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作者：Luk, Jonathan; Oh, Sung-Tin
摘要：This is the first and main paper of a two-part series, in which we prove the C-2-formulation of the strong cosmic censorship conjecture for the Einstein-Maxwell-(real)-scalar-field system in spherical symmetry for two-ended asymptotically flat data. For this model, it is known through the works of Dafermos and Dafermos-Rodnianski that the maximal globally hyperbolic future development of any admissible two-ended asymptotically flat Cauchy initial data set possesses a non-empty Cauchy horizon, ...
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作者：Land, Markus; Tamme, Georg
摘要：To any pullback square of ring spectra we associate a new ring spectrum and use it to describe the failure of excision in algebraic K-theory. The construction of this new ring spectrum is categorical and hence allows us to determine the failure of excision for any localizing invariant in place of K-theory. As immediate consequences we obtain an improved version of Suslin's excision result in K-theory, generalizations of results of Geisser and Hessel-holt on torsion in (bi)relative K-groups, an...
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作者：Chattopadhyay, Eshan; Zuckerman, David
摘要：We explicitly construct an extractor for two independent sources on n bits, each with min-entropy at least log(C) n for a large enough constant C. Our extractor outputs one bit and has error n(-Omega(1)). The best previous extractor, by Bourgain, required each source to have min-entropy .499n. A key ingredient in our construction is an explicit construction of a monotone, almost-balanced Boolean function on n bits that is resilient to coalitions of size n(1-delta) for any delta > 0. In fact, o...
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作者：Hwang, Jun-Muk
摘要：A conjecture of Hirschowitz's predicts that a globally generated vector bundle W on a compact complex manifold A satisfies the formal principle; i.e., the formal neighborhood of its zero section determines the germ of neighborhoods in the underlying complex manifold of the vector bundle W. By applying Cartan's equivalence method to a suitable differential system on the universal family of the Douady space of the complex manifold, we prove that this conjecture is true if A is a Fano manifold, o...
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作者：Buckmaster, Tristan; Vicol, Vlad
摘要：For initial datum of finite kinetic energy, Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this paper we prove that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy. Moreover, we prove that Holder continuous dissipative weak solutions of the 3D Euler equations may be obtained as a strong vanishing viscosity limit of a sequence of fin...
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作者：Matveev, Konstantin
摘要：We prove the classification of homomorphisms from the algebra of sym- metric functions to R with non-negative values on Macdonald symmetric functions P-lambda, which was conjectured by S. V. Kerov in 1992.
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作者：Huang, Hao
摘要：In this paper, we show that every (2(n-)(1) + 1)-vertex induced subgraph of the n-dimensional cube graph has maximum degree at least root n. This is the best possible result, and it improves a logarithmic lower bound shown by Chung, Furedi, Graham and Seymour in 1988. As a direct consequence, we prove that the sensitivity and degree of a boolean function are polynomially related, solving an outstanding foundational problem in theoretical computer science, the Sensitivity Conjecture of Nisan an...
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作者：Wu, Meng
摘要：We prove the following conjecture of Furstenberg (1969): if A, B subset of [0,1] are closed and invariant under xp mod 1 and xq mod 1, respectively, and if log p/log q is not an element of Q, then for all real numbers u and v, dim(H) (uA+ v) boolean AND B <= max {0, dim(H) A + dim(H) B - 1}. We obtain this result as a consequence of our study on the intersections of incommensurable self-similar sets on R. Our methods also allow us to give upper bounds for dimensions of arbitrary slices of plan...
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作者：Duminil-Copin, Hugo; Raoufi, Aran; Tassion, Vincent
摘要：We prove an inequality on decision trees on monotonic measures which generalizes the OSSS inequality on product spaces. As an application, we use this inequality to prove a number of new results on lattice spin models and their random-cluster representations. More precisely, we prove that For the Potts model on transitive graphs, correlations decay exponentially fast for beta < beta(c.) For the random-cluster model with cluster weight q >= 1 on transitive graphs, correlations decay exponential...
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作者：Frisch, Joshua; Hartman, Yair; Tamuz, Omer; Ferdowsi, Pooya Vahidi
摘要：A countable discrete group G is called Choquet-Deny if for every non-degenerate probability measure mu on G, it holds that all bounded mu-harmonic functions are constant. We show that a finitely generated group G is Choquet-Deny if and only if it is virtually nilpotent. For general countable discrete groups, we show that G is Choquet-Deny if and only if none of its quotients has the infinite conjugacy class property. Moreover, when G is not Choquet-Deny, then this is witnessed by a symmetric, ...
