You may have seen the words "topological recursion" mentioned in papers on matrix models, Hurwitz theory, Gromov-Witten theory, topological string theory, knot theory, topological field theory, JT gravity, cohomological field theory, free probability theory, gauge theories, to name a few. The goal of these lecture notes is certainly not to explain all these applications of the topological recursion framework. Rather, the intention is to provide a down-to-earth (and hopefully accessible) introduction to topological recursion itself, so that when you see these words mentioned, you can understand what it is all about. These lecture notes accompanied a series of lectures at the Les Houches school "Quantum Geometry (Mathematical Methods for Gravity, Gauge Theories and Non-Perturbative Physics)" in Summer 2024.

Given a continuous linear cocycle A over a homeomorphism f of a compact metric space X, we investigate its set R of Lyapunov-Perron regular points, that is, the collection of trajectories of f that obey the conclusions of the Multiplicative Ergodic Theorem. We obtain results roughly saying that the set R is of first Baire category (i.e., meager) in X, unless some rigid structure is present. In some settings, this rigid structure forces the Lyapunov exponents to be defined everywhere and to be independent of the point; that is what we call complete regularity.

Context. To date, three nuclear transients have been associated with high-energy neutrino events. These transients are generally thought to be powered by tidal disruptions of stars (TDEs) by massive black holes. However, AT2019aalc, hosted in a Seyfert-1 galaxy, was not yet classified due to a lack of multiwavelength observations. Interestingly, the source has re-brightened 4 years after its discovery. Aims. Our aim is to constrain the physical mechanism responsible for the second optical flare, which may also provide clues to the origin of the initial event. Methods. We conducted a multi-wavelength monitoring program (from radio to X-rays) of AT2019aalc during its re-brightening in 2023/2024. Results. The observations revealed multiple X-ray flares during the second optical flaring episode of the transient and a uniquely bright UV counterpart. The second flare, similarly to the first one, is also accompanied by IR dust echo emission. A long-term radio flare is found with an inverted spectrum. Optical spectroscopic observations reveal the presence of Bowen Fluorescence lines and strong high-ionization coronal lines indicating an extreme level of ionization in the system. Conslusions. The results suggest that the transient can be classified as a Bowen Fluorescence Flare (BFF), a relatively new sub-class of flaring active galactic nuclei (AGN). AT2019aalc can be also classified as an extreme coronal line emitter (ECLE). We found that, in addition to AT2019aalc, another BFF AT2021loi is spatially coincident with a high-energy neutrino event. We propose a repeating TDE scenario within an AGN framework to explain the multi-wavelength properties of AT2019aalc and suggest a possible connection among ECLEs, BFFs, and TDEs occurring in AGNs.

The CPT theorem states that a unitary and Lorentz-invariant theory must also be invariant under a discrete symmetry $\mathbf{CRT}$ which reverses charge, time, and one spatial direction. In this article, we study a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry group, in which two of the nontrivial symmetries (``Reflection Reality'' and a 180 degree rotation) are implied by Unitarity and Lorentz Invariance respectively, while the third is $\mathbf{CRT}$. (In cosmology, Scale Invariance plays the role of Lorentz Invariance.) This naturally leads to converses of the CPT theorem, as any two of the discrete $\mathbb{Z}_2$ symmetries will imply the third one. Furthermore, in many field theories, the Reflection Reality $\mathbb{Z}_2$ symmetry is actually sufficient to imply the theory is fully unitary, over a generic range of couplings. Building upon previous work on the Cosmological Optical Theorem, we derive non-perturbative reality conditions associated with bulk Reflection Reality (in all flat FLRW models) and $\mathbf{CRT}$ (in de Sitter spacetime), in arbitrary dimensions. Remarkably, this $\mathbf{CRT}$ constraint suffices to fix the phase of all wavefunction coefficients at future infinity (up to a real sign) -- without requiring any analytic continuation, or comparison to past infinity -- although extra care is required in cases where the bulk theory has logarithmic UV or IR divergences. This result has significant implications for de Sitter holography, as it allows us to determine the phases of arbitrary $n$-point functions in the dual CFT.

