Motivated by quantum gravity on spacetimes with multi-scale geometry, we analyze quantum field theories with a self-adjoint fractional power $(\Box^2)^{γ/2}$ of the d'Alem\-bert\-ian in the kinetic term, for any real $γ>0$. Selecting a particularly simple version of the kinetic term which we call hermitian polynomial, we study the spectral decomposition of the propagator and, when $γ>1$, obtain the standard mass singularity $-k^2=m^2$. This is the only mode in the perturbative spectrum of asymptotic states, since the only other content of the theory is a cloud of purely virtual particles with complex masses. We also show that other versions of the self-adjoint fractional kinetic term lead to a different distribution of the virtual complex modes but to the same physical spectrum for $0<γ<3$, thus addressing the issue of uniqueness in this class of nonlocal theories. The non-hermitian version of the theory has the $-k^2=m^2$ particle plus a continuum of standard massive modes. Finally, we prove that unitarity of scalar, gauge and gravity models is respected at all perturbative orders if, in the hermitian cases, one adopts the fakeon prescription on scattering amplitudes or, in the non-hermitian case, $0<γ<1$ or $2<γ<3$ with the standard Feynman prescription. These results drastically simplify previous characterizations of fractional quantum gravity, which is super-renormalizable for $γ>2$.

We study the anomalous thermoelectric Hall response of two-dimensional massive Dirac fermions to first order in the electron-electron interaction. We compute both the Nernst response to a Luttinger-type gravitational potential and the particle magnetization, the latter being required to remove spurious non-transport contributions. We show that, for arbitrary interactions, the magnetization is described by a remarkably simple formula. Surprisingly, and contrary to expectations, subtracting the magnetization currents does not make the thermoelectric Hall coefficient vanish in the zero-temperature limit. We attribute this to violation of locality on the smallest length scales, which is inevitable in a quantized field theory, that happens to manifest itself in infrared physics.

Standard Model predictions for Majorana neutrino transition magnetic moments (TMMs) are subject to severe chiral and GIM-like suppressions, rendering them vanishingly small. To dynamically generate a macroscopic TMM, we propose a dark sector framework featuring a $U(1)_D$ gauge symmetry, a vector-like lepton doublet, and two complex dark scalars. We demonstrate that while fermion-radiated loop amplitudes identically cancel due to Majorana self-conjugacy, a chirally enhanced dark TMM is successfully generated exclusively through scalar-radiated loops. This mechanism safely shifts the required chirality flip onto the heavy internal fermion line and utilizes a misaligned double-scalar mixing in flavor space to evade the Majorana antisymmetry prohibition. We systematically confront this tensor portal framework with multi-frontier experimental constraints. Since the dark TMM generation is inextricably linked to charged lepton flavor violation, the internal Yukawa couplings are stringently capped by the latest $μ\to e γ$ limits from MEG II. Concurrently, the visible-dark kinetic mixing portal is heavily bottlenecked by missing energy and mono-photon searches at NA64 and BaBar. Our global phenomenological analysis reveals that the synergistic theoretical upper bound dictated by these indirect high-energy probes completely eclipses the direct scattering constraints from Borexino. This establishes a strict phenomenological hierarchy: high-intensity cLFV probes and accelerator-based dark sector searches jointly possess the overwhelmingly dominant exclusionary power over direct solar neutrino limits for such microscopic magnetic moment models.

We show that the wavefunction of the universe in theories of conformally coupled scalars in power-law Friedmann-Robertson-Walker (FRW) cosmologies satisfies a graphical coaction, by means of which we can understand its complete analytic structure in terms of the acyclic minors of Feynman graphs. Our construction extends to all particle multiplicities and any loop order, and if we isolate certain weight-one contributions, it reproduces the ``kinematic flow'' that encodes the differential equation of the wavefunction coefficients. Similarly, any discontinuity of the wavefunction coefficient is easily extracted from the coaction.

