Valerio Proietti

• Marra, Pasquale; Proietti, Valerio & Sheng, Xiaobing (2024). Hofstadter-Toda spectral duality and quantum groups. Journal of Mathematical Physics. ISSN 0022-2488. 65(7). doi: 10.1063/5.0202635. Full text in Research Archive Show summary

  The Hofstadter model allows to describe and understand several phenomena in condensed matter such as the quantum Hall effect, Anderson localization, charge pumping, and flat-bands in quasiperiodic structures, and is a rare example of fractality in the quantum world. An apparently unrelated system, the relativistic Toda lattice, has been extensively studied in the context of complex nonlinear dynamics, and more recently for its connection to supersymmetric Yang-Mills theories and topological string theories on Calabi-Yau manifolds in high-energy physics. Here we discuss a recently discovered spectral relationship between the Hofstadter model and the relativistic Toda lattice which has been later conjectured to be related to the Langlands duality of quantum groups. Moreover, by employing similarity transformations compatible with the quantum group structure, we establish a formula parametrizing the energy spectrum of the Hofstadter model in terms of elementary symmetric polynomials and Chebyshev polynomials. The main tools used are the spectral duality of tridiagonal matrices and the representation theory of the elementary quantum group.

• Proietti, Valerio & Yamashita, Makoto (2024). Homology and K-theory of dynamical systems IV. Further structural results on groupoid homology. Ergodic Theory and Dynamical Systems. ISSN 0143-3857. 45(1). doi: 10.1017/etds.2024.37. Full text in Research Archive Show summary

  We consider the homology theory of étale groupoids introduced by Crainic and Moerdijk [A homology theory for étale groupoids. J. Reine Angew. Math. 521 (2000), 25–46], with particular interest to groupoids arising from topological dynamical systems. We prove a Künneth formula for products of groupoids and a Poincaré-duality type result for principal groupoids whose orbits are copies of an Euclidean space. We conclude with a few example computations for systems associated to nilpotent groups such as self-similar actions, and we generalize previous homological calculations by Burke and Putnam for systems which are analogues of solenoids arising from algebraic numbers. For the latter systems, we prove the HK conjecture, even when the resulting groupoid is not ample.

• Bönicke, Christian & Proietti, Valerio (2024). A CATEGORICAL APPROACH to the BAUM-CONNES CONJECTURE for ÉTALE GROUPOIDS. Journal of the Institute of Mathematics of Jussieu. ISSN 1474-7480. 23(5). doi: 10.1017/S1474748023000531. Full text in Research Archive Show summary

  We consider the equivariant Kasparov category associated to an étale groupoid, and by leveraging its triangulated structure we study its localization at the ‘weakly contractible’ objects, extending previous work by R. Meyer and R. Nest. We prove the subcategory of weakly contractible objects is complementary to the localizing subcategory of projective objects, which are defined in terms of ‘compactly induced’ algebras with respect to certain proper subgroupoids related to isotropy. The resulting ‘strong’ Baum–Connes conjecture implies the classical one, and its formulation clarifies several permanence properties and other functorial statements. We present multiple applications, including consequences for the Universal Coefficient Theorem, a generalized ‘going-down’ principle, injectivity results for groupoids that are amenable at infinity, the Baum–Connes conjecture for group bundles, and a result about the invariance of K-groups of twisted groupoid C∗-algebras under homotopy of twists.

• Guo, Hao; Proietti, Valerio & Wang, Hang (2024). A geometric Elliott invariant and noncommutative rigidity of mapping tori. Journal of Functional Analysis. ISSN 0022-1236. 287(11). doi: 10.1016/j.jfa.2024.110625. Full text in Research Archive Show summary

  We prove a rigidity property for mapping tori associated to minimal topological dynamical systems using tools from noncommutative geometry. More precisely, we show that under mild geometric assumptions, a leafwise homotopy equivalence of two mapping tori associated to Zd-actions on a compact space can be lifted to an isomorphism of their foliation C*-algebras. This property is a noncommutative analogue of topological rigidity in the context of foliated spaces whose space of leaves is singular, where isomorphism type of the C*-algebra replaces homeomorphism type. Our technique is to develop a geometric approach to the Elliott invariant that relies on topological and index-theoretic data from the mapping torus. We also discuss how our construction can be extended to slightly more general homotopy quotients arising from actions of discrete cocompact subgroups of simply connected solvable Lie groups, as well as how the theory can be applied to the magnetic gap-labelling problem for certain Cantor minimal systems.

• Proietti, Valerio & Yamashita, Makoto (2024). Homology and K-theory of dynamical systems III. Beyond stably disconnected Smale spaces. Transactions of the American Mathematical Society. ISSN 0002-9947. 378(3), p. 2129–2155. doi: 10.1090/tran/9353. Full text in Research Archive Show summary

  We study homological invariants of étale groupoids arising from Smale spaces, continuing our previous work, but going beyond the stably disconnected case by incorporating resolutions in the space direction. We show that the homology groups defined by Putnam are isomorphic to the Crainic–Moerdijk groupoid homology with integer coefficients. We also show that the K-groups of C*-algebras of stable and unstable equivalence relations have finite rank. For unstably disconnected Smale spaces, we provide a cohomological spectral sequence whose second page shows Putnam’s (stable) homology groups, and converges to the K-groups of the unstable C*-algebra.

• Proietti, Valerio & Yamashita, Makoto (2023). Homology and K-theory of dynamical systems II. Smale spaces with totally disconnected transversal. Journal of Noncommutative Geometry. ISSN 1661-6952. 17(3), p. 957–998. doi: 10.4171/JNCG/494. Full text in Research Archive Show summary

  We apply our previous work on the relation between groupoid homology and K-theory to Smale spaces. More precisely, we consider the unstable equivalence relation of a Smale space with totally disconnected stable sets and prove that the associated spectral sequence shows Putnam's stable homology groups on the second sheet. Moreover, this homology is in fact isomorphic to the groupoid homology of the unstable equivalence relation.

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