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Luigi Lombardi

Publications and source records attributed to Luigi Lombardi.

At least 19 recordsLinked to original sources
arXiv2025

Aspherical Complex Surfaces, the Singer Conjecture, and Gromov-Lück Inequality $χ\geq |σ|$

We discuss the Singer conjecture and Gromov-Lück inequality $χ\geq |σ|$ for aspherical complex surfaces. We give a proof of the Singer conjecture for aspherical complex surface with residually finite fundamental group that does not rely on Gromov's Kähler groups theory. Without the residually finiteness assumption, we observe that this conjecture can be proven for all aspherical complex surfaces except possibly those in Class $\mathrm{VII}_0^+$ (a positive answer to the global spherical shell conjecture would rule out the existence of aspherical surfaces in this class). We also sharpen Gromov's inequality for aspherical complex surfaces that are not in Class $\mathrm{VII}_0^+$. This is achieved by connecting the circle of ideas of the Singer conjecture with the study of Reid's conjecture.

math.DG↗
arXiv2025

Reconstruction theorems in the supported case

We show that any equivalence of bounded derived categories of coherent sheaves on a smooth projective complex variety supported in a closed algebraic subset preserves the dimension of the support in two cases: (i) the restriction of the (anti)canonical bundle to the support is ample; (ii) the supports are irreducible and the equivalence sends a skyscraper sheaf of a closed point to a skyscraper sheaf of a closed point. Moreover, in the first case the equivalence recovers the set of closed points of the support up to homeomorphism.

math.AG↗
arXiv2025

A network psychometric analysis of maths anxiety factors in Italian psychology students

Dealing with mathematics can induce significant anxiety, strongly affecting psychology students' academic performance and career prospects. This phenomenon is known as maths anxiety and several scales can measure it. Most scales were created in English and abbreviated versions were translated and validated among Italian populations (e.g. Abbreviated Maths Anxiety Scale). This study translated the 3-factor MAS-UK scale in Italian to produce a new tool, MAS-IT, validated specifically in a sample of Italian undergraduates enrolled in psychology or related BSc programmes. A sample of 324 Italian undergraduates completed the MAS-IT. The data were analysed using confirmatory Factor Analysis (CFA), testing the original MAS-UK 3-factor model. CFA results revealed that the original MAS-UK 3-factor model did not fit the Italian data. A subsequent Exploratory Graph Analysis (EGA) identified 4 distinct components/factors of maths anxiety detected by MAS-IT. The items relative to "Passive Observation maths anxiety" factor remained stable across the analyses, whereas "Evaluation maths anxiety" and "Everyday/Social maths anxiety" items showed a reduced or poor item stability. Quantitative findings indicated potential cultural or contextual differences in the expression of maths anxiety in today's psychology undergraduates, underlining the need for more appropriate tools to be used among psychology students.

stat.AP↗
arXiv2025

Irregular fibrations of derived equivalent varieties

We study the behavior of irregular fibrations of a variety under derived equivalence of its bounded derived category. In particular we prove the derived invariance of the existence of an irregular fibration over a variety of general type, extending the case of irrational pencils onto curves of genus $g\geq 2$. We also prove that a derived equivalence of such fibrations induces a derived equivalence between their general fibers.

math.AG↗
arXiv2024

A novel CFA+EFA model to detect aberrant respondents

Aberrant respondents are common but yet extremely detrimental to the quality of social surveys or questionnaires. Recently, factor mixture models have been employed to identify individuals providing deceptive or careless responses. We propose a comprehensive factor mixture model for continuous outcomes that combines confirmatory and exploratory factor models to classify both the non-aberrant and aberrant respondents. The flexibility of the proposed {classification model} allows for the identification of two of the most common aberrant response styles, namely faking and careless responding. We validated our approach by means of two simulations and two case studies. The results indicate the effectiveness of the proposed model in dealing with aberrant responses in social and behavioural surveys.

stat.ME↗
arXiv2023

Singular hermitian metrics and the decomposition theorem of Catanese, Fujita, and Kawamata

We prove that a torsion-free sheaf $\mathcal F$ endowed with a singular hermitian metric with semi-positive curvature and satisfying the minimal extension property admits a direct-sum decomposition $\mathcal F \simeq \mathcal U \oplus \mathcal A$ where $\mathcal U$ is a hermitian flat bundle and $\mathcal A$ is a generically ample sheaf. The result applies to the case of direct images of relative pluricanonical bundles $f_* ω_{X/Y}^{\otimes m}$ under a surjective morphism $f\colon X \to Y$ of smooth projective varieties with $m\geq 2$. This extends previous results of Fujita, Catanese--Kawamata, and Iwai.

