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作者：De Simoi, Jacopo; Kaloshin, Vadim; Wei, Qiaoling
摘要：We show that any sufficiently (finitely) smooth Z(2) -symmetric strictly convex domain sufficiently close to a circle is dynamically spectrally rigid; i.e., all deformations among domains in the same class that preserve the length of all periodic orbits of the associated billiard flow must necessarily be isometric deformations. This gives a partial answer to a question of P. Sarnak.
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作者：Levine, Lionel; Pegden, Wesley; Smart, Charles K.
摘要：We prove that the set of quadratic growths attainable by integer-valued superharmonic functions on the lattice Z(2) has the structure of an Apollonian circle packing. This completely characterizes the PDE that determines the continuum scaling limit of the Abelian sandpile on the lattice Z(2).
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作者：Naber, Aaron; Valtora, Daniele
摘要：In this paper we study the regularity of stationary and minimizing harmonic maps f: B-2(p) subset of M -> N between Riemannian manifolds. If Sk(f)={x is an element of M: no tangent map at x is k+1-symmetric} is kth-stratum of the singular set of f, then it is well known that dimSk=k, however little else about the structure of Sk(f) is understood in any generality. Our first result is for a general stationary harmonic map, where we prove that Sk(f) is k-rectifiable. In the case of minimizing ha...
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作者：Tikuisis, Aaron; White, Stuart; Winter, Wilhelm
摘要：We prove that faithful traces on separable and nuclear C*-algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear C*-algebras of finite nuclear dimension which satisfy the UCT is now complete. Secondly, our result links the finite to the general version of the Toms-Winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dim...
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作者：Labourie, Francois
摘要：We prove that given a Hitchin representation in a real split rank 2 group G(0), there exists a unique equivariant minimal surface in the corresponding symmetric space. As a corollary, we obtain a parametrization of the Hitchin components by a Hermitian bundle over Teichm\uller space. The proof goes through introducing holomorphic curves in a suitable bundle over the symmetric space of G(0). Some partial extensions of the construction hold for cyclic bundles in higher rank.
