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PubblicazioneOn the lower bound of the curvature exponent on step-two Carnot groups( 2025)In this work, we show that there exists a step-two Carnot group on which the new lower bound of the curvature exponent given in Golo and Zhang [Anal. Geom. Metr. Spaces 12 (2024), p. 30] can be strictly less than the curvature exponent by studying the convergence of the structure constants of Lie algebra.
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PubblicazionePerceptual Decision Making of Nonequilibrium Fluctuations(SISSA, 2026-01-30)How does the brain recover a weak signal that is submerged in intense stochastic fluctuations to make fast yet accurate choices? We recast perceptual decision-making as the inference of a nonzero drift (v) in the presence of large diffusivity (D): the observer must determine the direction of motion when trajectories are dominated by diffusion. A concrete analogy is reading wind while hunting: turbulent gusts scramble moment-to-moment cues, yet a subtle, consistent drift in air motion carries actionable information about wind direction. In this perspective, the core problem is signal discovery under diffusion—extracting the sign and magnitude of a weak drift from time-resolved fluctuations. This framing contrasts with the traditional random-dot motion (RDM) paradigm, where “coherence” (the percentage of dots moving consistently) is an effective but unitless control of difficulty. Although 50% coherence is intuitively “more signal” than 10%, its relationship to the quantity of evidence is ambiguous: is it five times more, or something else? Because coherence does not specify the physical statistics generating samples, an ideal-observer analysis (e.g., a Sequential Probability Ratio Test, SPRT) cannot be uniquely grounded in first principles. By instead specifying a physics-based stimulus with known parameters, this thesis makes the evidence quantified and the ideal-observer well-posed. We generated visual motion stimuli from a drifted Brownian process. To index the distance from equilibrium, we employed an interpretable nonequilibrium measure (entropy production Σ) proportional to drift–diffusion contrast, which increases as directional drive overwhelms diffusivity. In this generative setting, the momentary log-likelihood ratio (LLR) for direction decisions and the optimal stopping boundaries of the SPRT are derived analytically from (v, D), providing a physics-grounded benchmark for ideal performance. Across three behavioral experiments, we asked: i) whether human observers (N = 67) detect and exploit graded nonequilibrium dynamics; ii) how closely their choices approach an ideal-observer benchmark; iii) how evidence integration adapts as Σ varies; and iv) whether such adaptation depends on task structure and the spatiotemporal layout of the stimuli. Results showed that stimulus dynamics (Σ, v, D) robustly shaped decision metrics, demonstrating that observers are indeed sensitive to graded changes in Σ. An analytical SPRT captured these effects and quantified deviations from ideal performance. Complementarily, an Evidence Integration Model (EIM) fitted to the data revealed a systematic adjustment of the temporal integration window with Σ: in each trial, observers assigned greater weight to the most recent portion of the trajectory (a recency effect) whose strength scaled with Σ. Observers were also sensitive to salient changes in trajectory directionality, consistent with adaptive weighting under nonstationary drift. Crucially, these effects were stronger in a blocked design—where Σ was held constant within blocks—than in an intermixed design, where Σ varied from trial to trial, indicating that stable nonequilibrium statistics facilitate calibration of integration timescales. Finally, sensitivity to the nonequilibrium structure was modulated not only by the physical parameters (v, D) but also by the spatial and temporal layout of the stimuli. Overall, by embedding perceptual evidence in a physics-based process that specifies its quantity, this work refines the characterization of variables that govern perceptual decisions and clarifies the temporal dynamics underlying efficient sensory evidence integration. It shows that when evidence is measured—rather than merely manipulated—ideal-observer analyses become principled, enabling precise tests of how the brain detects weak directional signals under high diffusion.