math.AG↗
arXiv2023

Derived equivalence and fibrations over curves and surfaces

We prove that the bounded derived category of coherent sheaves on a smooth projective complex variety reconstructs the isomorphism classes of fibrations onto smooth projective curves of genus $g\geq 2$. Moreover, in dimension at most four, we prove that the same category reconstructs the isomorphism classes of fibrations onto normal projective surfaces with positive holomorphic Euler characteristic and admitting a finite morphism to an abelian variety. Finally, we study the derived invariance of a class of fibrations with minimal base-dimension under the condition that all the Hodge numbers of type $h^{0,p}(X)$ are derived invariant.

math.AG↗
arXiv2023

L^2-Betti Numbers and Convergence of Normalized Hodge Numbers via the Weak Generic Nakano Vanishing Theorem

We study the rate of growth of normalized Hodge numbers along a tower of abelian covers of a smooth projective variety with semismall Albanese map. These bounds are in some cases optimal. Moreover, we compute the $L^2$-Betti numbers of irregular varieties that satisfy the weak generic Nakano vanishing theorem e.g., varieties with semismall Albanese map). Finally, we study the convergence of normalized plurigenera along towers of abelian covers of any irregular variety. As an application, we extend a result of Kollár concerning the multiplicativity of higher plurigenera of a smooth projective variety of general type, to a wider class of varieties. In the Appendix, we study irregular varieties for which the first Betti number diverges along a tower of abelian covers induced by the Albanese variety.

math.AG↗
arXiv2023

Cognitive network science reveals bias in GPT-3, ChatGPT, and GPT-4 mirroring math anxiety in high-school students

Large language models are becoming increasingly integrated into our lives. Hence, it is important to understand the biases present in their outputs in order to avoid perpetuating harmful stereotypes, which originate in our own flawed ways of thinking. This challenge requires developing new benchmarks and methods for quantifying affective and semantic bias, keeping in mind that LLMs act as psycho-social mirrors that reflect the views and tendencies that are prevalent in society. One such tendency that has harmful negative effects is the global phenomenon of anxiety toward math and STEM subjects. Here, we investigate perceptions of math and STEM fields provided by cutting-edge language models, namely GPT-3, Chat-GPT, and GPT-4, by applying an approach from network science and cognitive psychology. Specifically, we use behavioral forma mentis networks (BFMNs) to understand how these LLMs frame math and STEM disciplines in relation to other concepts. We use data obtained by probing the three LLMs in a language generation task that has previously been applied to humans. Our findings indicate that LLMs have an overall negative perception of math and STEM fields, with math being perceived most negatively. We observe significant differences across the three LLMs. We observe that newer versions (i.e. GPT-4) produce richer, more complex perceptions as well as less negative perceptions compared to older versions and N=159 high-school students. These findings suggest that advances in the architecture of LLMs may lead to increasingly less biased models that could even perhaps someday aid in reducing harmful stereotypes in society rather than perpetuating them.

cs.CY↗
arXiv2023

Derived invariance of the Albanese relative canonical ring

We show the derived invariance of various geometric invariants of smooth complex projective varieties governed by the Albanese map, including the relative canonical ring and the class of the relative canonical model in a suitable variant of the Grothendieck ring of varieties. Then we derive some applications to the derived invariance of Hodge numbers.

math.AG↗
arXiv2023

Multi-fidelity error estimation accelerates greedy model reduction of complex dynamical systems

Model order reduction usually consists of two stages: the offline stage and the online stage. The offline stage is the expensive part that sometimes takes hours till the final reduced-order model is derived, especially when the original model is very large or complex. Once the reduced-order model is obtained, the online stage of querying the reduced-order model for simulation is very fast and often real-time capable. This work concerns a strategy to significantly speed up the offline stage of model order reduction for large and complex systems. In particular, it is successful in accelerating the greedy algorithm that is often used in the offline stage for reduced-order model construction. We propose multi-fidelity error estimators and replace the high-fidelity error estimator in the greedy algorithm. Consequently, the computational complexity at each iteration of the greedy algorithm is reduced and the algorithm converges more than 3 times faster without incurring noticeable accuracy loss.

math.NA↗
arXiv2022

A probabilistic tree model to analyze fuzzy rating data

In this contribution we provide initial findings to the problem of modeling fuzzy rating responses in a psychometric modeling context. In particular, we study a probabilistic tree model with the aim of representing the stage-wise mechanisms of direct fuzzy rating scales. A Multinomial model coupled with a mixture of Binomial distributions is adopted to model the parameters of LR-type fuzzy responses whereas a binary decision tree is used for the stage-wise rating mechanism. Parameter estimation is performed via marginal maximum likelihood approach whereas the characteristics of the proposed model are evaluated by means of an application to a real dataset.

