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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. Generating Sets and Algebraic Properties of Pure Mapping Class Groups of Infinite Graphs

   https://doi.org/10.5802/ahl.238  

   Domat, George; Hoganson, Hannah; Kwak, Sanghoon (January 2025, Annales Henri Lebesgue)

   We completely classify the locally finite, infinite graphs with pure mapping class groups admitting a coarsely bounded generating set. We also study algebraic properties of the pure mapping class group. We establish a semidirect product decomposition, compute first integral cohomology, and classify when they satisfy residual finiteness and the Tits alternative. These results provide a framework and some initial steps towards quasi-isometric and algebraic rigidity of these groups. 

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