In this paper, we introduce an algebraic-topological invariant for commutative pm-rings, termed the spectral fundamental group, which is denoted by $π_{k}^{alg}(A)$. This group is defined via homotopy classes of loops within the space of induced spectral maps, which are generated by the $k$-algebra endomorphism monoid of the ring. We establish foundational properties of this invariant, proving that $π_{k}^{alg}(A)$ is an abelian group that naturally respects direct products and admits natural morphisms with respect to fully invariant subrings. Further, we establish an explicit isomorphism between the spectral fundamental group of certain continuous function rings and the classical fundamental group of their associated topological mapping spaces. Finally, utilizing a generalized dual number construction, we present an explicit example of a pm-ring that cannot be embedded into any function ring over a field of characteristic zero, yet possesses a nontrivial spectral fundamental group. This demonstrates that $π_{k}^{alg}(A)$ captures homotopical dynamics that are intrinsically algebraic.

Anderson localization (AL) and the non-Hermitian skin effect (NHSE) represent two paradigmatic localization phenomena driven, respectively, by disorder and non-Hermiticity. In one-dimensional (1D) non-Hermitian systems, these factors are known to compete and provide a smooth crossover between AL and NHSE upon parameter tuning. Here, we show that this interplay is fundamentally enriched in spinful systems, where an external magnetic field acts as an additional degree to manipulate the localization behavior. By investigating a disordered 1D spinful non-Hermitian chain, we demonstrate that under appropriately correlated disorder configurations across spin sectors, the magnetic field enhances the AL $\rightarrow$ NHSE crossover. Interestingly, this facilitates the Anderson delocalization transition even in strongly disordered systems where states would otherwise be Anderson localized. By analyzing the inverse participation ratio and the mean center of mass, we map the resulting triple interplay between disorder, non-Hermiticity, and the magnetic field strength, identifying regimes of Anderson localization and skin accumulation. We further reveal that this magnetic field driven delocalization phenomenon originates from an effective suppression of disorder strength via Zeeman-induced inter-chain coupling across the spin sectors.

We study the coaction of cosmological wavefunction coefficients of conformally coupled scalars in FRW background of a two-site example, which turns out to have an elegant diagrammatic interpretation. We show how the coaction acts on the twisted integrals for wavefunction coefficients, decomposing them into contributions associated with subtopologies and cuts, with the subtopologies admitting an interpretation as time-ordered integrals. This provides a clear interpretation of their analytic structure and suggests a broader applicability to more general cosmological diagrams.

Agricultural pricing policies are crucial for farm profitability and food security in India. This study analysed how input and output prices significantly influence the profitability of cereals in Karnataka, with the strategic support prices playing a crucial role in maintaining the price parity. The average annual TFP growth was 1.041 per cent. Rising input costs, particularly for human labour, led to reduced profitability for Jowar (6.12 per cent) and Ragi (4.89 per cent). The net effect was adverse for Jowar (-1.50 per cent) and Ragi (-0.86 per cent) due to rising input costs outpacing output prices. The study recommended increasing the MSP for Jowar (60 per cent) and Ragi (46.24 per cent) above the existing levels. A strategic price adjusted for changing input costs can stabilise farm incomes and promote sustainable production, enabling efficient pricing policies.

In this paper, we provide an overview and evaluation of different types of age assurance technologies (AAT). We describe and analyse 1) different approaches to age assurance online (age verification, age estimation, age inference, and parental control and consent), as well as 2) different age assurance architectures (online, offline device-based, offline credential-based), and assess their various combinations with regards to their respective a) effectiveness, b) side effects, and c) acceptance. We then discuss general limitations of AAT's effectiveness stemming from the possibility of circumvention and outline the most important side effects, in particular regarding privacy and anonymity of all users; bias, discrimination, and exclusion; as well as censorship and related concerns. We conclude our analyses by offering some recommendations on which types of AAT are better or less suited to protect minors online. Guiding our assessment is a weighing of effectiveness against side effects, resulting in a graduated hierarchy of acceptable AAT mechanisms.

