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PubblicazioneConfigurational entropy of random trees( 2025)We present a graph theoretical approach to the configurational statistics of random treelike objects, such as randomly branching polymers. In particular, for ideal trees we show that Prüfer labeling provides (i) direct access to the exact configurational entropy as a function of the tree composition, (ii) computable exact expressions for partition functions and important experimental observables for tree ensembles with controlled branching activity, and (iii) an efficient sampling scheme for corresponding tree configurations and arbitrary static properties.
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PubblicazioneIntegrable Magnetic Flows on the Two-Torus: Zoll Examples and Systolic Inequalities( 2021)In this paper, we study some aspects of integrable magnetic systems on the two-torus. On the one hand, we construct the first non-trivial examples with the property that all magnetic geodesics with unit speed are closed. On the other hand, we show that those integrable magnetic systems admitting a global surface of section satisfy a sharp systolic inequality.
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PubblicazioneThe Lusternik-Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles( 2016)Let M be a closed manifold and consider the Hamiltonian flow associated to an autonomous Tonelli Hamiltonian H: T∗M → R and a twisted symplectic form. In this paper we study the existence of contractible periodic orbits for such a flow. Our main result asserts that if M is not aspherical, then contractible periodic orbits exist for almost all energies above the maximum critical value of H.
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PubblicazioneStatistical signatures of integrable and non-integrable quantum hamiltonians( 2026)Integrability has long served as a cornerstone concept in classical mechanics, where it possesses a precise and unambiguous definition. Extending this notion to the quantum domain, however, remains a far more subtle and elusive problem. In particular, deciding whether a given quantum Hamiltonian – viewed simply as a matrix of its elements – does or does not define an integrable system is far from obvious. Yet this question is crucial: it bears directly on non-equilibrium dynamics, spectral correlations, the behaviour of cor- relation functions, and other fundamental properties of many-body quantum systems. In this work, we develop a statistical framework for addressing quantum integrability from a purely probabilistic standpoint. Our approach begins with the observation that a necessary signature of integrability is the finite probability of encountering vanishing energy gaps in the spectrum. On this basis, we formulate a twofold protocol capable of distinguishing be- tween integrable and non-integrable Hamiltonians. The first step consists of a systematic Monte Carlo decimation of the spectrum, designed to reveal the emergence (or absence) of Poissonian level spacing statistics. The iterative decimation compresses the Hilbert space exponentially, and its termination point determines whether the spectrum is governed by a mixed distribution or approaches the Poisson limit. In the second step, the distinction is instead obtained by analysing the distributions of k-step gaps which can help in discrim- inating between Poisson and mixed statistics. This procedure applies to Hamiltonians of arbitrary finite size, regardless of whether their structure involves a finite number of blocks or an exponentially fragmented Hilbert space. As a concrete benchmark, we implement the protocol on a class of quantum Hamiltonians constructed from the permutation group SN , thereby demonstrating both its effectiveness and its broad applicability.
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PubblicazioneVirtual invariants of Quot schemes of points on threefolds(SISSA, 2026-05-14)In this thesis, we study the deformation theory and the enumerative geometry of the Quot scheme of points Quot_Y(F, n) parametrizing zero-dimensional quotients of length n of a locally free sheaf F on a smooth (quasi)-projective 3-fold Y. We construct on Quot_Y(F, n) an almost perfect obstruction theory of virtual dimension zero. The notion of an almost perfect obstruction theory recently introduced by Kiem--Savvas is a weaker notion of a perfect obstruction theory in the sense of Behrend--Fantechi which still gives a virtual class. We therefore obtain a Chow class [Quot_Y(F, n)]^vir in degree zero, which in the projective case allows one to define the virtual invariants by taking the degree. We compute the virtual invariants of Quot_Y(F, n) with Y projective proving a conjectural formula of Ricolfi. We first perform the computation in the toric case by reducing it to the one of Fasola–Monavari–Ricolfi. We have to reprove the torus localization formula and the Siebert formula for almost perfect obstruction theories under suitable assumptions. To this end, we exploit the Jouanolou trick. The computation in the general case follows from the toric case via the theory of double point cobordism of Lee–Pandharipande and a degeneration argument similar to the one in Li–Wu. In the course of the proof, we also introduce virtual invariants of the Quot scheme of points on a 3-fold relative to a smooth divisor D in Y. We compute these invariants and generalize results of Levine--Pandharipande to the higher rank case. The proof crucially uses the computation of the invariants for local surfaces. We also define the virtual invariants via localization for any smooth quasi-projective 3-fold Y, which is acted on by a torus T, and such that the locally free sheaf F has a T-equivariant structure, and the fixed locus Y^T is proper. We compute these invariants extending again the work of Levine–Pandharipande. The strategy is to reduce the computation to the case of local geometries using deformation to the normal cone. In summary, the thesis provides a complete solution of the virtual enumerative geometry of the Quot scheme of points of a locally free sheaf on a smooth quasi-projective 3-fold, generalizing the classical case of the Hilbert scheme of points on a 3-fold.