Context: In the context of Galactic archaeology, the outer halo remains relatively unexplored with respect to its metallicity distribution, merger debris, and the abundance of known very/extremely metal-poor ([Fe/H]<-2.5) stars. Aim: We utilize the Pristine survey's publicly available, Pristine data release 1 (PDR1) and Pristine-Gaia synthetic (PGS) catalogues of photometric metallicities, to select Red Giant Branch (RGB) stars in the outer halo. Methods: The RGB selection pipeline selects giants based on the absence of a well-measured parallax in the brightness range where dwarfs have reasonable parallax estimate from Gaia DR3 data. The photometric distances are calculated using a BaSTI-isochrone fitting code and the photometric metallicities. Results: Photometric distances derived from PDR1- and PGS-giants show typical uncertainties of 12% and a scatter of up to 20% and 40% respectively, when validated against inverted-parallax and Starhorse-code distances. The PDR1-giants catalogue provides a low-to-no bias view of the metallicity structure versus distance compared to the PGS-giants catalogue (with a distance-metallicity selection bias), while the PGS-giants catalogue provides an all-sky view of the outer halo, especially in the VMP end. The PDR1-giants catalogue is used to study the metallicity distribution function of the halo out to ~100 kpc, using Gaussian Mixture Model decomposition. Additionally, we use the PDR1-giants with radial velocities from spectroscopic surveys to map the metallicity view of the integrals-of-motion space where accreted dwarf galaxy debris conserve their orbital parameters for a long time. Using the PGS-giants catalogue, we associate 41 stars tentatively to the stellar counterpart of the Magellanic stream in the VMP end, out to 70 kpc. Conclusions: We publish two RGB catalogues between -4.0<[Fe/H]<+0.1 with reliable photometric distances and metallicities.

Rational functions are exceptionally powerful tools in scientific computing, yet their abilities to advance quantum algorithms remain largely untapped. In this paper, we introduce effective implementations of rational transformations of a target operator on quantum hardware. By leveraging suitable integral representations of the operator resolvent, we show that rational transformations can be performed efficiently with Hamiltonian simulations using a linear-combination-of-unitaries (LCU). We formulate two complementary LCU approaches, discrete-time and continuous-time LCU, each providing unique strategies to decomposing the exact integral representations of a resolvent. We consider quantum rational transformation for the ubiquitous task of approximating functions of a Hermitian operator, with particular emphasis on the elementary signum function. For illustration, we discuss its application to the ground and excited state problems. Combining rational transformations with observable dynamic mode decomposition (ODMD), our recently developed noise-resilient quantum eigensolver, we design a fully real-time approach for resolving many-body spectra. Our numerical demonstration on spin systems indicates that our real-time framework is compact and achieves accurate estimation of the low-lying energies.

Countable $\mathcal{L}$-structures $\mathcal{N}$ whose isomorphism class supports a permutation invariant probability measure in the logic action have been characterized by Ackerman-Freer-Patel to be precisely those $\mathcal{N}$ which have no algebraicity. Here we characterize those countable $\mathcal{L}$-structure $\mathcal{N}$ whose isomorphism class supports a quasi-invariant probability measure. These turn out to be precisely those $\mathcal{N}$ which are not "highly algebraic" -- we say that $\mathcal{N}$ is highly algebraic if outside of every finite $F$ there is some $b$ and a tuple $\bar{a}$ disjoint from $b$ so that $b$ has a finite orbit under the pointwise stabilizer of $\bar{a}$ in $\mathrm{Aut}(\mathcal{N})$. As a bi-product of our proof we show that whenever the isomorphism class of $\mathcal{N}$ admits a quasi-invariant measure, then it admits one with continuous Radon--Nikodym cocycles.