Characterizing the surface and atmosphere of Earth-like planets in reflected light is a key goal for upcoming direct imaging surveys. NASA's next flagship-class astrophysics mission concept, the Habitable Worlds Observatory (HWO), is a space-based Ultraviolet/Optical/Near-Infrared observatory with a mission design requirement to reach the $10^{-10}$ contrast necessary to characterize Earth-like planets around Sun-like stars. While reflected light from planetary surfaces provides a unique opportunity to constrain the coverage of surface materials and biopigments, detailed predictions of HWO's ability to retrieve surface fractions are necessary but have not been conducted. Here, we model photon-counting noise from astrophysical, instrumental, and post-processing sources for the HWO Exploratory Analytic Case 5 design equipped with a charge-6 vector-vortex coronagraph. By combining our photon-counting noise with five distinct modern Earth models at quadrature, we simulate single-visit HWO observations and perform spectral retrievals using the open-source code $\texttt{POSEIDON}$ to assess our ability to constrain both the surface and atmospheric composition. We find that degeneracies between planetary radius, surface pressure, surface material, and cloud coverage in reflected-light retrievals can significantly complicate the classification of surface features. These degeneracies can complicate the detection of surface biopigments, such as the chlorophyll-induced red edge on modern Earth. Our work shows that developing concrete strategies for detecting surface features and breaking degeneracies in reflected-light observations of Earth-like planets is a critical priority for mission design and data analysis.

We present joint measurements of the pre- and post-reconstruction power spectra, $P_{\rm pre}$ and $P_{\rm post}$, together with their cross-power spectrum, $P_{\rm cross}$, for the Luminous Red Galaxies (LRGs) in the DESI Data Release 1 (DR1). We jointly analyse these observables with an emulator-based full-shape modeling framework, thereby, for the first time, we extract complementary nonlinear information from the galaxy density field before and after reconstruction in real survey data. Specifically, including $P_{\rm post}$ and $P_{\rm cross}$ in addition to $P_{\rm pre}$ (hereafter $P_{\rm all}$) yields an improvement of approximately $18$-$27\%$ in the $σ_8$ constraint in both $Λ$CDM and $w$CDM, depending on the redshift bin, relative to the $P_{\rm pre}$-only analysis with the cosmic microwave background distance priors (hereafter CMB). In $w$CDM, the joint CMB+$P_{\rm all}$ analysis can tighten the constraints on $w$ by approximately $5$-$15\%$ across the two LRG redshift bins, compared to the CMB+$P_{\rm pre}$ combination. Further incorporating the Type Ia supernova dataset and comparing the cosmological constraints in $w$CDM from each individual power-spectrum component with those from the full combination, we find that $P_{\rm all}$ consistently provides the tightest constraints. From the joint CMB+$P_{\rm all}$+DES-Dovekie dataset, we obtain $Ω_m = 0.314 \pm 0.0048$ and $w = -0.988 \pm 0.023$ for the \texttt{LRG1} sample, and $Ω_m = 0.318 \pm 0.0046$ and $w = -0.988 \pm 0.025$ for \texttt{LRG2}. These results demonstrate that combining pre- and post-reconstruction power spectra with their cross-correlation enables DESI to harvest additional nonlinear information, leading to tighter constraints on cosmological parameters.

Tensor decompositions are a fundamental tool in scientific computing and data analysis. In many applications -- such as simulation data on irregular grids, surrogate modeling for parameterized PDEs, or spectroscopic measurements -- the data has both discrete and continuous structure, and may only be observed at scattered sample points. The CP-HIFI (hybrid infinite-finite) decomposition generalizes the Canonical Polyadic (CP) tensor decomposition to settings where some factors are finite-dimensional vectors and others are functions drawn from infinite-dimensional spaces. The decomposition can be applied to a fully observed tensor (aligned) or, when only scattered observations are available, to a sparsely sampled tensor (unaligned). Current methods compute CP-HIFI factors by solving a sequence of dense linear systems arising from regularized least-squares problems to fit reproducing Kernel Hilbert space (RKHS) representations to the data, but these direct solves become computationally prohibitive as problem size grows. We propose new algorithms that achieve the same accuracy while being orders of magnitude faster. For aligned tensors, we exploit the Kronecker structure of the system to efficiently compute its eigendecomposition without ever forming the full system, reducing the solve to independent scalar equations. For unaligned tensors, we introduce a preconditioned conjugate gradient method, exploiting the problem's structure for fast matrix-vector products and efficient preconditioning. In our experiments, the proposed methods speed up the solution up to 500x compared to the prior naive direct methods, in line with the reduction in the theoretical computational complexity.

