I will describe how to deal with single-particle dynamics over complex aperiodic discrete geometries using C*-algebras. Using numerical examples, I will first convey that any local Hamiltonian over such a geometry falls into one and same algebra, completely determined by the underlying discrete geometry. I will then describe this algebra based on the notion of phason spaces, and compute it for particular cases (e.g. twisted bilayers). K-theory of this algebra tells us what topological invariants are available for a given discrete geometric structure. For example, a twisted bilayer can host up to second-Chern numbers but nothing above that. The punch line of the talk would be a C*-formulation of ordinary and higher order bulk-defect correspondences. The claim is that, given a geometry with defects (surfaces, hinges, corners, dislocations, disclinations, etc), the C*-algebraic framework supplies a device to enumerate all possible non-trivial defect-boundary correspondences, without missing one. I will include numerical demonstrations of these principles. References: P. D. Ojito, E. Prodan, T. Stoiber, C*-framework for higher-order bulk-boundary correspondences, Comm. Math. Phys 406, 233 (2025). B. Mesland, E. Prodan, Classifying the dynamics of architected materials by groupoid methods, Journal of Geometry and Physics 196, 105059 (2024).
My talk centers on an “ordinary-looking” ordinary differential equation—the prolate spheroidal wave operator—which has played a surprisingly rich and unexpected role across several fields. It first arose as a “lucky accident” in the 1960s solution by Slepian, Landau, and Pollak of the problem of time-and band-limiting of signals, raised by Claude Shannon. The same operator reappeared in the late 1990s through a cutoff mechanism in Alain Connes’ trace formula reformulation of the Weil explicit formula in number theory. More recently, in joint work with Connes (2022), we found that when this operator is extended from a finite interval to the whole line, its spectrum reproduces—with striking accuracy—the squares of the imaginary parts of the zeros of the Riemann zeta function. I will conclude with a refinement of this phenomenon based on analyzing the operator in the complex domain.
Gapped phases of two-dimensional free-fermionic systems without any symmetry are classified by an integer-valued topological invariant called the Chern number, which counts the net chirality of boundary modes. It has long been conjectured that systems with distinct Chern numbers cannot be deformed into each other without a phase transition, even in the presence of arbitrarily strong interactions. In this talk, I will present a construction of a many-body index that generalizes the Chern number to the interacting setting and allows us to prove this conjecture. When the edge modes can be described by a conformal field theory, the index provides a microscopic characterization of the chiral central charge.
There are now many examples of gapped fracton models, which are defined by the presence of restricted-mobility excitations above the quantum ground state. This complex landscape of examples is far from being mapped out. In this talk, I will describe recent progress on characterization and classification of fracton orders, and on related problems for subsystem symmetries.
