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作者：Zemke, Ian
摘要：We prove that the map on knot Floer homology induced by a ribbon concordance is injective. As a consequence, we prove that the Seifert genus is monotonic under ribbon concordance. Generalizing theorems of Gabai and Scharlemann, we also prove that the Seifert genus is super-additive under band connected sums of arbitrarily many knots. Our results give evidence for a conjecture of Gordon that ribbon concordance is a partial order on the set of knots.
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作者：Du, Xiumin; Zhang, Ruixiang
摘要：We show that, for n >= 3, limt(t -> 0) e(it Delta) f (x) = f (x) holds almost everywhere for all f is an element of H-s (R-n) provided that s > n/2(n+1). Due to a counterexample by Bourgain, up to the endpoint, this result is sharp and fully resolves a problem raised by Carleson. Our main theorem is a fractal L-2 restriction estimate, which also gives improved results on the size of the divergence set of the Schrodinger solutions, the Falconer distance set problem and the spherical average Fou...
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作者：Prywes, Eden
摘要：We show that a closed, connected and orientable Riemannian manifold M of dimension d that admits a nonconstant quasiregular mapping from R-d must have bounded dimension of the cohomology independent of the distortion of the map. The dimension of the degree l de Rham cohomology of M is bounded above by ((d)(l)). This is a sharp upper bound that proves the Bonk-Heinonen conjecture. A corollary of this theorem answers an open problem posed by Gromov in 1981. He asked whether there exists a d-dime...
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作者：Schneiderman, Rob; Teichner, Peter
摘要：We compute the group LM2,24 of link homotopy classes of link maps of two 2-spheres into 4-space. It turns out to be free abelian, generated by geometric constructions applied to the Fenn-Rolfsen link map and detected by two self-intersection invariants introduced by Kirk in this setting. As a corollary, we show that any link map with one topologically embedded component is link homotopic to the unlink. Our proof introduces a new basic link homotopy, which we call a Whitney homotopy, that shrin...
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作者：Anantharaman, Nalini; Sabri, Mostafa
摘要：We prove a quantum-ergodicity theorem on large graphs, for eigenfunctions of Schrodinger operators in a very general setting. We consider a sequence of finite graphs endowed with discrete Schrodinger operators, assumed to have a local weak limit. We assume that our graphs have few short loops, in other words that the limit model is a random rooted tree endowed with a random discrete Schrodinger operator. We show that an absolutely continuous spectrum for the infinite model, reinforced by a goo...
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作者：Gamburd, Alex; Magee, Michael; Ronan, Ryan
摘要：We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation x(1)(2) + x(2)(2) + ... +x(n)(2) = ax(1)x(2) ... x(n) + k. When n >= 4, the previous best result is by Baragar (1998) that gives an exponential rate of growth with exponent beta that is not in general an integer when n >= 4. We give a new interpretation of this exponent of growth in terms of the unique parameter for which there exists a certain conformal measure on projective space.
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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：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...