Most vulnerability detection studies focus on datasets of vulnerabilities in C/C++ code, offering limited language diversity. Thus, the effectiveness of deep learning methods, including large language models (LLMs), in detecting software vulnerabilities beyond these languages is still largely unexplored. In this paper, we evaluate the effectiveness of LLMs in detecting and classifying Common Weakness Enumerations (CWE) using different prompt and role strategies. Our experimental study targets six state-of-the-art pre-trained LLMs (GPT-3.5- Turbo, GPT-4 Turbo, GPT-4o, CodeLLama-7B, CodeLLama- 13B, and Gemini 1.5 Pro) and five programming languages: Python, C, C++, Java, and JavaScript. We compiled a multi-language vulnerability dataset from different sources, to ensure representativeness. Our results showed that GPT-4o achieves the highest vulnerability detection and CWE classification scores using a few-shot setting. Aside from the quantitative results of our study, we developed a library called CODEGUARDIAN integrated with VSCode which enables developers to perform LLM-assisted real-time vulnerability analysis in real-world security scenarios. We have evaluated CODEGUARDIAN with a user study involving 22 developers from the industry. Our study showed that, by using CODEGUARDIAN, developers are more accurate and faster at detecting vulnerabilities.

In evidence synthesis, multilevel modeling approaches (MMAs) are commonly employed to combine aggregate data (AD) and individual participant data (IPD). These approaches rely on an aggregate outcome model that is ideally obtained by integrating the prespecified individual- level outcome model over the covariate distribution observed in each eligible study. In non- linear settings, such an integration may however be analytically intractable and requires ap- proximations. In this paper, we propose a novel method for incorporating AD into causal meta-analysis of IPD studies that can overcome this challenge. Rather than relying on an ag- gregate outcome model that is difficult to be correctly formulated, we propose modeling the trial membership as a function of baseline covariates. This model allows one to estimate the individual-level outcome model in each AD study by leveraging IPD available in other trials, and then to transport the treatment effects estimated from both AD and IPD trials to an external target population, even when only aggregate covariate data are available for that population. Unlike previous proposals, we do not require pseudo-IPD to be generated from the aggregate data, which helps minimize bias due to incomplete information on the covariate distribution in each AD trial and in the target population.

We study fibrations of the projective model for the symmetric space associated with $\text{SL}(2n,\mathbb{R})$ by codimension $2$ projective subspaces, or pencils of quadrics. In particular we show that if such a smooth fibration is equivariant with respect to a representation of a closed surface group, the representation is quasi-isometrically embedded, and even Anosov if the pencils in the image contain only non-degenerate quadrics. We use this to characterize maximal representations among representations of a closed surface group into $\text{Sp}(2n,\mathbb{R})$ by the existence of an equivariant continuous fibration of the associated symmetric space, satisfying an additional technical property. These fibrations extend to fibrations of the projective structures associated to maximal representations by bases of pencils of quadrics.

We propose a detector concept, LAYered CAvern Surface Tracker (LAYCAST), to be installed on the ceiling and the wall of the cavern hosting the main experiment of future electron-positron colliders such as CEPC and FCC-ee. With detailed and realistic considerations of the design of such a new experiment, the proposed detector is dedicated to extending the sensitivity reach of the main detector to various theoretical scenarios of long-lived particles (LLPs). We study carefully four such scenarios involving a light scalar boson $X$, the heavy neutral lepton $N$, the lightest neutralino $\tildeχ^0_1$ in the R-parity-violating supersymmetry, and the axion-like particle $a$. Long-lived light scalar bosons are considered to be produced from the Standard-Model (SM) Higgs boson's decay ($h \to X X$) at the center-of-mass energy $\sqrt{s} =$ 240 GeV, while the other three types of LLPs are produced either from $Z$-boson decays (viz. $Z \to ν\, N, ~\tildeχ^0_1\, \tildeχ^0_1 $) or direct scattering process ($ e^- e^+ \to ~γ\, a$) at $\sqrt{s} =$ 91.2 GeV, where $γ$ and $ν$ denote the SM photon and neutrino, respectively. With Monte-Carlo simulations, we derive the sensitivities of the proposed experiment to these LLPs and the corresponding signal-event numbers. We also provide a dedicated estimate of a potentially important SM background from long-lived neutral kaons in hadronic $Z$ decays, and show that it is strongly suppressed by the combined requirements of the main detector and LAYCAST. Our findings show that LAYCAST can probe large new parameter space beyond both current bounds and the expected reach of the main experiments at CEPC and FCC-ee. Comparison with existing works in similar directions is also made.

