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作者：Ryabogin, Dmitry
摘要：We give a negative answer to Ulam's Problem 19 from the Scottish Book asking is a solid of uniform density which will float in water in every position a sphere? Assuming that the density of water is 1, we show that there exists a strictly convex body of revolution K subset of R-3 of uniform density 1/2, which is not a Euclidean ball, yet floats in equilibrium in every orientation. We prove an analogous result in all dimensions d >= 3.
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作者：Bhattacharya, Prasit; Kitchloo, Nitu
摘要：In this paper, we provide a new proof of the stable Adams conjecture. Our proof constructs a canonical null-homotopy of the stable J-homomorphism composed with a virtual Adams operation, by applying the K-theory functor to a multinatural transformation. We also point out that the original proof of the stable Adams conjecture is incorrect and present a correction. This correction is crucial to our main application. We settle the question on the height of higher associative structures on the mod...
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作者：Bhatt, Bhargav; Scholze, Peter
摘要：We introduce the notion of a prism, which may be regarded as a deperfection of the notion of a perfectoid ring. Using prisms, we attach a ringed site - the prismatic site - to a p-adic formal scheme. The resulting cohomology theory specializes to (and often refines) most known integral p-adic cohomology theories. As applications, we prove an improved version of the almost purity theorem allowing ramification along arbitrary closed subsets (without using adic spaces), give a co-ordinate free de...
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作者：Hezari, Hamid; Zelditch, Steve
摘要：We show that if the eccentricity of an ellipse is sufficiently small, then up to isometries it is spectrally unique among all smooth domains. We do not assume any symmetry, convexity, or closeness to the ellipse, on the class of domains.In the course of the proof we also show that for nearly circular domains, the lengths of periodic orbits that are shorter than the perimeter of the domain must belong to the singular support of the wave trace. As a result we also obtain a Laplace spectral rigid...
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作者：Buzzi, Jerome; Crovisier, Sylvain; Sarig, Omri
摘要：We show that C-infinity-surface diffeomorphisms with positive topological entropy have finitely many ergodic measures of maximal entropy in general, and exactly one in the topologically transitive case. This answers a question of Newhouse, who proved that such measures always exist. To do this we generalize Smale's spectral decomposition theorem to non-uniformly hyperbolic surface diffeomorphisms, we introduce homoclinic classes of measures, and we study their properties using codings by irred...
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作者：Brown, Aaron; Hertz, Federico; Wang, Zhiren
摘要：We consider smooth actions of lattices in higher-rank semisimple Lie groups on manifolds. We define two numbers r(G) and m(G) associated with the roots system of the Lie algebra of a Lie group G. If the dimension of the manifold is smaller than r(G), then we show the action preserves a Borel probability measure. If the dimension of the manifold is at most m(G), we show there is a quasi-invariant measure on the manifold such that the action is measurably isomorphic to a relatively measure-prese...
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作者：Judge, Chris; Mondal, Sugata
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作者：Raskin, Sam
摘要：We prove the rank 1 case of a conjecture of Frenkel-Gaitsgory: critical level Kac-Moody representations with regular central characters localize onto the affine Grassmannian. The method uses an analogue in local geometric Langlands of the existence of Whittaker models for most representations of GL(2) over a non-Archimedean field.
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作者：Merle, Frank; Raphael, Pierre; Rodnianski, Igor; Szeftel, Jeremie
摘要：In this paper and its sequel, we construct a set of finite energy smooth initial data for which the corresponding solutions to the compressible three-dimensional Navier-Stokes and Euler equations implode (with infinite density) at a later time at a point, and we completely describe the associated formation of singularity. This paper is concerned with existence of smooth self-similar profiles for the barotropic Euler equations in dimension d >= 2 with decaying density at spatial infinity. The p...
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作者：Hahn, Jeremy; Wilson, Dylan
摘要：We equip BP(n) with an E3-BP-algebra structure for each prime p and height n. The algebraic K-theory of this ring is of chromatic height exactly n + 1, and the map K(BP(n))(p)-+ Lfn+1K(BP(n))(p) has bounded above fiber.