In Einstein gravity, matter with an arbitrarily small density can be a black hole. Pressure in the star diverges if size of the star is smaller than 9/8 of the Schwarzschild radius, implying the gravitational collapse into a black hole. By taking quantum effects of matter, however, pressure is bounded from above, and a core with negative energy appears instead. Density of matter increases and eventually reaches the cut-off scale as size of the star approaches the Schwarzschild radius. This result implies that density must be very large as the Planck scale if the star is put inside the Schwarzschild radius.

Network topology matrices are algebraic representations of graphs that are widely used in modeling and analysis of various applications including electrical circuits, communication networks and transportation systems. In this paper, we propose to use Higher-Order-Logic (HOL) based interactive theorem proving to formalize network topology matrices. In particular, we formalize adjacency, degree, Laplacian and incidence matrices in the Isabelle/HOL proof assistant. Our formalization is based on modelling systems as networks using the notion of directed graphs (unweighted and weighted), where nodes act as components of the system and weighted edges capture the interconnection between them. Then, we formally verify various classical properties of these matrices, such as indexing and degree. We also prove the relationships between these matrices in order to provide a comprehensive formal reasoning support for analyzing systems modeled using network topology matrices. To illustrate the effectiveness of the proposed approach, we formally analyze the Kron reduction of the Laplacian matrix and verify the total power dissipation in a generic resistive electrical network, both commonly used in power flow analysis.

Observations with the Canadian Hydrogen Intensity Mapping Experiment (CHIME) have been used to measure the 21 cm intensity mapping auto power spectrum, at $z\sim 1$, over a frequency range from 608.2 MHz to 707.8 MHz at wavenumbers $0.4~h~{\rm Mpc}^{-1} \lesssim k \lesssim 1.5~h~{\rm Mpc}^{-1}$. In this paper, we present the results of two different approaches to interpreting this measurement. In the first approach, we use a parametric power spectrum model to constrain an amplitude parameter, defined as $\mathcal{A}^2_{\rm HI} \equiv 10^6 Ω_{\rm HI}^2(b^2_{\rm HI}+\langle f μ^2\rangle)^2$, where $Ω_{\rm HI}$ is the cosmological density parameter for atomic hydrogen ($\rm HI$), $b_{\rm HI}$ is the linear bias for $\rm HI$, and $\langle f μ^2\rangle$ incorporates the dominant large-scale impact of redshift-space distortions on the angle-averaged power spectrum. Imposing an additional prior on either $Ω_{\rm HI}$ or $b_{\rm HI}$, based on values in the literature, allows us to break the pairwise degeneracy between those two parameters. In the second approach, we compare CHIME's measurement with predictions for the power spectrum of $\rm HI$ from the IllustrisTNG simulations, finding that the measurement disagrees with the TNG100 run at $3.1σ$ and the TNG300 run at $4.0σ$. This disagreement is most likely attributable to the strength of nonlinear redshift-space clustering of $\rm HI$ in the simulations, rather than the total abundance of $\rm HI$, and invites further investigation of the physical processes in the simulations that determine the behavior of $\rm HI$ at nonlinear scales. These results exemplify the ability of 21 cm intensity mapping to provide astrophysical information using measurements at nonlinear scales.

Small, trade-dependent economies often exhibit limited maritime connectivity, yet empirical evidence on the structural configuration of their container systems remains limited. This study analyzes route concentration and node distributions in Mauritania's maritime container system during 2019-2022 using shipment-level data measured in forty-foot equivalent units (FFE). Routes, origin nodes, destination nodes, and industries are represented as FFE-weighted probability distributions, and concentration and divergence metrics are used to assess structural properties. The results show strong corridor concentration across the seven observed routes (HHI = 0.296), with the top three accounting for approximately 84% of total FFE. Node structures differ by direction: imports are associated with a highly concentrated set of destination nodes (HHI = 0.848), while exports originate from only two origin nodes (HHI = 0.567) and are distributed across a large number of destinations (HHI = 0.053). Industry distributions are more concentrated for exports (HHI = 0.352) than for imports (HHI = 0.096), with frozen fish and seafood accounting for more than 53% of export volume. Temporal analysis shows that route concentration remains stable over time (HHI ~ 0.293-0.303), while node distributions exhibit measurable variation, particularly for export destinations (JSD ~ 0.395) and import origins (JSD ~ 0.250).