stat.ME↗
arXiv2021

A psychometric modeling approach to fuzzy rating data

Modeling fuzziness and imprecision in human rating data is a crucial problem in many research areas, including applied statistics, behavioral, social, and health sciences. Because of the interplay between cognitive, affective, and contextual factors, the process of answering survey questions is a complex task, which can barely be captured by standard (crisp) rating responses. Fuzzy rating scales have progressively been adopted to overcome some of the limitations of standard rating scales, including their inability to disentangle decision uncertainty from individual responses. The aim of this article is to provide a novel fuzzy scaling procedure which uses Item Response Theory trees (IRTrees) as a psychometric model for the stage-wise latent response process. In so doing, fuzziness of rating data is modeled using the overall rater's pattern of responses instead of being computed using a single-item based approach. This offers a consistent system for interpreting fuzziness in terms of individual-based decision uncertainty. A simulation study and two empirical applications are adopted to assess the characteristics of the proposed model and provide converging results about its effectiveness in modeling fuzziness and imprecision in rating data.

stat.AP↗
arXiv2021

Modeling random and non-random decision uncertainty in ratings data: A fuzzy beta model

Modeling human ratings data subject to raters' decision uncertainty is an attractive problem in applied statistics. In view of the complex interplay between emotion and decision making in rating processes, final raters' choices seldom reflect the true underlying raters' responses. Rather, they are imprecisely observed in the sense that they are subject to a non-random component of uncertainty, namely the decision uncertainty. The purpose of this article is to illustrate a statistical approach to analyse ratings data which integrates both random and non-random components of the rating process. In particular, beta fuzzy numbers are used to model raters' non-random decision uncertainty and a variable dispersion beta linear model is instead adopted to model the random counterpart of rating responses. The main idea is to quantify characteristics of latent and non-fuzzy rating responses by means of random observations subject to fuzziness. To do so, a fuzzy version of the Expectation-Maximization algorithm is adopted to both estimate model's parameters and compute their standard errors. Finally, the characteristics of the proposed fuzzy beta model are investigated by means of a simulation study as well as two case studies from behavioral and social contexts.

stat.AP↗
arXiv2020

A Bayesian brain model of adaptive behavior: An application to the Wisconsin Card Sorting Task

Adaptive behavior emerges through a dynamic interaction between cognitive agents and changing environmental demands. The investigation of information processing underlying adaptive behavior relies on controlled experimental settings in which individuals are asked to accomplish demanding tasks whereby a hidden state or an abstract rule has to be learned dynamically. Although performance in such tasks is regularly considered as a proxy for measuring high-level cognitive processes, the standard approach consists in summarizing response patterns by simple heuristic scoring measures. With this work, we propose and validate a new computational Bayesian model accounting for individual performance in the established Wisconsin Card Sorting Test. We embed the new model within the mathematical framework of Bayesian Brain Theory, according to which beliefs about the hidden environmental states are dynamically updated following the logic of Bayesian inference. Our computational model maps distinct cognitive processes into separable, neurobiologically plausible, information-theoretic constructs underlying observed response patterns. We assess model identification and expressiveness in accounting for meaningful human performance through extensive simulation studies. We further apply the model to real behavioral data in order to highlight the utility of the proposed model in recovering cognitive dynamics at an individual level. Practical and theoretical implications of our computational modeling approach for clinical and cognitive neuroscience research are finally discussed, as well as potential future improvements.

q-bio.NC↗
arXiv2020

Pushforwards of pluricanonical bundles under morphisms to abelian varieties

Let $f \colon X \to A$ be a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). We show that the sheaves $f_* ω_X^{\otimes m}$ become globally generated after pullback by an isogeny. We use this to deduce a decomposition theorem for these sheaves when $m \ge 2$, analogous to that obtained by Chen-Jiang when $m = 1$. This is in turn applied to effective results for pluricanonical linear series on irregular varieties with canonical singularities.

math.AG↗
arXiv2020

Moving Seshadri Constants, and Coverings of Varieties of Maximal Albanese Dimension

Let $X$ be a smooth projective complex variety of maximal Albanese dimension, and let $L \to X$ be a big line bundle. We prove that the moving Seshadri constants of the pull-backs of $L$ to suitable finite abelian étale covers of $X$ are arbitrarily large. As an application, given any integer $k\geq 1$, there exists an abelian étale cover $p\colon X' \to X$ such that the adjoint system $\big|K_{X'} + p^*L \big|$ separates $k$-jets away from the augmented base locus of $p^*L$, and the exceptional locus of the pull-back of the Albanese map of $X$ under $p$.

math.AG↗