There has been considerable effort to mimic analog black holes and wormholes in solid state systems. Lattice realizations in particular present specific challenges. One of those is that event horizons in general have both white and black hole (grey hole) character, a feature guaranteed by the Nielsen-Ninomiya theorem. We here explore and extend the capability of superconducting circuit hardware to implement on-demand spacetime geometries on lattices, combining nonreciprocity of gyrators with the nonlinearity of Josephson junctions. We demonstrate the possibility of the metric sharply changing within a single lattice point, thus entering a regime where the modulation of system parameters is "trans-Planckian", and the Hawking temperature ill-defined. Instead of regular Hawking radiation, we find an instability in the form of an exponential burst of charge and phase quantum fluctuations over short time scales - a robust signature even in the presence of an environment. Moreover, we present a loop-hole for the typical black/white hole ambiguity in lattice simulations: exceptional points in the dispersion relation allow for the creation of pure black (or white) hole horizons, at the expense of a radical change in the dynamics of the wormhole interior.

Competition during range expansions is of great interest from both practical and theoretical view points. Experimentally, range expansions are often studied in homogeneous Petri dishes, which lack spatial anisotropy that might be present in realistic populations. Here, we analyze a model of anisotropic growth, based on coupled Kardar-Parisi-Zhang and Fisher-Kolmogorov-Petrovsky-Piskunov equations that describe surface growth and lateral competition. Compared to a previous study of isotropic growth, anisotropy relaxes a constraint between parameters of the model. We completely characterize spatial patterns and invasion velocities in this generalized model. In particular, we find that strong anisotropy results in a distinct morphology of spatial invasion with a kink in the displaced strain ahead of the boundary between the strains. This morphology of the out-competed strain is similar to a shock wave and serves as a signature of anisotropic growth.

We compute the leading order asymptotic of the maximum of the characteristic polynomial for i.i.d. matrices with real or complex entries. In particular, this result is new even for real Ginibre matrices, which was left as an open problem in [arXiv:2303.09912]; the complex Ginibre case was covered in [arXiv:1902.01983]. These are the first universality results for the non--Hermitian analog of the first order term of the Fyodorov--Hiary--Keating conjecture. Our methods are based on constructing a coupling to the branching random walk via Dyson Brownian motion. In particular, we find a new connection between real i.i.d. matrices and inhomogeneous branching random walk.

Order and disorder constitute two fundamental and opposite themes in condensed matter physics and materials science. Crystals are considered the epitome of order, characterised by long-range translational order. The discovery of quasicrystals, which exhibit rotational symmetries forbidden in crystals and lack periodicity, led to a paradigm shift in solid-state physics. Moving one step forward, it is intriguing to ask whether ordered matter can exist without apparent symmetry breaking. The same question arises considering how ordered amorphous (noncrystalline) solids can be structured. Here, we present the discovery of ideal noncrystals in two dimensions, which are disordered in the conventional sense, lacking Bragg peaks, but exhibit high orderliness based on the steric order, i.e., they are ideally packed. A path-integral-like scheme reveals the underlying long-range structural correlation. We find that these ideal noncrystals are characterised by phononic vibrational modes following the Debye law, fully affine elastic responses, and suppressed density fluctuations at longer wavelengths, reminiscent of hyperuniformity -- all characteristics typically associated with crystals. Therefore, ideal noncrystals represent a peculiar form of matter with a mixed nature -- noncrystalline yet possessing crystal-like properties. Notably, these states are found to be thermodynamically favourable, indicating them as a possible new class of ordered matter without apparent symmetry breaking. Our findings significantly broaden the conceptualization of ordered states of matter and may contribute to a deeper understanding of entropy-driven ordering, particularly in generic amorphous materials.

We present the first results of the search for sub-MeV fermionic dark matter absorbed by electron targets of germanium using the 205.4~kg$\cdot$day data collected by the CDEX-10 experiment, with the analysis threshold of 160~eVee. No significant dark matter (DM) signals over the background are observed. Results are presented as limits on the cross section of DM--electron interaction. We present new constraints of cross section in the DM range of 0.1--10 keV/$c^2$ for vector and axial-vector interaction. The upper limit on the cross section is set to be $\rm 6.8\times10^{-46}~cm^2$ for vector interaction, and $\rm 2.3\times10^{-46}~cm^2$ for axial-vector interaction at DM mass of 5 keV/$c^2$.