We define a unified categorical framework for studying six subproblems arising from the classical Four Subspace Problem. For each subproblem, we construct a functor from its associated category to the category of representations of the quiver corresponding to the Four Subspace Problem. This approach gives a common structural setting for the six cases considered and allows a simultaneous and coherent analysis via functorial methods. We prove that the six functors are additive and fully faithful, and we show that none of them is dense. As a consequence, each functor induces an equivalence between the corresponding source category and a well-identified full subcategory of the target category. These equivalences provide an effective mechanism for transferring classification results and structural properties, thereby clarifying the structural interrelations among the categories studied.

As superconducting processors scale, understanding how physical layout shapes qubit interactions is essential for architectural reliability. Existing methods offer limited insight into how electromagnetic design choices translate into execution-level behavior. We present EPAR, an electromagnetic-to-architecture framework that predicts robustness early directly from physical design by reconstructing how design distortion modifies the effective Hamiltonian, reroutes mediated connectivity, and influences control-pulse response. Across all tested layouts, EPAR's structural scores show 100% agreement with two-qubit error trends yet reveal over 10X robustness differences among edges with identical calibrated error rates, going beyond conventional metrics to provide improved and actionable compiler guidance.

The adiabatic equation of state $P \propto n^Γ$ describes the pressure evolution of highly collisional, isotropic plasmas in terms of their density, providing a possible closure of the fluid moment hierarchy in the absence of heat fluxes and dissipation. An analogous closure exists for collisionless, magnetised plasmas, whose pressure tensor is anisotropic with respect to the magnetic field, and the closure is therefore double-adiabatic, prescribing the evolution of the parallel and perpendicular pressures in terms of the magnetic-field strength and density. Here, we present a general first-principle formalism to derive adiabatic laws using the symmetries of the system. With this theory we recover the adiabatic equation of state $P \propto n^Γ$ for isotropic plasmas and the double-adiabatic equations of state for collisionless, magnetised plasmas. We extend the latter to the relativistic regime, finding that their exact functional form depends on the pressure anisotropy and is not a simple power law. Our double-adiabatic equations of state describe simple geometries, like magnetic mirrors or compressed homogeneous plasmas, as well as complex high-energy astrophysical processes, such as the evolution of plasmoid structures formed during magnetic reconnection.

We introduce a generalized Bayesian method for multiple changepoint analysis with a loss function inspired by multinomial logistic regression. The method does not require a specification of the data-generating process and avoids restrictive assumptions on the nature of changepoints. From the joint posterior distribution, we can make simultaneous inference on the locations of changepoints and the coefficients of a multinomial logistic regression model for distinguishing data across homogeneous segments. The multinomial logistic regression coefficients provide a familiar means of interpreting potentially complex changes. To select the number of changepoints, we leverage posterior summaries that measure whether the multinomial logistic classifier can distinguish data from either side of a potential changepoint. To simulate from the generalized posterior distribution, we present a Gibbs sampler based on Pólya-Gamma data augmentation. We assess the accuracy and flexibility of our method through simulation studies featuring different types of changes and demonstrate its interpretability through applications to financial network data and topological data derived from nanoparticle videos.

Lattice systems are effective for modeling heterogeneous materials, but their computational cost is often prohibitive. The QuasiContinuum (QC) method reduces this cost by interpolating the lattice response over a coarse finite-element mesh, yet material interfaces in heterogeneous systems still require fine discretizations. Enrichment strategies from the eXtended Finite Element Method (XFEM) address this by representing interfaces on nonconforming meshes. In this work, we combine Heaviside enrichment with meshless Local Maximum Entropy (LME) interpolation in the QC framework for heterogeneous lattice systems. We systematically investigate the role of the LME locality parameter and its optimization. The results show that optimized LME interpolation improves displacement accuracy by about one order of magnitude over QC with linear interpolation at the same number of degrees of freedom. In addition, the optimal locality-parameter fields are nonuniform near interfaces and exhibit systematic spatial structure. Based on these observations, we derive simple pattern-based rules that retain much of the benefit of full optimization at a fraction of the computational cost. The approach is demonstrated on three numerical examples.

