@article{8422, abstract = {The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of an ellipse of small eccentricity which preserves integrability near the boundary, is itself an ellipse. This extends the result in Avila et al. (Ann Math 184:527–558, ADK16), where integrability was assumed on a larger set. In particular, it shows that (local) integrability near the boundary implies global integrability. One of the crucial ideas in the proof consists in analyzing Taylor expansion of the corresponding action-angle coordinates with respect to the eccentricity parameter, deriving and studying higher order conditions for the preservation of integrable rational caustics.}, author = {Huang, Guan and Kaloshin, Vadim and Sorrentino, Alfonso}, issn = {1016-443X}, journal = {Geometric and Functional Analysis}, keywords = {Geometry and Topology, Analysis}, number = {2}, pages = {334--392}, publisher = {Springer Nature}, title = {{Nearly circular domains which are integrable close to the boundary are ellipses}}, doi = {10.1007/s00039-018-0440-4}, volume = {28}, year = {2018}, } @article{8421, abstract = {The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a small integrable perturbation of an ellipse must be an ellipse. This extends and completes the result in Avila-De Simoi-Kaloshin, where nearly circular domains were considered. One of the crucial ideas in the proof is to extend action-angle coordinates for elliptic billiards into complex domains (with respect to the angle), and to thoroughly analyze the nature of their complex singularities. As an application, we are able to prove some spectral rigidity results for elliptic domains.}, author = {Kaloshin, Vadim and Sorrentino, Alfonso}, issn = {0003-486X}, journal = {Annals of Mathematics}, keywords = {Statistics, Probability and Uncertainty, Statistics and Probability}, number = {1}, pages = {315--380}, publisher = {Annals of Mathematics, Princeton U}, title = {{On the local Birkhoff conjecture for convex billiards}}, doi = {10.4007/annals.2018.188.1.6}, volume = {188}, year = {2018}, } @article{8420, abstract = {We show that in the space of all convex billiard boundaries, the set of boundaries with rational caustics is dense. More precisely, the set of billiard boundaries with caustics of rotation number 1/q is polynomially sense in the smooth case, and exponentially dense in the analytic case.}, author = {Kaloshin, Vadim and Zhang, Ke}, issn = {0951-7715}, journal = {Nonlinearity}, keywords = {Mathematical Physics, General Physics and Astronomy, Applied Mathematics, Statistical and Nonlinear Physics}, number = {11}, pages = {5214--5234}, publisher = {IOP Publishing}, title = {{Density of convex billiards with rational caustics}}, doi = {10.1088/1361-6544/aadc12}, volume = {31}, year = {2018}, } @article{8426, abstract = {For any strictly convex planar domain Ω ⊂ R2 with a C∞ boundary one can associate an infinite sequence of spectral invariants introduced by Marvizi–Merlose [5]. These invariants can generically be determined using the spectrum of the Dirichlet problem of the Laplace operator. A natural question asks if this collection is sufficient to determine Ω up to isometry. In this paper we give a counterexample, namely, we present two nonisometric domains Ω and Ω¯ with the same collection of Marvizi–Melrose invariants. Moreover, each domain has countably many periodic orbits {Sn}n≥1 (resp. {S¯n}n⩾1) of period going to infinity such that Sn and S¯n have the same period and perimeter for each n.}, author = {Buhovsky, Lev and Kaloshin, Vadim}, issn = {1560-3547}, journal = {Regular and Chaotic Dynamics}, pages = {54--59}, publisher = {Springer Nature}, title = {{Nonisometric domains with the same Marvizi-Melrose invariants}}, doi = {10.1134/s1560354718010057}, volume = {23}, year = {2018}, } @article{9053, abstract = {The development of strategies to assemble microscopic machines from dissipative building blocks are essential on the route to novel active materials. We recently demonstrated the hierarchical self-assembly of phoretic microswimmers into self-spinning microgears and their synchronization by diffusiophoretic interactions [Aubret et al., Nat. Phys., 2018]. In this paper, we adopt a pedagogical approach and expose our strategy to control self-assembly and build machines using phoretic phenomena. We notably introduce Highly Inclined Laminated Optical sheets microscopy (HILO) to image and characterize anisotropic and dynamic diffusiophoretic interactions, which cannot be performed by conventional fluorescence microscopy. The dynamics of a (haematite) photocatalytic material immersed in (hydrogen peroxide) fuel under various illumination patterns is first described and quantitatively rationalized by a model of diffusiophoresis, the migration of a colloidal particle in a concentration gradient. It is further exploited to design phototactic microswimmers that direct towards the high intensity of light, as a result of the reorientation of the haematite in a light gradient. We finally show the assembly of self-spinning microgears from colloidal microswimmers and carefully characterize the interactions using HILO techniques. The results are compared with analytical and numerical predictions and agree quantitatively, stressing the important role played by concentration gradients induced by chemical activity to control and design interactions. Because the approach described hereby is generic, this works paves the way for the rational design of machines by controlling phoretic phenomena.}, author = {Aubret, Antoine and Palacci, Jérémie A}, issn = {1744-6848}, journal = {Soft Matter}, keywords = {General Chemistry, Condensed Matter Physics}, number = {47}, pages = {9577--9588}, publisher = {Royal Society of Chemistry }, title = {{Diffusiophoretic design of self-spinning microgears from colloidal microswimmers}}, doi = {10.1039/c8sm01760c}, volume = {14}, year = {2018}, }