In this article, we investigate the strongly robust property of toric ideals associated with weighted oriented graphs. We establish that the toric ideals of a broad class of monomial ideals are strongly robust; this class encompasses the edge ideals of weighted oriented graphs in which every edge is incident to a vertex of degree $2$.

Regularization of control policies using entropy can be instrumental in adjusting predictability of real-world systems. Applications benefiting from such approaches range from, e.g., cybersecurity, which aims at maximal unpredictability, to human-robot interaction, where predictable behavior is highly desirable. In this paper, we consider entropy regularization for interval Markov decision processes (IMDPs). IMDPs are uncertain MDPs, where transition probabilities are only known to belong to intervals. Lately, IMDPs have gained significant popularity in the context of abstracting stochastic systems for control design. In this work, we address robust minimization of the linear combination of entropy and a standard cumulative cost in IMDPs, thereby establishing a trade-off between optimality and predictability. We show that optimal deterministic policies exist, and devise a value-iteration algorithm to compute them. The algorithm solves a number of convex programs at each step. Finally, through an illustrative example we show the benefits of penalizing entropy in IMDPs.

For connected reductive groups together with a Frobenius root $F$, we show that the cohomology of the structure sheaf and respectively the canonical sheaf for compactified Deligne--Lusztig varieties associated to an element in the free monoid generated by the simple reflections is isomorphic to that of a minimal length element in an $F$-conjugacy class in the Weyl group.

Projected Dynamical Systems (PDSs) form a class of discontinuous constrained dynamical systems, and have been used widely to solve optimization problems and variational inequalities. Recently, they have also gained significant attention for control purposes, such as high-performance integrators, saturated control and feedback optimization. In this work, we establish that locally Lipschitz continuous dynamics, involving Control Barrier Functions (CBFs), namely CBF-based dynamics, approximate PDSs. Specifically, we prove that trajectories of CBF-based dynamics uniformly converge to trajectories of PDSs, as a CBF-parameter approaches infinity. Towards this, we also prove that CBF-based dynamics are perturbations of PDSs, with quantitative bounds on the perturbation. Our results pave the way to implement discontinuous PDS-based controllers in a continuous fashion, employing CBFs. We demonstrate this on numerical examples on feedback optimization and synchronverter control. Moreover, our results can be employed to numerically simulate PDSs, overcoming disadvantages of existing discretization schemes, such as computing projections to possibly non-convex sets. Finally, this bridge between CBFs and PDSs may yield other potential benefits, including novel insights on stability.

Context. The presence of dark matter in neutron stars is of growing interest due to its potential impact on the structure and observable properties of these objects. Among the various candidates, the hypothetical sexaquark has emerged as a promising bosonic dark matter particle, potentially forming under extreme conditions in neutron star cores. Aims. We investigate whether a hybrid neutron star model that includes hyperons, bosonic dark matter (in the form of sexaquarks), and deconfined quark matter can satisfy all current observational constraints. We particularly focus on identifying the range of sexaquark masses consistent with mass-radius measurements and the tidal deformability limit. Methods. We used the DD2Y-T model for the hadronic phase, which includes hyperons, and a nonlocal Nambu-Jona-Lasinio model for the deconfined quark phase. The phase transition was modeled as a smooth crossover using the replacement interpolation construction method. Sexaquark-baryon interactions were introduced via an effective mass shift representing repulsion. We incorporated the full set of current observational data, including NICER measurements of PSRs J0437-4715 and newly published J0614-3329 data, and performed a Bayesian analysis to constrain the sexaquark mass. Results. Our results show that the presence of the sexaquark softens the equation of state, enabling the hybrid model to satisfy both the radius and tidal deformability constraints around the canonical 1.4 M_\odot neutron stars. We find that hybrid EOSs with a sexaquark mass around 1900 MeV are in agreement with all available constraints, including those from HESS J1731-347 and PSR J0952-0607, which represent the lowest and highest mass neutron stars observed to date. The Bayesian analysis favors a sexaquark mass range of 1885-1935 MeV, supporting the potential relevance of this exotic particle in neutron star interiors.

