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1. Multifractality of light in photonic arrays based on algebraic number theory

   https://doi.org/10.1038/s42005-020-0374-7  

   Sgrignuoli, Fabrizio; Gorsky, Sean; Britton, Wesley A.; Zhang, Ran; Riboli, Francesco; Dal Negro, Luca (June 2020, Communications Physics)

   Abstract Many natural patterns and shapes, such as meandering coastlines, clouds, or turbulent flows, exhibit a characteristic complexity that is mathematically described by fractal geometry. Here, we extend the reach of fractal concepts in photonics by experimentally demonstrating multifractality of light in arrays of dielectric nanoparticles that are based on fundamental structures of algebraic number theory. Specifically, we engineered novel deterministic photonic platforms based on the aperiodic distributions of primes and irreducible elements in complex quadratic and quaternions rings. Our findings stimulate fundamental questions on the nature of transport and localization of wave excitations in deterministic media with multi-scale fluctuations beyond what is possible in traditional fractal systems. Moreover, our approach establishes structure–property relationships that can readily be transferred to planar semiconductor electronics and to artificial atomic lattices, enabling the exploration of novel quantum phases and many-body effects. 

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2. Classification of the Anyon Sectors of Kitaev’s Quantum Double Model

   https://doi.org/10.1007/s00220-025-05363-w  

   Bols, Alex; Vadnerkar, Siddharth (July 2025, Communications in Mathematical Physics)

   Abstract We give a complete classification of the anyon sectors of Kitaev’s quantum double model on the infinite triangular lattice and for finite gauge groupG, including the non-abelian case. As conjectured, the anyon sectors of the model correspond precisely to equivalence classes of irreducible representations of the quantum double algebra ofG. 

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3. Bertini theorems for differential algebraic geometry

   https://doi.org/10.1090/proc/13588  

   Freitag, James (January 2024, Proceedings of the American Mathematical Society)

   We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the differential analogue of Bertini’s theorem, namely that for an arbitrary geometrically irreducible differential algebraic variety which is not an algebraic curve, generic hypersurface sections are geometrically irreducible and codimension one. Surprisingly, we prove a stronger result in the case that the order of the differential hypersurface is at least one; namely that the generic differential hypersurface sections of an irreducible differential algebraic variety are irreducible and codimension one. We also calculate the Kolchin polynomials of the intersections and prove several other results regarding intersections of differential algebraic varieties. 

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4. Integral Points on Varieties With Infinite Étale Fundamental Group

   https://doi.org/10.1093/imrn/rnad147  

   Achenjang, Niven T; Morrow, Jackson S (July 2023, International Mathematics Research Notices)

   Abstract We study integral points on varieties with infinite étale fundamental groups. More precisely, for a number field $$F$$ and $X/F$ a smooth projective variety, we prove that for any geometrically Galois cover $$\varphi \colon Y \to X$$ of degree at least $$2\dim (X)^{2}$$, there exists an ample line bundle $$\mathscr{L}$$ on $$Y$$ such that for a general member $$D$$ of the complete linear system $$|\mathscr{L}|$$, $$D$$ is geometrically irreducible and any set of $$\varphi (D)$$-integral points on $$X$$ is finite. We apply this result to varieties with infinite étale fundamental group to give new examples of irreducible, ample divisors on varieties for which finiteness of integral points is provable. 

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5. Nested gausslet basis sets

   https://doi.org/10.1063/5.0180092  

   White, Steven R; Lindsey, Michael J (December 2023, The Journal of Chemical Physics)

   We introduce nested gausslet bases, an improvement on previous gausslet bases that can treat systems containing atoms with much larger atomic numbers. We also introduce pure Gaussian distorted gausslet bases, which allow the Hamiltonian integrals to be performed analytically, as well as hybrid bases in which the gausslets are combined with standard Gaussian-type bases. All these bases feature the diagonal approximation for the electron–electron interactions so that the Hamiltonian is completely defined by two Nb × Nb matrices, where Nb ≈ 104 is small enough to permit fast calculations at the Hartree–Fock level. In constructing these bases, we have gained new mathematical insight into the construction of one-dimensional diagonal bases. In particular, we have proved an important theorem relating four key basis set properties: completeness, orthogonality, zero-moment conditions, and diagonalization of the coordinate operator matrix. We test our basis sets on small systems with a focus on high accuracy, obtaining, for example, an accuracy of 2 × 10−5 Ha for the total Hartree–Fock energy of the neon atom in the complete basis set limit. 

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