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1. Rademacher type and Enflo type coincide

   Ivanisvili, Paata; van Handel, Ramon; Volberg, Alexander. (September 2020, Annals of mathematics)

   null (Ed.)

   A nonlinear analogue of the Rademacher type of a Banach space was introduced in classical work of Enflo. The key feature of Enflo type is that its definition uses only the metric structure of the Banach space, while the definition of Rademacher type relies on its linear structure. We prove that Rademacher type and Enflo type coincide, settling a long-standing open problem in Banach space theory. The proof is based on a novel dimension-free analogue of Pisier’s inequality on the discrete cube. 

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2. Rademacher type and Enflo type coincide

   https://doi.org/10.4007/annals.2020.192.2.8  

   Ivanisvili, Paata; Van Handel, Ramon; Volberg, Alexander (February 2020, Annals of mathematics)

   A nonlinear analogue of the Rademacher type of a Banach space was introduced in classical work of Enflo. The key feature of Enflo type is that its definition uses only the metric structure of the Banach space, while the definition of Rademacher type relies on its linear structure. We prove that Rademacher type and Enflo type coincide, settling a long-standing open problem in Banach space theory. The proof is based on a novel dimension- free analogue of Pisier’s inequality on the discrete cube. 

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3. Twinning stress of type I and type II deformation twins

   https://doi.org/10.1016/j.actamat.2019.07.004  

   Mullner, Peter (July 2019, Acta materialia)

   Recent reports on highly mobile type II twin boundaries challenge the established understanding of deformation twinning and motivate this study. We consider the motion of twin boundaries through the nucleation and growth of disconnection loops and develop a mechanism-based model for twin boundary motion in the framework of isotropic linear elasticity. While such mechanisms are well established for type I and compound twins, we demonstrate based on the elastic properties of crystals that type II twin boundaries propagate in a similar way. Nucleation of a type I twinning disconnection loop occurs in a discrete manner. In contrast, nucleation of a type II twinning disconnection loop occurs gradually with increasing Burgers vector. The gradual nucleation of a type II disconnection loop accounts for the higher mobility of type II twin boundaries compared with type I twin boundaries. We consider the homogeneous nucleation of a disconnection loop, which is adequate for twinning in shape memory alloys with a low-symmetry crystal lattice. For the magnetic shape memory alloy Ni-Mn-Ga, the model predicts twinning stresses of 0.33 MPa for type II twinning and 4.7 MPa for type I twinning. Over a wide temperature range, the twinning stress depends on temperature only through the temperature dependence of the elastic constants, in agreement with experimental results. 

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4. Type I and type II superconductivity in a quasi-2D Dirac metal

   https://doi.org/10.1039/D5MA00022J  

   Lygouras, Chris J; Zhang, Junyi; Gautreau, Jonah; Pula, Mathew; Sharma, Sudarshan; Gao, Shiyuan; Berry, Tanya; Halloran, Thomas; Orban, Peter; Grissonnanche, Gael; et al (March 2025, Materials Advances)

   Dirac material LaCuSb2 shows anisotropic superconducting response to applied magnetic fields. 

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5. Rational Homotopy Type and Computability

   https://doi.org/10.1007/s10208-022-09582-8  

   Manin, Fedor (October 2023, Foundations of Computational Mathematics)

   Given a simplicial pair (X, A), a simplicial complex Y, and a map f:A -> Y, does f have an extension to X? We show that for a fixed Y, this question is algorithmically decidable for all X, A, and f if Y has the rational homotopy type of an H-space. As a corollary, many questions related to bundle structures over a finite complex are likely decidable. Conversely, for all other Y, the question is at least as hard as certain special cases of Hilbert's tenth problem which are known or suspected to be undecidable. 

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