In this paper, we study geometric points in tensor triangular geometry. In doing so, we construct a counter-example to Balmer's Nerves of Steel conjecture using free constructions in higher Zariski geometry. We then go on to introduce and discuss constructible spectra in the context of tensor triangular geometry. For tensor triangulated categories satisfying a mild enhancement condition, we use these spectra to construct geometric incarnations of (homological or triangular) primes via maps to "pointlike" tensor triangulated categories.

The electrification of energy demand across sectors, powered by solar and wind generation, is the best strategy for achieving carbon neutrality. Carbon dioxide removal (CDR) strategies are also expected to play a crucial role by providing net-negative emissions that can offset residual CO2 emissions, including those from cement manufacturing. While previous studies have assessed the role of CDRs in Europe's decarbonisation, most either focus solely on combinations of biogenic point-source capture and direct air capture (DAC) coupled with underground sequestration, or consider multiple CDR strategies at low spatial and temporal resolution, thereby limiting the representation of linkages amongst technologies. In this study, the sector-coupled European energy system model PyPSA-Eur is extended to include afforestation, perennialisation, biochar, and enhanced rock weathering (ERW) as additional CDR strategies. Using this model with a 3-hourly resolution and a network comprising 90 nodes, results show that a climate-neutral energy system equipped with these CDR strategies is 9% less expensive. Afforestation, perennialisation, and ERW potentials are fully utilised across regions, whereas biochar is not selected due to limited solid biomass feedstock being allocated to other higher-value processes. Furthermore, when these CDR strategies are combined with underground sequestration and a continental CO2 transport network, DAC is no longer required to achieve climate neutrality in Europe.

We prove that if $A$ and $B$ are daisy cubes whose $τ$-graphs are forests, then $A$ and $B$ are isomorphic if and only if their $τ$-graphs are isomorphic. The result is applied to show that a daisy cube with at least one edge is the resonance graph of a plane bipartite graph $G$ if and only if its $τ$-graph is a forest which is isomorphic to the inner dual of the subgraph of $G$ obtained by removing all forbidden edges. As a consequence, some well known properties of Fibonacci cubes and Lucas cubes are provided as examples with different proofs.

SHAPR (Solo Human-Centred and AI-Assisted Practice) is a framework for research software development that integrates human-centred decision-making with AI-assisted capabilities. While prior work introduced SHAPR as a conceptual framework, this paper focuses on its operationalisation as a structured, traceable, and knowledge-generating approach to AI-assisted research practice. We present a set of interconnected models describing how research activities are organised through iterative cycles (Explore-Build-Use-Evaluate-Learn), how artefacts evolve through development and use, and how empirical evidence is transformed into conceptual knowledge. Central to this process are Structured Knowledge Units (SKUs), which provide modular and reusable representations of insights derived from practice, supporting knowledge accumulation across cycles. The framework introduces evidence and traceability as a cross-cutting mechanism linking human decisions, AI-assisted development, and artefact evolution to enable transparency, reproducibility, and systematic refinement. SHAPR is also positioned as an AI-executable research framework, as its structured processes and documentation can be interpreted by generative AI systems to guide research workflows. Simultaneously, SHAPR supports a continuum of AI involvement, allowing researchers to balance control, learning, and automation across different contexts. Beyond individual workflows, SHAPR is conceptualised as an integrated research system combining LLM workspaces, development environments, cloud storage, and version control to support scalable, knowledge-centred research practices. Overall, SHAPR provides a practical and theoretically grounded foundation for conducting rigorous, transparent, and reproducible research in AI-assisted environments, contributing to the development of scalable and methodologically sound research practices.

