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PubblicazioneLarge Scale Response of Gapless 1d and Quasi-1d Systems( 2025)We consider the transport properties of non-interacting, gapless one-dimensional quantum systems and of the edge modes of two-dimensional topological insulators, in the presence of time-dependent perturbations. We prove the validity of Kubo formula, in the zero temperature and infinite volume limit, for a class of perturbations that are weak and slowly varying in space and in time, in an Euler-like scaling. The proof relies on the representation of the real-time Duhamel series in imaginary time, which allows to prove its convergence uniformly in the scaling parameter and in the size of system, at low temperatures. Furthermore, it allows to exploit a suitable cancellation for the scaling limit of the model, related to the emergent anomalous chiral gauge symmetry of relativistic one-dimensional fermions. The cancellation implies that as the temperature and the scaling parameter are sent to zero, the linear response is the only contribution to the full response of the system. The explicit form of the leading contribution to the response function is determined by lattice conservation laws. In particular, the method allows to prove the quantization of the edge conductance of 2d quantum Hall systems from quantum dynamics.
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Pubblicazione∂ -problem for the focusing nonlinear Schrödinger equation and soliton shielding( 2025)We consider soliton gas solutions of the focusing nonlinear Schrödinger (NLS) equation, where the point spectrum of the Zakharov-Shabat linear operator condenses in a bounded domain D in the upper half-plane. We show that the corresponding inverse scattering problem can be formulated as a ∂-problem on the complex plane. We prove that the function of the N soliton solution converges in the limit N→∞ to the function (a Fredholm determinant) of the ∂-problem. Furthermore, we prove that such a function is non-vanishing for all values of x and t, thus showing the existence of a solution of the ∂-problem. Then we show that, when the domain D is an ellipse and the soliton gas spectral data are analytic, the inverse problem reduces to the soliton spectra concentrating on the segment connecting the foci of the ellipse (soliton shielding). The NLS solution for fixed times is asymptotically step-like oscillatory, and it is described by a periodic elliptic function as x→-∞ while it vanishes exponentially fast as x→+∞.
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PubblicazioneSoliton synchronization with randomness: rogue waves and universality( 2025)We consider an N-soliton solution of the focusing nonlinear Schr & ouml;dinger equations. We give conditions for the synchronous collision of these N solitons. When the solitons velocities are well separated and the solitons have equal amplitude, we show that the local wave profile at the collision point scales as the sinc(x) function. We show that this behaviour persists when the amplitudes of the solitons are i.i.d. sub-exponential random variables. Namely the central collision peak exhibits universality: its spatial profile converges to the sinc(x) function, independently of the distribution. We derive Central Limit Theorems for the fluctuations of the profile in the near-field regime (near the collision point) and in the far-field regime.
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PubblicazioneShielding of breathers for the focusing nonlinear Schrödinger equation( 2025)The authors investigate the focusing nonlinear Schrödinger equation through the lens of breather dynamics. They construct a deterministic breather gas by considering the N-breather solution in the limit N→∞, where the discrete scattering spectrum fills a compact domain of the complex plane and the norming constants are smoothly interpolated and scaled as 1/N. This framework generalizes recent results on soliton gases to the breather setting. A central contribution is the demonstration of a shielding effect: under suitable choices of spectral domains (including quadrature domains) and interpolating functions, the breather gas reproduces a finite breather configuration, effectively ``hiding'' the infinite background. The analysis is carried out using Riemann-Hilbert techniques, and connections are drawn with special classes of breather solutions such as Kuznetsov-Ma and Akhmediev breathers.
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PubblicazioneReal Algebraic Geometry in Manifold Learning(SISSA, 2026-04-01)Lower-dimensional representations of data play an essential role in modern machine learning. This thesis studies the geometric structure underlying data from the perspective of real algebraic geometry, with the goal of developing testing procedures for a generalized manifold hypothesis, here called the geometric hypothesis. The dissertation is based on the following two works, listed below in chronological order of development. - Testing Variety Hypothesis (joint with A.~Lerario, P.~Roos Hoefgeest, and M.~Scolamiero). This work studies the geometric hypothesis for real algebraic varieties of bounded degree, allowing singularities and stratified structures. We show that the testing problem can be reduced to a semialgebraic decision problem and derive explicit sample complexity bounds. - Testing Algebraic Complete Intersections. This work introduces an effective testing procedure for regression by real algebraic complete intersections with controlled geometry. The method yields explicit bounds on sample complexity together with arithmetic complexity estimates for the algorithmic implementation. (Currently in preparation.) In the thesis, the two works are presented in the opposite order, reflecting a conceptual progression from the regular to the singular setting.