Quantum Machine Learning (QML) has emerged as a promising intersection of quantum computing and classical machine learning, anticipated to drive breakthroughs in computational tasks. This paper discusses the question which security concerns and strengths are connected to QML by means of a systematic literature review. We categorize and review the security of QML models, their vulnerabilities inherent to quantum architectures, and the mitigation strategies proposed. The survey reveals that while QML possesses unique strengths, it also introduces novel attack vectors not seen in classical systems. We point out specific risks, such as cross-talk in superconducting systems and forced repeated shuttle operations in ion-trap systems, which threaten QML's reliability. However, approaches like adversarial training, quantum noise exploitation, and quantum differential privacy have shown potential in enhancing QML robustness. Our review discuss the need for continued and rigorous research to ensure the secure deployment of QML in real-world applications. This work serves as a foundational reference for researchers and practitioners aiming to navigate the security aspects of QML.

An Event-Related Potential (ERP)-based Brain-Computer Interface (BCI) Speller System assists people with disabilities to communicate by decoding electroencephalogram (EEG) signals. A P300-ERP embedded in EEG signals arises in response to a rare, but relevant event (target) among a series of irrelevant events (non-target). Different machine learning methods have constructed binary classifiers to detect target events, known as calibration. The existing calibration strategy uses data from participants themselves with lengthy training time. Participants feel bored and distracted, which causes biased P300 estimation and decreased prediction accuracy. To resolve this issue, we propose a Bayesian signal matching (BSM) framework to calibrate EEG signals from a new participant using data from source participants. BSM specifies the joint distribution of stimulus-specific EEG signals among source participants via a Bayesian hierarchical mixture model. We apply the inference strategy. If source and new participants are similar, they share the same set of model parameters; otherwise, they keep their own sets of model parameters; we predict on the testing data using parameters of the baseline cluster directly. Our hierarchical framework can be generalized to other base classifiers with parametric forms. We demonstrate the advantages of BSM using simulations and focus on the real data analysis among participants with neuro-degenerative diseases.

This work proposes an experimental framework to characterize a cohesive law for mode-II interlaminar fracture and demonstrates its implementation. For a size effect study, geometrically scaled end-notched flexure specimens were tested using microscopic and macroscopic digital image correlation (DIC) systems. The fracture energy was characterized using a compliance calibration method and Bažant's type-II size effect law for comparison. In the proposed experimental framework, the DIC data were post-processed using three steps: coordinate transformation, curve fitting, and through-thickness deformation analysis. Different magnitudes of separation values were measured from different sizes at fracture loads, implying size effect and partial development of cohesive laws. Modeling and simulations were intended to validate the proposed method and demonstrate the utilization of the experimental data. Additionally, challenges related to finding a single cohesive law for geometrically scaled specimens of a single material were exposed. A single cohesive law for the scaled specimens was developed and proposed as a material property of the specimen material. The fracture energy of the single law was smaller than the energy obtained from the size effect analysis, while the sizes of fracture process zones at fracture loads were smaller than the experimental measurements. However, the global fracture behaviors of the models showed good agreement with the experimental data of the mid-size specimen while showing reasonable agreement with the other sizes. Furthermore, the single law successfully captured local fracture behaviors by showing partial cohesive zone development at the fracture loads and matching the microscopic measurement of the separation values.

