We demonstrate this framework first by synthetic data and then by analyzing the avalanche statistics of a two dimensional, Random Field Ising Model (RFIM) at zero temperature. This method can lead to possible use in estimating microscopic heterogeneity of a system from analysis of an estimated Lorenz curve, particularly in socio-economic and physical contexts where the full probability distribution function is unavailable.

In the generality of a rigidly-compactly generated tensor triangulated category, we introduce semi-Bousfield classes in terms of the vanishing of the tensor product in positive degrees with respect to a fixed reasonable $t$-structure. We show that semi-Bousfield classes provide a common generalisation of Bousfield classes and compactly generated tensor-compatible $t$-structures. Then we specialise to the setting of the unbounded derived category $\mathcal{D}_{\mathrm{qc}}(X)$ of a Noetherian scheme $X$ and show that the stratification bijection naturally extends to an assignment which takes a (not necessarily monotone) perversity on $X$ to a semi-Bousfield class in $\mathcal{D}_{\mathrm{qc}}(X)$. If $X$ is regular, this assignment constitutes a stratification of the whole semi-Bousfield lattice, while in the singular case, its image consists precisely of those semi-Bousfield classes arising from objects of finite Tor-dimension. Restricting this bijection to monotone perversities recovers the recent classification of compactly generated tensor-compatible $t$-structures of Dubey and Sahoo, (arXiv:2204.05015).