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PubblicazioneA Bayesian Need for Simplicity: A study of how the brain selects and implements internal models, using a normative approach and Bayesian data analysis methods(SISSA, 2026-01-28)A prominent hypothesis in cognitive neuroscience is that the brain operates as an inference engine, relying on internal models to interpret sensory input and guide decision-making. While a great variety of behaviors and biases are explained through this approach, it is less clear how the brain builds and, most importantly, chooses among different models to interpret its environment. A general bias toward simpler interpretations is well-established, but critical gaps remain regarding the computational mechanisms driving this preference. Specifically, it is unclear whether the brain assesses the simplicity (or complexity) of an interpretation through optimal, theory-aligned computations or if it relies on heuristic approximation that can be skewed by perceptual biases. Furthermore, we don't know if the measure of complexity is fixed or adapted to the detailed features of the models at hand. The first part of this work examines the criteria humans use to select between alternative interpretations of noisy data. First, investigating the effect of a variable amount of data, we show that human interpretations, while qualitatively following the principle of parsimony, diverge from optimal behavior. Our results suggest that intuitive model selection is based on a perceptual evaluation of the sample size, rather than a principled computation of the model fit. Second, we address which properties of the model affect the perception of its complexity. We show that the brain does not rely on a generic complexity metric, as some model selection criteria do. Instead, characteristics unique to the models considered have a direct effect on the choice between interpretations. Broadening the scope from intuitive model selection to biological implementation, the latter half of this work investigates how internal models shape sensory perception and how they are instantiated in neural populations, leveraging the same probabilistic frameworks used in the first half. In the domain of sensory processing, we apply Bayesian inference to explain perceptual biases in rats. We show that the influence of task-irrelevant sounds on visual discrimination is best explained by an internal model affected by a compressive warping of visual representations caused by auditory inputs, providing behavioral evidence for direct sensory interaction. Finally, we develop a novel Bayesian method for analyzing neural population activity. Applied to data from mice performing a value-based decision-making task, this technique allows for the decoding of latent variables required to construct and update an internal model of reward contingency. Taken together, these findings offer a multi-level perspective on internal models, from cognition to perception to neural implementation, shedding light on the algorithms the brain uses to manage uncertainty and providing methodological tools to investigate them further.
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PubblicazioneFluctuations and Correlations of Local Topological Order Parameters in Disordered Two-Dimensional Topological Insulators( 2025)Two-dimensional topological insulators are characterized by an insulating bulk and conductive edge states protected by the nontrivial topology of the bulk electronic structure. They remain robust against moderate disorder until Anderson localization occurs and destroys the topological phase. Interestingly, disorder can also induce a topological phase - known as a topological Anderson insulator - starting from an otherwise pristine trivial phase. While topological invariants are generally regarded as global quantities, we argue that space-resolved topological markers can act as local order parameters, revealing the role of fluctuations and correlations in the local topology under Anderson disorder and vacancies. With this perspective, we perform numerical simulations of disorder-driven topological phase transitions in the Haldane and Kane-Mele models, using supercells with both open and periodic boundary conditions. We find that short-scale fluctuations of topological markers vanish upon coarse graining, except at the topological phase transition, where their correlation length peaks and large-scale fluctuations remain. Notably, such a topological correlation function is characterized by critical exponents that appear universal across disorder types, yet they can resolve different topological phase transitions.
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PubblicazioneUniqueness of Gauge Covariant Renormalisation of Stochastic 3D Yang–Mills–Higgs( 2026)Local solutions to the 3D stochastic quantisation equations of Yang-Mills-Higgs were constructed in Chandra (Invent Math 237:541-696, 2024), and it was shown that, in the limit of smooth mollifications, there exists a mass renormalisation of the Yang-Mills field such that the solution is gauge covariant. In this paper we prove the uniqueness of the mass renormalisation that leads to gauge covariant solutions. This strengthens the main result of Chandra (Invent Math 237:541-696, 2024), and is potentially important for the identification of the limit of other approximations, such as lattice dynamics. Our proof relies on systematic short-time expansions of singular stochastic PDEs and of regularised Wilson loops. We also strengthen the recently introduced state spaces of Cao (Comm Part Diff Equ 48:209-251, 2023); Cao (Comm Math Phys 405:3, 2024); Chandra (Invent Math 237:541-696, 2024) to allow for finer control on line integrals appearing in expansions of Wilson loops.