The co-evolution of galaxies and supermassive black holes (SMBHs) underpins our understanding of galaxy evolution, but different methods to measure SMBH masses have only infrequently been cross-checked. We attempt to identify targets to cross-check two of the most accurate methods, megamaser and cold molecular gas dynamics. Three promising galaxies are selected from all those with existing megamaser SMBH mass measurements. We present Atacama Large Millimeter/sub-millimeter Array (ALMA) 12CO(2-1) and 230-GHz continuum observations with angular resolutions of about 0.5". Every galaxy has an extended rotating molecular gas disc and 230-GHz continuum source(s), but all also have irregularities and/or non-axisymmetric features: NGC1194 is highly inclined and has disturbed and lopsided central 12CO(2-1) emission; NGC3393 has a nuclear disc with fairly regular but patchy 12CO(2-1) emission with little gas near the kinematic major axis, faint emission in the very centre and two brighter structures reminiscent of a nuclear ring and/or spiral; NGC5765B has a strong bar and very bright 12CO(2-1) emission concentrated along two bisymmetric offset dust lanes and two bisymmetric nuclear spiral arms. 12CO(2-1) and 12CO(3-2) observations with the James Clerk Maxwell Telescope are compared with the ALMA observations. Because of the disturbed gas kinematics and the impractically long integration times required for higher angular resolution observations, none of the three galaxies is suitable for a future SMBH mass measurement. Nonetheless, increasing the number of molecular gas observations of megamaser galaxies is valuable, and the ubiquitous disturbances suggest a link between large-scale gas properties and the existence of megamasers.

Recently, G. Navarro introduced a new conjecture that unifies the Alperin Weight Conjecture and the Glauberman correspondence into a single statement. In this paper, we reduce this problem to simple groups and prove it for several classes of groups and blocks. Our reduction can be divided into two steps. First, we show that assuming the so-called Inductive (Blockwise) Alperin Weight Condition for finite simple groups, we obtain an analogous statement for arbitrary finite groups, that is, an automorphism-equivariant version of the Alperin Weight Conjecture inducing isomorphisms of modular character triples. Then, we show that the latter implies Navarro's conjecture for each finite group.

We study the $L^2$-gradient flows, $\partial_t u-\mathrm{div}(\mathrm{D}f(x,\mathbb{A}u))=0$, of functionals of the type $\int_Ωf(x,\mathbb{A}u)\,\mathrm{d}x$, where $f$ is a convex function of linear growth and $\mathbb{A}$ is some first-order linear constant-coefficient differential operator. To this end, we identify the relaxation of the functional to the space $\mathrm{BV}^{\mathbb{A}}\cap L^2$, identify its subdifferential, and show pointwise representation formulas for the relaxation and the subdifferential, both with and without Dirichlet boundary conditions. The existence and uniqueness then follow from abstract semigroup theory. We further show that our solutions can be obtained as limits of the corresponding flows with $p$-growth as $p\searrow 1$.

We prove the existence of "murmurations" in the family of holomorphic modular forms of level $1$ and weight $k\to\infty$, that is, correlations between their root numbers and Hecke eigenvalues at primes growing in proportion to the analytic conductor. This is the first demonstration of murmurations in an archimedean family.

We introduce semiframes (an algebraic structure) and investigate their duality with semitopologies (a topological one). Both semitopologies and semiframes are relatively recent developments, arising from a novel application of topological ideas to study decentralised computing systems. Semitopologies generalise topology by removing the condition that intersections of open sets are necessarily open. The motivation comes from identifying the notion of an actionable coalition in a distributed system -- a set of participants with sufficient resources for its members to collaborate to take some action -- with open set; since just because two sets are actionable (have the resources to act) does not necessarily mean that their intersection is. We define notions of category and morphism and prove a categorical duality between (sober) semiframes and (spatial) semitopologies, and we investigate how key well-behavedness properties that are relevant to understanding decentralised systems, transfer (or do not transfer) across the duality.

In this paper, we study the relationship between systems controlled via Control Barrier Function (CBF) approaches and a class of discontinuous dynamical systems, called Projected Dynamical Systems (PDSs). In particular, under appropriate assumptions, we show that the vector field of CBF-controlled systems is a Krasovskii-like perturbation of the set-valued map of a differential inclusion, that abstracts PDSs. This result provides a novel perspective to analyze and design CBF-based controllers. Specifically, we show how, in certain cases, it can be employed for designing CBF-based controllers that, while imposing safety, preserve asymptotic stability and do not introduce undesired equilibria or limit cycles. Finally, we briefly discuss about how it enables continuous implementations of certain projection-based controllers, that are gaining increasing popularity.