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2. Topology Optimization of Conformal Structures Using Extended Level Set Methods and Conformal Geometry Theory

   https://doi.org/10.1115/DETC2018-85655  

   Ye, Qian ; Guo, Yang ; Chen, Shikui ; Gu, Xianfeng David ; Lei, Na ( August 2018 , Proceedings of Proceedings of the International Design Engineering Technical Conferences & Computers & Information in Engineering Conference)

   In this paper, we propose a new method to approach the problem of structural shape and topology optimization on manifold (or free-form surfaces). A manifold is conformally mapped onto a 2D rectangle domain, where the level set functions are defined. With conformal mapping, the corresponding covariant derivatives on a manifold can be represented by the Euclidean differential operators multiplied by a scalar. Therefore, the topology optimization problem on a free-form surface can be formulated as a 2D problem in the Euclidean space. To evolve the boundaries on a free-form surface, we propose a modified Hamilton-Jacobi equation and solve it on a 2D plane following the conformal geometry theory. In this way, we can fully utilize the conventional level-set-based computational framework. Compared with other established approaches which need to project the Euclidean differential operators to the manifold, the computational difficulty of our method is highly reduced while all the advantages of conventional level set methods are well preserved. We hope the proposed computational framework can provide a timely solution to increasing applications involving innovative structural designs on free-form surfaces in different engineering fields. 

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3. Vector Field Map Representation for Near Conformal Surface Correspondence: Vector Field Map Representation for Near Conformal Surface Correspondence

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4. Evolution operators in conformal field theories and conformal mappings: Entanglement Hamiltonian, the sine-square deformation, and others

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5. Conformal blocks from celestial gluon amplitudes

   https://doi.org/10.1007/JHEP05(2021)170  

   Fan, Wei ; Fotopoulos, Angelos ; Stieberger, Stephan ; Taylor, Tomasz R. ; Zhu, Bin ( May 2021 , Journal of High Energy Physics)

   null (Ed.)

   A bstract In celestial conformal field theory, gluons are represented by primary fields with dimensions ∆ = 1 + iλ , λ ∈ ℝ and spin J = ±1, in the adjoint representation of the gauge group. All two- and three-point correlation functions of these fields are zero as a consequence of four-dimensional kinematic constraints. Four-point correlation functions contain delta-function singularities enforcing planarity of four-particle scattering events. We relax these constraints by taking a shadow transform of one field and perform conformal block decomposition of the corresponding correlators. We compute the conformal block coefficients. When decomposed in channels that are “compatible” in two and four dimensions, such four-point correlators contain conformal blocks of primary fields with dimensions ∆ = 2 + M + iλ , where M ≥ 0 is an integer, with integer spin J = −M, −M + 2 , … , M − 2 , M . They appear in all gauge group representations obtained from a tensor product of two adjoint representations. When decomposed in incompatible channels, they also contain primary fields with continuous complex spin, but with positive integer dimensions. 

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