The dynamics of interacting particles in orbital magnetic fields are notoriously difficult to study, as this physics is inherently connected to electronic correlations in two-dimensional systems, for which no straightforward theoretical methods are available. Here, we report on the diffusive relaxation dynamics of two-dimensional interacting fermionic systems under a uniform magnetic field in the infinite temperature regime. We first show that the fermionic truncated Wigner approximation captures the equilibration dynamics unexpectedly well for intermediate interaction strengths when going beyond one dimension. This high accuracy holds at least for relatively small ladder systems, which are accessible to the Lanczos method that we use to benchmark the reliability of the Wigner approximation. We find that strong interactions, which exceed the hopping energy, suppress magnetic-field effects on diffusive transport. However, when the interactions are comparable to the kinetic energy, the diffusion is significantly reduced by the magnetic flux. This is observed for sufficiently large systems (above approximately 400 lattice sites), where finite-size effects weakly affect particle transport. We suggest that our results should be directly accessible on current optical lattice platforms.

This set of lecture notes is an expanded version of a mini-course the author gave in March of 2025 for the program ``Representation Theory \& Noncommutative Geometry" at the Institut Henri Poincaré, Paris. The goal is to provide a survey of the main properties of the theta correspondence over finite fields of odd characteristic, including its compatibility with Harish-Chandra and Lusztig series, and with the Jordan decomposition of representations, as well as its full explicit description.

We study the unconstrained minimization of a smooth and strongly convex population loss function under a stochastic oracle that introduces both additive and multiplicative noise; this is a canonical and widely-studied setting that arises across operations research, signal processing, and machine learning. We begin by showing that standard approaches such as sample average approximation and robust (or averaged) stochastic approximation can lead to suboptimal -- and in some cases arbitrarily poor -- performance with realistic finite sample sizes. In contrast, we demonstrate that a carefully designed variance reduction strategy, which we term VISOR for short, can significantly outperform these approaches while using the same sample size. Our upper bounds are complemented by finite-sample, information-theoretic local minimax lower bounds, which highlight fundamental, instance-dependent factors that govern the performance of any estimator. Taken together, these results demonstrate that an accelerated variant of VISOR is instance-optimal, achieving the best possible sample complexity up to logarithmic factors while also attaining optimal oracle complexity. We apply our theory to generalized linear models and improve upon classical results. In particular, we obtain the best-known non-asymptotic, instance-dependent generalization error bounds for stochastic methods, even in linear regression.

We study linear operators on a finite-dimensional space whose Kippenhahn curves consist of concentric circles centered at the origin. We say that such operators have Circularity property. One class of examples is rotationally invariant operators. To every operator with norm at most one, we associate an infinite sequence of partial isometries and study when Circularity property can be passed back and forth along that sequence. In particular, we introduce a class of operators for which every partial isometry in the aforementioned sequence has Circularity property, and show that this class is broader than the class of rotationally invariant operators. As a consequence, every such an operator provides a counterexample to the Gau--Wang--Wu conjecture about nilpotent partial isometries. We also discuss possible refinements of the conjecture. Finally, we propose a way to check whether a matrix is rotationally invariant, suitable for numerical experiments.

As a unique probe for geophysical research, geoneutrinos can reveal the distribution of internal heat sources in the Earth by detecting electron antineutrinos produced by the radioactive decay of $^{238}$U, $^{232}$Th, and $^{40}$K. However, commercial nuclear power plants continuously produce the same type of electron antineutrinos, which constitute a primary background difficult to eliminate in geoneutrino experiments. As geoneutrino measurements and reactor background modeling approach sub-percent precision, even small matter-induced corrections to reactor antineutrino propagation require quantitative assessment. In this paper, we develop a high-precision prediction framework for reactor neutrino fluxes at underground labs, using global reactor operating data, reactor-to-detector distances, and matter effects (MSW) on neutrino propagation through the Earth. To solve the three-flavor MSW evolution efficiently, we implement a second-order Strang-splitting solver in the vacuum mass basis. Within this framework, we have calculated the reactor neutrino oscillation probabilities, including the MSW effect under one-dimensional (spherically symmetric) and three-dimensional (including lateral inhomogeneities) Earth models, and compared them with the vacuum oscillation scenario, to assess the impact of Earth's structural features on the accuracy of reactor neutrino flux predictions.